Category Theory: Applications to Algebra, Logic and Topology. Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

962 Category Theory Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981

Edited by K.H. Kamps, D. Pumplen, and W. Tholen

Springer-Verlag Berlin Heidelberg New York 1982

Editors

Klaus Heiner Kamps Dieter Pumpl0n Walter Tholen Fachbereich Mathematik und Informatik Fernuniversit~t - Gesamthochschule L0tzowstr. 125, 5800 Hagen Federal Republic of Germany

AMS Subject Classifications (1980): 18-06, 03D, 05C, 06D, 08A, 13C, 13E, 16A, 18A, 18B, 18C, 18D, 18F, 18G, 20L, 26E, 46A, 46B, 46G, 46M, 54B, 54D, 54E, 55F, 55N, 55P, 57M ISBN 3-540-11961-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-11961-2 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

During

the last stages

this volume

the editors

death of our colleague Her personality remembered

of the p r e p a r a t i o n learnt Graciela

of the tragic Salicrup.

and her work will

by all of us.

of

always

be

PREFACE

The I n t e r n a t i o n a l to Algebra, 1981;

C o n f e r e n c e on C a t e g o r y T h e o r y - A p p l i c a t i o n s

Logic and T o p o l o g y - was h e l d in G u m m e r s b a c h , J u l y

it was a t t e n d e d by 93 m a t h e m a t i c i a n s

6-10,

from 19 d i f f e r e n t coun-

tries. Financial

support

for this c o n f e r e n c e was p r o v i d e d by a grant of

the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t a d d i t i o n a l means of the M i n i s t e r Landes N o r d r h e i n - W e s t f a l e n . their sincere thanks

(grant no.

4851/140/80)

fur W i s s e n s c h a f t

and by

und F o r s c h u n g des

The o r g a n i z e r s w o u l d like to e x p r e s s

for this f i n a n c i a l

assistance,

without which

this c o n f e r e n c e w o u l d not have been possible. The c o n f e r e n c e had been d i v i d e d into three sections: c a t e g o r y theory,

c a t e g o r y theory and logic,

c a t e g o r y theory to analysis,

General

and a p p l i c a t i o n s

t o p o l o g y and c o m p u t e r science.

very m u c h a p p r e c i a t e d by the o r g a n i z e r s

of It was

that John Gray agreed to

be c h a i r m a n of this c o n f e r e n c e

and special thanks are due to him

for his e s s e n t i a l

to its success.

contribution

also very g r a t e f u l

to Horst H e r r l i c h

the section on a p p l i c a t i o n s logy and c o m p u t e r

The o r g a n i z e r s w o u l d

ning of the c o n f e r e n c e

of c a t e g o r y theory to analysis,

like to express Prof.

Peters,

its p r e p a r a t i o n e s s e n t i a l

D u r i n g the c o n f e r e n c e

and e f f e c t i v e help was given and this help has

b e e n g r a t e f u l l y a c k n o w l e d g e d by the o r g a n i z e r s . from the u n i v e r s i t y a d m i n i s t r a t i o n

Thanks

E s p e c i a l l y Mr.

should be m e n t i o n e d

for

for this conference.

are due to the F a c h b e r e i c h M a t h e m a t i k und I n f o r m a t i k of

the F e r n u n i v e r s i t ~ t Many c o l l e a g u e s its preparation. Mrs.

for his ope-

and for the w e l c o m e he e x t e n d e d to the par-

by the a d m i n i s t r a t i o n of the F e r n u n i v e r s i t ~ t ,

Bl0mel

topo-

t h e i r thanks to the Rektor

Dr. Dr. b . c . O .

t i c i p a n t s on b e h a l f of the F e r n u n i v e r s i t ~ t .

his e n g a g e m e n t

are

science.

of the F e r n u n i v e r s i t ~ t ,

and d u r i n g

The o r g a n i z e r s

for his help as c h a i r m a n for

for s u p p o r t i n g this c o n f e r e n c e

in every respect.

a d v i s e d and a s s i s t e d us d u r i n g the c o n f e r e n c e We w o u l d

I. M U l l e r and Mrs.

like e s p e c i a l l y to

and

thank the s e c r e t a r i e s

K. T o p p for their m o s t e f f i c i e n t work.

VJ

Last, Dr.

b u t by no m e a n s

G. Greve,

T. MUller,

Dr. W.

least, Sydow,

all m e m b e r s

of the F e r n u n i v e r s i t ~ t forts

that

there w e r e

ference

and they

ference

feel

This of this

this

our s i n c e r e

Klaus

Heiner

to e x p r e s s

D. BrUmmer,

of the F a c h b e r e i c h for their

Dr.

und

It is due

difficulties

to m a k e

our t h a n k s

B. H o f f m a n n

Mathematik

engagement.

no o r g a n i z a t i o n a l

did t h e i r best

of S p r i n g e r

conference.

series.

Dr.

like

to

and Dr.

Informatik

to t h e i r

during

the p a r t i c i p a n t s

ef-

the con-

of the con-

at ease.

volume

ger L e c t u r e

we w o u l d

Notes All

Lecture

We w o u l d

like

in M a t h e m a t i c s contributions

thanks

Kamps

Notes

Dieter

volume

referees

PumplUn

the p r o c e e d i n g s

the e d i t o r s

for a c c e p t i n g

to this

go to all the

constitutes

to t ha n k

of the S p r i n -

the p r o c e e d i n g s have

been

for

refereed

for their work.

Walter

Tholen

and

PARTICIPANTS

M. A d e l m a n C. A n g h e l H. B a r g e n d a M. B a r r J.M. Beck H.L. B e n t l e y G.J. B i r d R. B S r g e r D. B o u r n H. B r a n d e n b u r g R.D. B r a n d t R. B r o w n C. C a s s i d y Y. Diers G. D u b r u l e A. Duma J.W. D u s k i n R. D y c k h o f f A. Frei P. F r e y d A. F r ~ l i c h e r J.W. Gray C. G r e i t h e r G. G r e v e R. G u i t a r t R. H a r t i n g M. H ~ b e r t H. H e r r l i c h P.J. H i g g i n s M. H ~ p p n e r B. H o f f m a n n R.-E. H o f f m a n n M. H u ~ e k J. Isbell B. Jay P.T. J o h n s t o n e K.H. K a m p s G.M. K e l l y H. K l e i s l i A. K o c k J. L a m b e k H. L i n d n e r F.E.J. L i n t o n H. L o r d R.B. LGs chow J. M a c D o n a l d S. M a c L a n e

L. M ~ r k i G Maury A MSbus T MGller C J. M u l v e y A Mysior R Nakagawa G Naud~ L.D. Nel S.B. N i e f i e l d A. O b t u ~ o w i c z B. P a r e i g i s J. P e n o n M. P f e n d e r A.M. Pitts H.-E. P o r s t T. P o r t e r A. P u l t r D. P u m p l G n R. R e i t e r G. R i c h t e r R. R o s e b r u g h J. R o s i c k ~ G. S a l i c r u p B.M. S c h e i n D. S c h u m a c h e r F. Schwarz Z. S e m a d e n i T. S p i r c u G.E. S t r e c k e r R. S t r e e t T. S w i r s z c z W. S y d o w M. T h i ~ b a u d T. T h o d e W. T h o l e n V.V. T o p e n t c h a r o v V. T r n k o v ~ K. U l b r i c h R.F.C. W a l t e r s H. W e b e r p a l s S. W e c k R. W i e g a n d t A. W i w e g e r R.J. W o o d O. Zurth

AUTHORS'

H.L.

Bentley

ADDRESSES

D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of T o l e d o Toledo, Ohio 43605 U.S.A.

R. Betti

Istituto Matematico U n i v e r s i t ~ di M i l a n o Via Saldini 50, M i l a n o Italy

F. B o r c e u x

U n i v e r s i t ~ C a t h o l i q u e de Louvain 1348-Louvain-La-Neuve Belgium

D. B o u r n

U n i v e r s i t ~ de P i c a r d i e U.E.R. de M a t h ~ m a t i q u e s 33, rue St Leu 80039 A m i e n s France

H. B r a n d e n b u r g

I n s t i t u t fur M a t h e m a t i k I Freie U n i v e r s i t ~ t B e r l i n A r n i m a l l e e 2-6 10OO B e r l i n 33 Fed. Rep. of G e r m a n y

R. B r o w n

School of M a t h e m a t i c s and Computer Science U n i v e r s i t y C o l l e g e of N o r t h Wales Bangor, G w y n e d d LL57 2UW U.K.

Y. Diers

D ~ p a r t e m e n t de ~ ""a t h e"m a t l"q u e s U.E.R. des S c i e n c e s U n i v e r s i t ~ de V a l e n c i e n n e s 59326 V a l e n c i e n n e s France

A. Frei

Mathematics Department U n i v e r s i t y of B r i t i s h C o l u m b i a V a n c o u v e r , B.C. C a n a d a V6T IY4

A. F r 6 1 i c h e r

S e c t i o n de M a t h ~ m a t i q u e s U n i v e r s i t ~ de G e n ~ v e 2-4, rue du Li~vre 1 2 1 1 G e n ~ v e 24 Switzerland

IX

J.W. Gray

Department of Mathematics University of Illinois Urbana, Ill. 61801 U.S.A.

G. Greve

Fachbereich Mathematik und Informatik Fernuniversit~t 5800 Hagen Fed. Rep. of Germany

P.J. Higgins

Department of Mathematics University of Durham Science Laboratories South Road Durham DHI 3LE U.K.

R.-E. Hoffmann

Fachbereich Mathematik Universit~t Bremen 2800 Bremen 33 Fed. Rep. of Germany

M. H~ppner

Fachbereich MathematikInformatik Universit~t-GesamthochschulePaderborn 4790 Paderborn Fed. ReD. of Germany

M. Husek

Matematick~ Ustav University Karlova Sokolovsk~ 83 18600 Praha Czechoslovakia

S. Kaijser

Uppsala University Uppsala Sweden

J. Lambek

Department of Mathematics McGill University 805 Sherbrooke St. West Montreal, PQ Canada H3A 2K6

J. MacDonald

Mathematics Department University of British Columbia Vancouver, B.C. Canada V6T IY4

L. M~rki

Mathematical Institute Hungarian Academy of Sciences Re~itanoda u. 13-15 1053 Budapest Hungary

A. M e l t o n

D e p a r t m e n t of C o m p u t e r S c i e n c e W i c h i t a State U n i v e r s i t y W i c h i t a , K a n s a s 67208 U.S.A.

A. M y s i o r

I n s t i t u t e of M a t h e m a t i c s U n i v e r s i t y of G d a n s k 80952 G d a n s k Poland

L.D. Nel

D e p a r t m e n t of M a t h e m a t i c s Carleton University Ottawa, O n t a r i o C a n a d a KIS 5B6

S.B. N i e f i e l d

Union C o l l e g e S c h e n e c t a d y , N.Y. U.S.A.

J.W.

Pelletier

12308

F a c u l t y of Arts York U n i v e r s i t y 4700 Keele Street Downsview, O n t a r i o C a n a d a M3J IP3

M. P f e n d e r

M A 7-I Technische Universit~t Berlin Str. des 17. Juni 135 1OOO B e r l i n Fed. Rep. of G e r m a n y

H.-E.

Fachbereich Mathemaik Universit~t Bremen 2800 B r e m e n 33 Fed. Rep. of G e r m a n y

Porst

T. P o r t e r

School of M a t h e m a t i c s and C o m p u t e r Science U n i v e r s i t y C o l l e g e of North Wales Bangor, G w y n e d d LL57 2UW U.K.

A. P u l t r

Matematick~ Ustav University Karlova S o k o l o v s k ~ 83 18600 P r a h a Czechoslovakia

R. Reiter

Fachbereich Mathematik Technische Universit~t Berlin Str. des 17. Juni 135 10OO B e r l i n Fed. Rep. of G e r m a n y

Xl

G. R i c h t e r

F a k u l t ~ t fHr M a t h e m a t i k Universit~t Bielefeld U n i v e r s i t ~ t s s t r . 25 4800 B i e l e f e l d I Fed. Rep. of G e r m a n y

M. S a r t o r i u s

Fachbereich Mathematik Technische Universit~t Berlin Str. des 17. Juni 135 1000 B e r l i n Fed. Rep. of G e r m a n y

T. S p i r c u

National Institute for S c i e n t i f i c and T e c h n i c a l Creation D e p a r t m e n t of M a t h e m a t i c s Bdul P~cii 220 79622 B u c h a r e s t Romania

A. Stone

Mathematics Department UC Davis Davis, C a l i f o r n i a U.S.A.

G.E.

D e p a r t m e n t of M a t h e m a t i c s K a n s a s State U n i v e r s i t y M a n h a t t a n , Kansas 66506 U.S.A.

Strecker

R. Street

S c h o o l of M a t h e m a t i c s and Physics Macquarie University N o r t h Ryde, N.S.W. 2113 Australia

W. S y d o w

F a c h b e r e i c h M a t h e m a t i k und Informatik Fernuniversit~t 5800 H a g e n Fed. Rep. of G e r m a n y

J. T a y l o r

D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of D u r h a m Science Laboratories South Road D u r h a m DHI 3LE U.K.

W. T h o l e n

F a c h b e r e i c h M a t h e m a t i k und Informatik Fernuniversit~t 5800 H a g e n Fed. Rep. of G e r m a n y

Xil

V. Trnkov~

Matematick9 Ustav University Karlova Sokolovsk~ 83 18600 Praha Czechoslovakia

R.F.C. Walters

Department of Pure Mathematics University of Sydney N.S.W. 2006 Australia

R. Wiegandt

Mathematical Institute Hungarian Academy of Sciences Re~itanoda u° 13-15 1053 Budapest Hungary

A. Wiweger

Institute of Mathematics Polish Academy of Sciences ~niadeckich 8 00-950 Warszawa Poland

CONTENTS

H.L.

Bentley A note

R. Betti The

on the h o m o l o g y

and R.F.C. symmetry

of r e g u l a r

nearness

spaces

Walters

of the C a u c h y - c o m p l e t i o n

of a c a t e g o r y

F. B o r c e u x On a l g e b r a i c

localizations

13

D. B o u r n A canonical to c o h e r e n t

H. B r a n d e n b u r g A remark

R. B r o w n

Y.

on c a r t e s i a n

An a p p l i c a t i o n 23

closedness

33

Higgins

complexes

and n o n - a b e l i a n

extensions

39

Diers Un c r i t ~ r e de r e p r ~ s e n t a b i l i t ~ de f a i s c e a u x

A.

limits.

and M. H u ~ e k

and P.J.

Crossed

a c t i o n on i n d e x e d homotopy

par

sections

continues 51

Frei Kan e x t e n s i o n s

and s y s t e m s

of i m p r i m i t i v i t y

62

A. F r ~ l i c h e r Smooth

J.W.

structures

69

Gray Enriched

algebras,

spectra

and h o m o t o p y

limits

82

G. Greve General construction topological, uniform

of m o n o i d a l c l o s e d s t r u c t u r e s and n e a r n e s s spaces

in 100

XlV

P.J.

H i g g i n s and J. T a y l o r The f u n d a m e n t a l g r o u p o i d and the h o m o t o p y c r o s s e d c o m p l e x of an o r b i t space

115

R.-E. H o f f m a n n Minimal

topological

completion

of ~ B a n I -->

~Vec

123

M. H ~ p p n e r On the freeness of W h i t e h e a d - d i a g r a m s

133

M. Hu~ek Applications

of c a t e g o r y t h e o r y to u n i f o r m structures

138

S. K a i j s e r and W. P e l l e t i e r A categorical

framework

for i n t e r p o l a t i o n t h e o r y

145

J. L a m b e k 153

T o p o s e s are m o n a d i c over c a t e g o r i e s

J. M a c D o n a l d and A. Stone Essentially monadic adjunctions

167

J. M a c D o n a l d and W. T h o l e n Decomposition factors

of m o r p h i s m s

into i n f i n i t e l y m a n y 175

L. M&rki and R. W i e g a n d t 190

R e m a r k s on r a d i c a l s in c a t e g o r i e s

A. M e l t o n and G.E.

Strecker

On the s t r u c t u r e of f a c t o r i z a t i o n

structures

197

A. M y s i o r A remark on s c a t t e r e d spaces

209

L.D. Nel B o r n o l o g i c a l L 1 - f u n c t o r s as Kan e x t e n s i o n s Riesz-like representations

and 213

XV

S.B. Niefield Exactness and projectivity

221

M. Pfender, R. Reiter, and M. Sartorius Constructive arithmetics

228

H.-E. Porst Adjoint diagonals for topological completions

237

T. Porter Internal categories and crossed modules

249

A. Pultr Subdirect irreducibility and congruences.

256

G. Richter Algebraic categories of topological spaces

263

T. Spircu Extensions of a theorem of P. Gabriel

272

R. Street Characterization of bicategories of stacks

282

W. Sydow On hom-functors and tensor products of topological vector spaces

292

V. Trnkov~ Unnatural isomorphisms of products in a category

302

A. Wiweger Categories of kits, coloured graphs, and games

312

A Note on the Homology of Regular Nearness Spaces H. L. Bentley Abstract:

I t is shown that the homology and cohomology groups of a regular near-

ness space can be defined by means of a v a r i a t i o n on the ~ech method, which uses nerves of uniform covers:

the v a r i a t i o n involves associating with each uniform

cover, not the nerve, but a complex, called the vein, defined by means of nearness In a recent paper, the author showed that the ~ech homology and cohomology groups (: Vietoris homology and Alexander cohomology groups) of merotopic and nearness spaces s a t i s f y , in a variant form, a l l the axioms of Eilenberg-Steenrod. For d e f i n i t i o n s of these groups and f o r h i s t o r i c a l information, the reader is referred to that paper [ I ] .

We are interested here in regular nearness spaces

(for the d e f i n i t i o n , see Herrlich [5]) and in the p o s s i b i l i t y of using what i s , formally, a d i f f e r e n t d e f i n i t i o n of the homology and cohomology groups, but a d e f i n i t i o n which we prove gives r i s e to the usual ~ech groups. By a pair (X, Y) of nearness spaces we mean a nearness space X together with a nearness subspace Y of X. where ~ I

A uniform cover of (X, Y) is a pair 0~= ( C ~ I , O ~ 2)

is a uniform cover of X, ~2C-011 , and ~ 2 L ) { X - Y} is a uniform

cover of Y.

~

The nerve K(01) of a uniform cover ~ =

( C)~I , (~I 2) of (X, Y) is a pair of

s i m p l i c i a l complexes K ( ~ I ) = ( K I ( O I ) , K 2 ( ~ ) ) . elements of ~ I ; ~-~C~ ~

•

a simplex of K I ( ~ I )

The vertices of K I ( ( ~ ) are the

is a f i n i t e subset C~of ~ I such that

The vertices of K2(OI ) are the elements of ~ 2 ;

K2((~) is a f i n i t e subset ( ~ o f

OI 2 such that Y ~ ~ C ~

a simplex of

~ ~o

Recall that a collection C)jL of subsets of a nearness space X is said to be near in X i f f o r each uniform cover ~ of X there exists C e ~ G e C~ ,

C F'~G #

~ .

such that for a l l

Recall also that i f Y is a nearness subspace of X then

a collection ( ~ o f subsets of Y is near in Y i f and only i f ~ is near in X. Now we are ready to make our main d e f i n i t i o n ; i t is a v a r i a t i o n on the d e f i n i t i o n of the nerve.

The vein J(01 ) of a uniform cover C~: ( (-~I' 012) of (X, Y) is a pair of simplicial complexes J( ~ ) = (Jl ( C)I ), J2 ( 0 ] ) ) . the elements of ~ l ;

The vertices of Jl ( 0 1 ) are

a simplex of Jl ( ~ ) is a finite subset (~of 011 such that

C~ is near in X. The vertices of J2 ( 01 ) are the elements of (#~2; a simplex of J2 (

) is a finite subset ~ o f

C)I2 such that

If ~ = ( (Y~I' 01 2) and ~ =

( ~l'

C~A {Y} is near in Y.

J~2 ) are uniform covers of the pair

(X, Y) of nearness spaces then we say that (#~ is a refinement of ~_~ i f refinement of ~Fl and 012 is a refinement of ~2"

('~l is a

Under this relation of re-

finement, the set of all uniform covers of a pair of nearness spaces becomes a directed set. Thus, there is a spectrum of complexes K(OI )

~. K( ~LF )

J( Ol )

"~J(~)

and of complexes

for ~ a refinement of ~J.

From these spectra there arise two spectra of homology

groups and two of cohomology groups. From now on, let G be a fixed abelian group. G will be the coefficient group of our homology and cohomology theories but explicit denotation of G will be suppressed. The direct spectrum of cohomology groups oC~

: Hn(K( ~J ))

~.Hn(K( 01 )) V

has for its limit group the n-dimensional Cech cohomology group of (X, Y) which we will denote by ~n(x, Y).

/~

The inverse spectrum of homology groups

: Hn(K(OI ))

!>Hn(K(~T ))

has for its limit group the n-dimensional ~ech homology group of (X, Y) which we will denote by ~n(X, Y). The direct spectrum of cohomology groups ~

: Hn(j( ~

))

> Hn(j( C)] ))

has for its l i m i t group the n-dimensional vascular cohomology group, of (X, Y) which we will denote by Hn(x, Y).

The inverse spectrum of homology groups

3

OI ~Zy

: Hn(J( 01 ))

-Hn(J(~

))

has for its limit group the n-dimensional vascular homology group of (X, Y) which we will denote by Hn(X, Y). We are now ready for the statement of our main result. Theorem.

I f (X, Y) is a pair of regular nearness spaces then the ~ech and

vascular homology, and cohomology, groups coincide, i.e. Hn(X, Y) = ~n(X, Y)

and

~n(x, Y) = ~n(x, Y) for a l l n. Proof: dual.

We give a proof only for the homology groups; the proof for cohomology is With each collection~u% of subsets of X, we associate the collection ~*

=

{E C X I for some D e ~

Of course, as usual we are using the notation uniform cover of X. write

E < D to mean that {D, X - E} is a

For each uniform cover ~ = ( ~ I '

~ * = ( (~ I * '

012) of (X, Y) we w i l l

~ 2 * ) ; note that because X is regular then ~ *

uniform cover of (X, Y). note that

, E < D},

(To show that

( CY~ 2 LJ {X - Y})*

refines

is again a

C~2" U{X - Y} is a uniform cover of X, (~2" ~ { x - Y}.)

For each uniform cover ~ of X, there exists a s i m p l i c i a l map

gc~ : J ( C ~ * ) which, on vertices E e ~ l * ' g~

satisfies

>K( L~ ) g~(E) e ~ l

and E < g~(E).

Of course,

is not determined by this condition but any two such simplicial maps have to

be contiguous and so, at the homology level, a unique homomorphism f~

= (g~),

:

H n ( J ( O * ) ) - - - ~ Hn(K( ~

))

is determined, which depends only on ~ and not on the p a r t i c u l a r choice of g ~ . Before going on, i t should be noted that the fact that g ~

is a s i m p l i c i a l map

arises from the fact that ~enever ~ is a f i n i t e subset of ~ I * ' and only i f the form

~{g~(E)

I Ee ~ }

# B.

~ is near i f

Also, since the set of a l l covers of

L~* is a cofinal subset of the set of a l l uniform covers of (X, Y), i t

follows that the fc~ form a homomorphism of the inverse spectrum.

For each uniform cover L~ of (X, Y), K ( ( ~ )

is a subcomplex of J( ~I ) so we

have the homomorphism k~ : Hn(K( ~ ))

>Hn(J( ~

))

induced by the inclusion map. Turning our attention now to the l i m i t groups, we have the projection homomorphisms u ~ : Hn(X, Y)

Hn(J( C~ ))

V~ : Hn(X, Y)

Hn(K((~I)),

and as well as the l i m i t homomorphisms f~: Hn(X, Y)

v X > Hn( , Y)

and k:

Hn(X, Y)

Hn(X, Y).

Consider the following diagram: Hn(X, Y)

Hn(J(~*))

f~

t~o~, > H n ( d ( ( ~ I ) )

kc~

----a.

Hn(K(651")) .

~c~

k~

k

> Hn(K( 01 ))

\

>

Hn(X, Y)

I t is clear that each of the inner triangles is commutative, because each homomorphism is induced either by a projection of refinements or by an inclusion map. To show that f o k = I, let x e vHn( X, Y) and compute as follows:

v

O)

f

k

x

=

=

x

f ~ k 6 ~ , VC~m x X

=

V ~ X.

Consequently, f k x = x. An equally pleasant computation shows that

k Ofoo = 1

and the proof

of the theorem is complete. For regular nearness spaces, the above theorem provides an a l t e r n a t i v e method v

of computing the Cech groups:

one can compute by means of the vascular theory.

I f X is a regular nearness space and Y is a dense nearness subspace of X and i f is a c o l l e c t i o n of subsets of Y which s a t i s f i e s is near in Y.

(3{clxA i A e ~

} ~ ~

then

This observation, together with the knowledge t h a t , in the above

s i t u a t i o n , the homology groups of X are the same as those of Y, indicates t h a t , instead of passing to the extension X and using the Cech theory, one could stay i n side Y and use the vascular theory. Of course, not every nearness space Y is a subspace of a topological nearness space X so, even i f X is the completion of Y, there may e x i s t c o l l e c t i o n s C~ of subsets of Y such that

dl {clxA I A e C21 } : ~ .

In such a case, i t might also be

advantageous to use the vascular theory. We w i l l now present an example using thevascular homology groups Hn(X, Y). Consider the Euclidean plane as a nearness subspace (= uniform subspace) of i t s Alexandrov one-point compactificaton.

Let X be the nearness subspace (= uniform

subspace) induced on the subset 1 1 { ( I , y)[ - I ~ y ~ I } l . ) { ( x , ~) [ 1 < x} t_J { ( x , - T) I 1 ~ x} , The completion of X is a c i r c l e S1 on a 2-sphere S2.

Thus, by the fact proved

in [ 2 ] , the homology of X is the same as that of SI . The point here though is that the homology of X can be computed without going outside X.

The d e t a i l s are as follows.

The set of a l l f i n i t e uniform covers of X is a cofinal subset of the set of a l l uniform covers of X. So, consider an a r b i t r a r y f i n i t e uniform cover ~ :~Y = {A e ~

of X.

Let

I A is unbounded}

and l e t ~ > 0 be such that {GC X I diam G < E }

refines ~ .

Let x+ be the supremum of the set { l } L W { x e R I f o r s ~ e y > 0 and f o r some A e 0 ) - ~ ,

(x, y ) e A}

and l e t x- be the supremum of the set {l}U{x Let ~

e R I for some y < 0 and f o r some A e ~ - ~ ,

be a set of i n t e r v a l s on X of diameter at most E such that

{(l,y)

I -I
I 1 <x~x +}~{(x,-~)

is covered by the union of the i n t e r v a l s in ~ ~

(x, y ) 6 A}.

~ ~ then ~

has cardinal at most 2.

and such that i f

I 1 ~x<x-} ~C~

with

Obviously, such a collection ~ of

intervals exists. Now consider the u n i f o ~ cover

~U~

of X.

i t is easy to see what the vein J ( C ~ U ~ (x +, I / x +)

Clearly i t refines ~ .

)looks like.

From ( x - , - I / x - )

And to

i t is j u s t a f i n i t e sequence of s t r a i g h t l i n e segments which touch at

consecutive endpoints and nowhere else. i t s e l f as a simplex because ~ i s

Furthe~ore, J ( ~ U ~

near in X.

) contains

Thus, p i c t o r i a l l y J ( { Y ~ U ~ )

looks

as follows:

So we see that J ( G ~ F ) c i r c l e SI . ~n(X)

is uniformly (= topologically) homotopic to a

Passing to the l i m i t , we have = Hn(S1).

In conclusion, we pose the following question: do we s t i l l

For non-regular nearness spaces,

have the is~orphisms between the ~ch and the vascular groups?

I f so,

a new proof w i l l need to be found since i t is clear that the proof presented above works only f o r regular nearness spaces.

(For non-regular spaces, there e x i s t

u n i f o ~ covers ~ f o r which the collection

~*

f a i l s to be a uniform cover.)

Bibliography 1.

H. L. Bentley, Homology and cohomology for merotopic and nearness spaces, preprint.

2.

D. Czarcinski, The Cech Homology Theory for Nearness Spaces, Dissertation, University of Toledo (1975).

3.

S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology,

Princeton

(1952). 4.

H. Herrlich, A concept of nearness, General Topology and Appl. 4 (1974) 191-212.

5.

H. Herrlich, Topological structures, In: Topological Structures, Math. Centre Tracts 52, Amsterdam (1974) 59-122.

THE S Y M M E T R Y OF THE C A U C H Y - C O M P L E T I O N OF A C A T E G O R Y R. Betti and R.F.C. W a l t e r s

C a u c h y - c o m p l e t i o n for e n r i c h e d categories was introduced b y F.W. Lawvere [33, g e n e r a l i z i n g the notion for metric spaces.

In this paper, we are concerned w i t h the

case of categories b a s e d on a b i c a t e g o r y w h i c h is locally p a r t i a l l y - o r d e r e d [i]).

([53,[63,

A natural q u e s t i o n that arises is w h e t h e r the C a u c h y - c o m p l e t i o n of a

s y m m e t r i c c a t e g o r y is again symmetric.

This is true for metric spaces

contrary is claimed in [43), b u t false in general. c a t e g o r y satisfies the "modular law" p r e s e r v e d by C a u c h y - c o m p l e t i o n .

(although the

We p r o v e that if the base bi-

(as d e f i n e d b y P. Freyd)

then symmetry is

A n application is the d e s c r i p t i o n of s h e a f i f i c a t i o n

i n terms of C a u c h y - c o m p l e t i o n . We refer to [53,[63 for d e f i n i t i o n s not given here.

i.

The modular law Let

B be a bicategory.

p,0,T, . . . .

Objects of B will be denoted u,v,w,..., and arrows

T h r o u g h o u t this paper w e will assume that B is locally a complete poset,

and that s u p r e m a in each B(u,v) c o m p o s i t i o n w i t h arrows

are p r e s e r v e d by i n t e r s e c t i o n in B(u,v)

(or b o t h sides).

and b y

We will also suppose given an i n v o l u t i o n

( )o: BoP + B (reversing arrows, b u t not order) w h i c h is the i d e n t i t y on objects. Examples.

The m a i n examples we have in mind are

r e g a r d e d as a b i c a t e g o r y w i t h one o b j e c t a n d relations, and, more g e n e r a l l y

(i) L a w v e r e ' s monoidal c a t e g o r y R

(see [33),

(ii) the bicategory, Rel, of sets

(iii) the b i c a t e g o r i e s Rel(C,J)

of relations aris-

ing from a c a t e g o r y C w i t h a topology J, as defined in [i] or [61. Definition.

(P. Freyd)

B satisfies the modular l ~

if~ for arrows p: u + v,

: v ÷ w and T: u + w, we have

"~^ (op) < c~(~°T^p). Remark.

The b i c a t e g o r i e s in examples

not example

(ii) and

(iii) satisfy the m o d u l a r law, though

(i).

We need the following technical

Lena.

If B satisfies the m o d u l a r law and

s a t i s f y i n g iu S

Proof.

V a i then i u ~ V i£I icI

(ai)i£ I is a family of arrows ai: u ÷ u

(a i A a ~ ) .

F i r s t we prove that if b -< 1 u : u + u then b = b °.

To see this note that

b = i u A (blu) < b ( b ° l u A lu)

(modular law)

-< lu(b° A l u)

(b < i u)

_< b e"

B u t b S i u implies b ° S i°u = lu and hence, Now applying

this

result

as above, b ° S b.

t o b = 1 A a i we g e t

that

o o = 1 u ^ a i < a. and 1

1u ^a i

o so i u A a i ~ a i ^ a i. Finally

notice

that

lu

N i~I

ai

implies

that

i u = l u A l u < I u A [iVi ai] = i V i ( l u A a i ) < iVI

2.

o (a i Aai)

.

Q.E.D.

A d j o i n t bimodules Let X be a B-category.

The

Caudhy-oo~et{on

PX of x (described in [i]) is

defined b y (i)

elements of PX over u are adjoint pairs of bimodules

(ii)

~: 6

I > X,

I >6, ~ --~ ~;

4: x

d((~i'~i)'(~2'42))

= 42 " ~i"

Now the main result of the paper depends

on the following

representation

theorem for adjoint pairs of bimodules: Theorem.

If B satisfies

bimodules, (i) and

(ii)

Proof of

the modular

law and ~ --J

4: u <--,-_ - 4 - ~ X is an adjoint pair of

then ~(X) =

V d(y,x) " (C°(y) A ~ ( y ) ) , y(X

4(X) = yVXt

(~(y) A~o(y))

. d(x,y).

(i).

Firstly,

~(x) = V d(y,x) Y

• ~(y)

>- V d(y,x)(4°(y)

(Yoneda)

A~(y)).

Y Now, 1

u

from the adjunction we have that i

-< V

(~(y)~(y)

A ~O(y)~O(y)).

u -< V ~(y)~(y), Y

and hence by our lemma that

Y Hence

~(x)

= ~(x) - i u -< ~(x) "V (~(y)~(y) A~°(y)4°(y)) Y -< ~(x) • V[~(y) (k~°(y)~°(y)4°(y) ^~(y)) ] . v

(modularity)

10

~O(y) 4 o(y) c o(y) = (~(y) 4(y)C(y))O

But

(adjunction)

_< (~ (y) d(y,y) ) °

<_ ~O(y). Hence

4(x) -< V ~(x)~(Y) (~°(Y) ^4(Y)) Y (adjunction)

-< V d(y,x) (C°(y) ^4(Y))Y The proof of

Remark.

Although

a very similar modules

the bicategory

(though simpler)

can be represented

R of example calculation

that includes both calculations.

3.

Symmetry

Theorem.

Proof.

8y;~netI~c

If B satisfies

Cauchy-completion

shows

we require

the modular

that d(y,x)

the ~ d u l a r

law,

that adjoint bi-

We do not have a natural

= d(x,y) ° for all x,y £ X.

law and X is symmetric,

then so is PX, the

of X.

It is sufficient

bimodules

(i) does not satisfy

to that above

in the same way in that case.

proof

For X to be

Q.E.D.

(ii) is similar.

to prove

then C = 4 °, because

that if ~ --@ ~ : u < ~

X is an adjoint pair of

then

d((41,C I) , (42,~2) ) = C2 • 41 = 42 • 41 (41 " ~2 )° = (~i " $2 )° = d((~2,~2),(41,CI)) °. Now from the theorem in §2, assuming 4°(x)

the symmetry

of X, we have

= IV d(y,x). (~O(y) ^ 4 ( y ) ) 3 o Y = V (~(y) ^4°(y)) -d(y,x) ° Y = V (4°(y) ^~(y)) Y =

That symmetry counterexample

$(x)

• d(x,y)

(syrmmetry of X)

.

is not always preserved

suggested by S. Kasangian.

Q.E.D.

is shown by the following

very simple

11

Counterexample.

Let G = {l,a,b} b e the group w i t h three elements.

b i c a t e g o r y w i t h one o b j e c t w h o s e arrows are subsets of G. Let I be the one e l e m e n t B-category.

Let

Let B be the

( )o be the identity.

Then a d j o i n t p a i r s of b i m o d u l e s I ~ -i = h g.

c o r r e s p o n d to elements of G, and under this c o r r e s p o n d e n c e d(g,h) PI, d(l,a)

4.

I H e n c e in

= a -I ~ a = d(a,l), so PI is not symmetric.

Sheafification In [11 and [5], sheaves on a site

complete Rel(C,J)-categories.

(C,J) are shown to be symmetric Cauchy-

In particular, p r e s h e a v e s on C are sheaves for a

topology J0 on C and there is an obvious functor R e I ( C , J 0) + ReI(C,J) w h i c h induces a functor Rel(C,J0)-cat + Rel(C,J)-cat This

(change of base)

p r e s e r v e s syrmnetry. N o w from the d e s c r i p t i o n of s h e a f i f i c a t i o n in [11, u s i n g the fact that symmetry

is p r e s e r v e d b y C a u c h y - c o m p l e t i o n , we get that s h e a f i f i c a t i o n is the composite

Preshv(C) ~ sym R e l ( C , J 0 ) - c a t ÷ sym R e l ( C , J ) - c a t

-~ Shv(C,J). Cauchy-completion

So we have a d e s c r i p t i o n of s h e a f i f i c a t i o n in terms of s t a n d a r d c o n s t r u c t i o n s of e n r i c h e d c a t e g o r y theory.

REFERENCES i.

R. Betti and A. Carboni, C a u c h y - c o m p l e t i o n and the a s s o c i a t e d sheaf, in

2.

to appear

Cahiers top. et g~om. diff.

Denis Higgs, A category approach to b o o l e a n v a l u e d set theory,

Lecture Notes,

University of Waterloo, 1973. 3.

F.W. Lawvere, Metric spaces, g e n e r a l i z e d logic, and closed categories,

Rendiconti del Seminario Matematico e Fisico di Milc~o 43 (1973) 135-166.

12

4.

H. Lindner, Morita equivalences of enriched categories,

diff. 5.

R.F.C.

Walters, Sheaves and Cauchy-complete categories,

diff. 6.

Cahiers top. et g~om.

15 (1974), 377-397.

Cahiers top. et g~om.

22 (1981), 283-286.

R.F.C. Walters, Sheaves on a site as Cauchy-complete categories, to appear in

Journal of Pure and Applied Algebra.

ON ALGEBRAIC LOCALIZATIONS by Francis BORCEUX Universit@ Catholique de Louvain 1348-Louvain-La-Neuve - Belgium

Throughout this paper, • denotes a topos of sheaves on a locale ~ and T denotes a finitary algebraic theory internally defined with respect to this topos ~.

(i.e.

a sheaf of finitary algebraic theories on ~). We are interested in studying the localizations of the c a t e g o r y ~ T of T-algebras in •

(i.e. the full exact reflective subcate-

gories o f ~ T ) .

For any integer n and any element ~ in ~, let us denote by Cn(~) the set of n-ary operations of the theory T(~).

Cn turns out to be a sheaf on H, i.e. an object i n ~ .

As the composite of two 1-ary operations is again a 1-ary operation, the object ¢I is in fact a monoid in ~.

We denote by ~ T the topos of objects in • on which the monoid ¢I acts.

There is

a trivial forgetful functor ~ T ~ ~ T which allows us to consider the object in ~ T W h i c h underlies a T-algebra.

We keep the same notation for a T-algebra and its underlying

¢1-object.

The topos ~ T has a subobject classifier ~T" inf-semi-lattice~Twith with respect to the ~T"

In this paper we define in ~ T

an

0 and I which plays, with respect to ~T, the same role as ~T ~T classifies the subobjects in ~ T and the topologies on mT

classify the localizations of ~ T

If A is some T-algebra, we define the notion of a characteristic map on A.

This

is a map A ~ ~T in the topos ~ T which satisfies some compatibility condition with respect to the theory T.

We prove that for a commutative theory T, the characteristic

maps on A in ~ T are in one to one correspondence with the subobjects of A in the algeT braic c a t e g o r y ~ .

~T is in ~ T an inf-semi-!attice with greatest element.

So it makes sense to

speak of a map j : ~T ~ ~T which satisfies the three Lawvere-Tierney conditions for a topology.

This is what we call a T-topology.

We prove that for a commutative

theory T, the T-topologies on ~ T in ~ T classify exactly the localizations of the algebraic c a t e g o r y ~ T.

14

It is also possible to generalize topology.

generated by a representable Grothendieck takes

in this context the notion of Gabriel-Grothendieck

The notion of crible becomes that of a subobject of the free algebra sheaf.

The three

conditions

for a Gabriel-

topology translate in this context but a fourth condition appears which

care of the fact that we start with a topo s ~ of sheaves and not of presheaves.

These generalized of the algebraic

Gabriel-Grothendieck

We treat explicitly an example groups.

topologies

classify again the localizations

c a t e g o r y ~ T.

: the topos of sets and the theory of abelian

In this case the classifying object w T is exactly the set of natural numbers.

We describe the characteristic map of a subgroup and we give Ab-topologies

some examples

of

on ~.

Finally we give a counterexample theory T is lacking .

to our results when the commutativity

of the

This is the case of sets and the theory of groups.

In this paper, the proofs are only sketched.

The details of them can be found

in [2].

§1 Some more notation.

U :

~T

- Some lemmas

~ •

is the forgetful funetor and F : ~ ~ ~ T is its

left adjoint which, of course, preserves monomorphisms. h a : H °p ~ Sets consisting

is the corresponding

representable

of all those B such that B < a.

corresponding

algebraic

category.

If ~ is some element in H,

sheaf and aS is the sublocale of H

aS is again a locale a n d ~ T ( a )

The object of this paragraph

relations between ~ T and the various ~T(~)

is the

is to describe the

for a 6 H.

LEMMA I. If A is any T-algebra and ~ E H, A(a) ~ ~T(Fhc~,A)

PROPOSITION

2.

~ET is a complete, cocomplete and regular category in which finite

limits co~nute with filtered colimits. set of generators of ~T.

The objects F h ,

for a 6 H, form a regular

15

PROPOSITION 3. For any ~ 6 H, the restriction functor

: ~T ~ T ( a )

~,

has a right adjoint a , and a left adjoint a!. ~! preserves and creates monomorphisms.

a* and a! are full and faithful and

Moreover, if A 6 E T and t o ~ A is monic,

the canonical morphism a! a* A ~ A is also monic.

a, is simply defined by (a,A)(8) for any A i n ~ T ( a )

and 8 in H.

(~'A)(~) = ~A(~) (Co(B) ~!(A) is the sheaf universally ful

: A(a ^ 8)

Now c o n s i d e r if ~ ~ a if not associated to the presheaf a'(A), a, is full and fai±

and thus the same holds for a! ([12] - 16 - 8 - 9).

(= the free algebra on 0) is the initial object in ET. morphisms

Now point out that @o The facts concerning mono-

follow easily from this remark and the component-by-component

description

of ~* and a ! .

o

LENYA 4. For any a 6 H, ¢o ~ F h

is monic.

[]

COROLLARY 5. I f

a ~ 8 in H, 8 ! 8 , ( F h a ) = Fh .

LEMMA 6. If o~ =

V a. in H, 'Fha = U Fh i i61 i i61

7. For any ~ in H, Fh a

"

[]

I

LEMNA 8. I f ¢o ~ A is monic i n E T, S ~

A is a subobject i n E T and a 6 H

c~!a* S = S fl a!~*A

LEMMA

9. :f ~o

A is monic i n E T, S ~

A and S. ~+ A are subobjects and a, ~. are i

l

elements in H, a!~*A D ( U i:I S a ( u i:I

a.! :

Si) m

~.*A) m u : i:I

LE~4A 10. Let f : A ~ B be a morphism and ~ o I=

v a. i n H . i6I :

A =

U i6I

u a ! a * S. i6I :

f-1

a.' :

a.* S :

B a monomorphism in E T.

(a.! :

a . * B) :

Let

16

COROLLARY 11. If I =

V a. in H and ¢ ~ A is monic in E T, i61 i o A = U ~.! a.* A. i6I l l

s

12. If T i8 commutative and A is any object in ~T, the canonical morphism

¢o ~ A is monic,

u

§2 - The classifyin5 pair O E T ~ . At each level a 6 H, the composite of two 1-aryoperations This provides

¢I with the structure of a monoid in ~.

is a 1-ary operation.

We denote by ~ T the topos of

¢1-objects i n ~ .

PROPOSITION

13. For any ~ in H

¢1(a ) ~ ~T (Fh ,Fh ). If 6 ~ a, the restriction map ¢i(a) * ¢i(~) is given by the action of 6!6". PROPOSITION

¢1-object.

m

14. Any algebra A 6 ~T is canonically provided with the structure of a

Any morphism in ~T becomes in this way a morphism in ~T.

For any ~ in H, define the action by composition

¢1 (~) x A(~) mE T (Fh ,Fh ) x~T (Fha,A) ~ T

(Fh ,A) m A(~).

[]

We turn now into the definition of the classifying object w T in E T not, in general, the~]Wobject of this topos ~ . ~T(a) = {subobjects of F h

in,T}.

(which is

For any ~ in H we define

Now if 6 ~ ~, we define a restriction map

~T(~) ~ ~T(S) by pulling back along the canonical inclusion Fh~ ~ F h . We also define an action of ¢I on w T

¢1(~) x ~T(a) mET(rha,Fha) x WT(a) ~ WT(a) again by pulling back along the morphisms F h

PROPOSITION

Fh .

[]

15. ~T is an inf-semi-lattioe in ET"

mT is a separated presheaf (corollary 11 applied to ET(e)

if e =

V ~. in H). i61 i

17

Now let ~ =

V ~. in H and R i 6 ~T(~i) a compatible family. Each R. is in particular i61 m l a subobjeet of Fh ; we define R as being the union of the R.l's, as subobjects of

Fh

For any j in I one has

.

Fh j n R = Fh j n (i6IU Ri)

=

u

(Fh ~j

i61

=

(Fh

U i61

=

fiR.) i n Rj)_

~i

U

a i ,

i61

" ~i*

(Rj)

=R.

J by lemmas 8, 9 and the compatibility of the R. 's. 1

So e'T is a sheaf and thus an object

i n IET . ~T

is provided with mappings in ]ET AT : ~T x ~T

-+60

T

t T : I -~ ~T where A T acts by intersection of subobjects and t T chooses the maximal subobject Fh

in each component.

So ~T is an inf-semi-lattice with I.

We are now able to introduce the notion of a characteristic map on an object of ~ T

; this is a map in ~ T with values in w T.

This technical notion will be useful

in classifying the loealizations of ~T.

DEFINITION 16. Let A be any object in ~T.

:A

A characteristic map on A is a morphism

~T i n ~ T such that for any ~ ~n H, n in~,

8 in @n(a) and x I ,. ..,x n in A(a)

n

^ ~ ( x i ) <~m(@(x I . . . . . Xn)). i=I THEOREM 17. If T is a co,~m~tative theory and A an object in ~T" there is a bijection

between (1) the subobjects of A in ~ T (2) the characteristic maps on A in ~T" Moreover, if some subobject S ~

A has a characteristic map ~ : A ~ ~T" the following

square is a pullback in ~T S,

A

~,1

kO

~ ~

T

18

If S~* A i n , T , we define ~ : A ~ , T the morphisms Fh

at the level ~ 6 H by pulling S back along

* A : A(a) ~ T ( F h

If 0 6 @n(~)

and fl,...f

: Fh

,A) ~ WT(~).

~ A are morphisms in ~T, the commutativity of the

n

theory allows us to define f = e( fl , "''fn ) : F h

~ A and for any subobject R ~ + A

one has n

N fU1(R) E f-1(R) • i=I l This implies that ~ is a characteristic map. Now if the characteristic map ~ : A ~ ~T is given, define S~* A i n ~ T by pulling back t T : I ~ ~T along ~ i n S T .

The conditions on ~ imply that S is in fact a

subob~ect of A in ~T. To check the one-to-one correspondance, it useful to point out that for ~ ~ B in H, any morphism F h and a morphism F h

Fh B in ~ T factors through the canonical inclusion F h ~ F h B ~ Fh , I.e. an element of ~i(~). u

EXAMPLE 18. Let • be the topos of sets and T the theory of abelian groups. the category of abelian groups and ~ T is the topos of (~,x)-sets.

~ T is

w T is the set

provided with the action ~x~

~

;

n z

,

n

=

-

-

n^Izl where ^ denotes the greatest common divisor.

If S ~+ A is an inclusion of abelian

groups, the corresponding characteristic map ~ : A ~ ~ is defined by ~(a)

= i n f {n ~ 11n a c s}

and ~(a) is zero if this set is empty.

Thus if ~(a) # 0,1 , ~(a) is exactly the

order of a in A/S.

§3 - Classifying the al~ebraic localizations. We generalize now the notions of "Lawvere-Tierney topology" and "GabrielGrothendieek topology" into our algebraic context.

We

prove that for a commutative

theory T, the localizations of ]ET can be exactly classified by these topologies and also by the universal closure operations on]E T. start with an arbitrary T.

(cfr. [ 8 ] - 3

- 13).

But let us

19

DEFINITION 19. A Lawvere-Tierney

T-topology on ~T is a morphism ~ : ~T ~ ~T i n ~ T

such that (1)

J

. tT

= tT

(2) J . j = j (S) j . A T = ^ T

" (J x j).

DEFINITION 20. A Gabriel-Grothendieck

T-topology on ~T consists in given, for any

6 H, a familyJ(a) of subobjects of Fh

in ~T such that

(I) F h 6 J(~) (~) R 6 J(~) and f : F~ s ~ F h (3) R 6 J(a) and S ~+ F h (4) R~+ Fh

and ~ =

~f-1(R) 6 J(S)

and Vf : Fh s ~ R f-1(R n S) 6 J(8) ~ S 6 J(a)

V ~. and R N Fh 6 J(ai ) ~ R 6 J(a) iEIi el " 1

PROPOSITION 21. Let ~ T

be any localization of ~T. J(e) = { R ~

Define

Fhallr iso }.

J is a Gabriel-Grothendieck T-topolo@y on ~T and this process describes an injection of the set of localizations into the set of Gabriel-Grothendieck T-topologies. Proof analogous to that given in the sets-case.

COUNTERE AMPLE 22. This counterexample shows that, for an arbitrary T, there is no bijection between the localizations of ~ T and the Gabriel-Grothendieek T-topologies. Consider the topos E of sets and the theory T of (abelian) groups. Fh

= (0) and Fh I = ~.

H = {0,1},

It follows that in this case the Gabrlel-Grothendieck T-topolo-

o gies coincide for both theories of groups and abelian groups.

Moreover axiom 4 of

definition 20 vanishes in this context and thus the T-topologies coincide with the usual Gabriel topologies on ~.

We know that there are infinitely many such topologies.

On the other hand, the category of groups admits only the two obvious localizations : (0) and itself.

Indeed, if ~ G r

is a localization and if n • £ J(1)

(n # 0), consider the canonical inclusion n ~ n ~ - - ~ w h i c h isomorphism. morphism ~

If x and y are the two generators of ~ , ~

which sends I on the word xy.

is taken by i into an pull n ~ n ~

back along the

You get an element of J(1) which

is nothing but (0); thus J(1) is the set of all subgroups of ~ and the localization is obvious. If we turn again to a commutative T, we get the main result of this paper which generalizes to an algebraic context various results on toposes.

20

THEOREM 23. If T is a commutative theory,

there is a bijection between

(1) the localizations of ~T (2) the universal closure operations on ~T (3) the Lawvere-Tierney T-topologies on ~T (4) the Gabriel-Grothendieck T-topologies o n ~ T. 1 If ~.~-r--~

is a localization and S ~+ A a subob~ect in ~T, define S by the

1

pullback ~i is

A

~ilA

where A ~ i IA is the canonical morphism arising from the adjunction i--4i. From a closure operation, define j : w T ~ ~T by ja(S) = S for any e in H. From a Lawvere-Tierney T-topology j define Fh } for any a in H.

i

If a Gabriel-Grothendieck topology J is given, define a localization ~ _ ¢ ~ T 1

as follows : A E I~I iff for any ~ E H and S E J(~), the canonical morphism~T(Fh~,A) ~ • T(s,A) is a bisection.

Now put l'A(a) = lim

~T(s,A) and IA = I'I'A.

This makes

scs(~) sense because T is commutative and t h u s ~ (Simms naturally provided with the structure of a T(~)-algebra. The proof is rather long; its sketch is very close to that of the usual proof in the case of the localizations of a topos of presheaves.

In order to write it

down, it is useful to point out the following facts. (I) condition (4) of definition 20 takes care of the fact that we are working with sheaves and not with presheaves. (2) if ~ : A ~ ~T is a characteristic map and j : ~T ~ ~T is a Law-¢ere-Tierney Ttopology, jo~ is again a characteristic map and thus defines a subobjeet of A in ~T. (3) if a Lawvere-Tierney T-topology ~ and a Gabriel-Grothendieck T-topology J correspond to each other by the constructions above, the mapping "J : H °p ~ Sets; ~ ~~ J(~)" is in fact a sheaf provided with an action of @I given by pulling back. a subobject of ~T i n ~ T

and the following diagram is a pullback in ~T"

Thus J is

21

J

)I

I

i

mT (4) let a, B E H with a % B. any subob~ect i n ~ T.

Let f : F h

Necessarily,

free presheaf of T-algebras on h . functor) with g : F ' h factors through ¢o"

j

~ ~T

~ Fh 8 be any morphism i n ~ T and S~* Fh a

f-1(S) = Fh .

To see thi~denote

by F'h~ the

If f is of the form ag (a = associated sheaf

~ F'hB, then F'hB(B) = Co(a) and therefore g, and thus f, So f-1(S) = Fh .

Now if f : Fh a ~ FhB is any morphism, it

can be described locally by morphisms of the form ag and

the result follows from

the glueing construction for sheaves,

m

EXAMPLE 24. Let us go back to the case of the topos of sets and the theory T of abelian groups (example 18). The localization of the category of abelian groups given by the usual localization process at some prime ideal p~ of ~ is classified by the Lawvere-Tierney T-topology j : ~ ~

given by : j(n) = the greatest power of p dividing n.

The localization of the category of abelian groups corresponding to the thick subcategory of abelian groups all of whose element have torsion is just the category of rational vector spaces. is given by j : ~ ~

The corresponding Lawvere-Tierney T-topology

: J(n) = lj if n # 0 ifn=O.

BIBLIOGRAPHY. [ I ] BORCEUX F., Sheaves of algebras for a commutative theory.

Ann. Soc. Scient.

Bruxelles; 95, I, pp. 3-19 (1981). [ 2 ] BORCEUX F., Sur les localisations

aJ[g~briques.

Preprint; Rapp. S@m. Math. Pures,

Universit@ de Louvain, Belgium (1981). [ 3] GABRIEL, Des cat$gories ab@liennes.

Bull. Soc. Math. France, 90, pp. 323-448,

(1962).

[ 4] GABRIEL-ULMER, Lokal pr~sentierbare Kategorien. Springer (1971).

Lect. Notes in Math., 221,

22

[ 5 ] GRILLET, Regular categories. [ 6] GROTHENDIECK,

Lect. Notes in Math., 226, Sringer (1972).

Th@orie des topos et cohomologie @tale des sh@mas.

Lect. Notes

in Math., 269, Springer (1972). [ 7] HELLER-ROWE, On the category of sheaves.

Am. Jour. of Math., 84, pp. 205-216,

(1962). [ 8] JOHNSTONE, Topos theory.

Academic Press (1977).

[ 9 ] LAWVERE, Functorial semantics of the algebraic theories.

Proe. Nat. Acad. Se.,

50, pp. 869-872, (1963). L 10] LINTON, Autonomous categories and duality of funetors.

Journ. alg., 2, pp. 315

349, (1965). [ 11 ] POPESCU, Abelian categories with applications to rings and modules.

Academic

Press (1973). [ 12] SCHUBERT, Categories.

Springer (1972).

[ 13[ TIERNEY, Axiomatic sheaf theory, eime, varenna (1971).

Ed. Cremoneze, pp. 249-

326, (1973). [ 14] ULMER, On the existence and. exactness of the associated sheaf functor. Appl. Algebra, 3, pp. 295-306, (1973).

J. Pure

A CANONICAL ACTION ON INDEXED LIMITS

AN APPLICATION TO COHERENT HOMOTOPY

by Dominique BOURN Universit~ de Picardie U.E.R. de Math~matiques 33, rue St Leu 80039 - Amiens - France

It is well known that the lax limit of a monad (seen as a 2-functor from the simplicial 2-category H) to Cat) is its category of algebras, and that there is a comonad on this category (i.e. an action of the 2-category ~co). There are other examples of such a situation. For instance, if ~

is the sub 2-category of

R) ,

generated by the 2-cell which gives the multiplication of the monads, then it is easy to see that there is an action of a certain 2-category on the lax limit of every 2-functor from ~

to Cat, namely the subv2-category •

of

©co

generated by

the 2-cell determining the co-unit of the comonads. The purpose of this work is to show that this fact is very general : given a V-category : ~ ---÷ I

~

and an indexation

there is a ~-mono~d which acts on each ~-indexed limit. The proof

is oE the same kind as the proof that Kan extensions are shape invariant. On the other hand, Bousfield-K~n homotopy limits are another example of indexed limits.

Recent

strong

shape

theory tries to preserve traces of higher homotopy

coherences. For instance Dwyer-Kan

[I0]

give a standard resolution in order to do

simplicial localizations. To each category C , they associate a simplicial category (i.e. enriched in the category of simplicial sets) F~ C

in order to take

higher simplicial coherences into account. A category being a particular simplicial set, via its nerve, a 2-category is a special

simplicial category. Actually their

resolution applied to the idempotent-category (with a single object and a single non trivial arrow

t

such that

I call a simplicial functor

F

t 2 = t) is just the 2-category ~. from ~

to Top (with its simplicial category struc-

ture) a coherent homotopy idempotent. Choosing the indexation ves rise to what Gray [14]

H ---÷ I] which gi-

considers as a generalization of the homotopy limits,

we describe the canonical action of the canonical simplicial monoid on this homotopy limit

of

F . In particular we show that the associated homotopy idempotent (i-

dempotent in the homotopy category Ho-Top) splits, although do not split in general (Freyd-Heller I choose to use

homotopy idempotents

[ 12 ] , Dydak-Hinc [ 11 ] ).

~-profunctors, introduced by Benabou

[2] and Lawvere [15] in this

work. Though it is necessary to recall some results at the beginning (Part

I : compo-

sition, right Kan extension, representability) the proofs of this paper are so much

24

easier in this context and the methods so intuitive and powerful that it seems justified to use them. They illuminate the notion of indexed limits, introduced by Auderset

[ I ] and Borceux-Kelly

[ 3 ] , to

whicb

Part

lar studies of the Bousfield-Kan homotopy limits [4]

II is devoted with particu-

[7] , the lax limits of Gray [13]

(we give a new profunctorial description of the associated indexation of Street

[~7 ]) and specially the lax limit of a semiad (a 2-functor from main tool for the application.

Part

III

contains

~

to Cat), the

the main result about the cano-

nical action and Part IV the application to coherent homotopy.

f. The ~z-pro.f~netors. Let

V

be a sy~maetric mono~dal closed category, complete and cocomplete. Let us re-

call from

[2]

and

[14]

Y-profunctors between the ¢ : mop® &÷~f

and

that if

# : A ---÷ ~

~f-category

A, ~ , C

and

~ : B ---+ C

are two

(i.e. : V-functors

~ : C°p @ B + Y) , there is a

Composition of profunctors. r

The composite ~ ® ¢ is defined by the coends : ~ @ ~(C, A) = J that are the cokernels of the maps

B ~(C,B) ® ~(B,A)

dO , d I :

J~- ~(C, B)®B(B, B')®¢(B', A) B,B'

do ) ~ ( C , a~ > B

B)®¢(B, A)

where d O and d I are respectively induced by : ~BB®~(B ',A) and ~(C,B) ® eBB

~(c,B) ~ ~(B,B') ~ ¢(B',A)

~ ~(C,B') ~ ¢(B',A)

~(C,B) ® ~(B,B') ~ ~(B',A)

> ~(C,B) ~ ¢(~',A)

the maps ~BB' and and ~ .

~BB'

being respectively the actions of B(B, B')

on

This composition can be clearly extended to natural transformations of profunctors, and is associative and unitary up to isomorphism, if the unit associated to is the

•-functor

~(-, -) : h o P ® ~ ÷ ~V.

Right Kan extensions of profunctors. The main property of the (bi-)category of profunctors, is that there always exist right Kan extensions. If

¢ : ~ ---> ~

then the right Kan extension [[¢, X~ where -~

[X, Y]

[[¢, X]]

(D, B) =fA

is the value at

and of

×: & --~ X

along

~) are two profunctors, ¢

is given by the ends :

[¢(B, A), x(D, A)] Y

of the right adjoint of the functor

X : V ÷~f ~ that are the kernels of the maps to 7~ [¢(B, A), x(D, A) ] t, ) ~

to, t I : [¢(B, A ) ® &(A,A'), x(D,A') ]

25

where

tO

and

tI

are respectively induced (via the above adjunction) by :

[¢(B,A), x(D,A)] ® 0(B,A)®/A(A,A')

ev®)/

x(D,A)®A(A,A')

XAA' x(D,A')

[~(B,A'), x(D,A')] ® ~(B,A) ®/A(A,A') I®0AA, )[0(B'A')' x(D,A')]®0(B,A') eV~x(D,A') Thus

[[~, X]]

(D, B) is nothing but \V-Nat(0(B, -), x(D, -)).

Representability of profunctors. There is an embedding from 113, to

if- prof (/k, 8)

~r(A, B), the V-category of

the if-category of

associates to each functor

~f-functors between

•-profunctors between

F : A ÷ B, the profunctor

~

and

B

A

and

which

~ F : & ---+ 8 , where

O F(B, A) = ~3(B, F A). Definition. A profunctor tor

F : A+

B

such that

0 : A ---+ ~ is cal led representable if there is a func~ F

is naturally isomorphic to

0 •

Hence we have the classical result : Proposition. Let

F : ~ + ~3 and

G : IA + D

be two ~7-functors, then

point wise (i.e. preserved by all representable functors

G

admits a

D(D, -) : D ÷ ~f)

right

Kan extension, if and only if the right Kan extension of profunctors [[0 F, 0 G I]

is representable.

II. The ~-indexed limits. Let

11 be the ~¢-category with only one object

unit object of

~ . If

A

• , and such that

A ÷ 11 in general, so no canonical notion of limit. Let functor (i.e. aV-functor

11 (~, ~)

is a ~¢-category, there is no canonical

~÷~f)

and

F : A ÷B

a

is the

~V-functor

~ : /A ---4 11 be a

~¢-pro-

~f-functor. Then ([I] , [31) ,

we have the following. Definition. A projective ~-indexed cone from an object element of [[qb, 0 F ~

(B, ~) . The ~f-functor

~-indexed limit, if the right Kan extension object of

B

~,

(B, ~) ~-~3(B, ~-limF).

~ F~

(denoted by

F

B

of

B

to

F

is an

admits a projective (or inverse)

[[0, 0 F]] is representable by an

0-1im F) ; that is, if and only if Likewise a~f-functor

ductive or direct) ~-indexed colimit (denoted by

L : A°P+~

0-colim L), if

has an (inL °p : A ÷ 8 °p

has a 0-indexed limit. We shall call Examples. If

~ F

the indexation of the limit. is a V-functor : A-~ if , then it is clear that

since

[ X,V-Nat(~, F) ]~V-Nat(X ® ~, F) for each object

X

If

is a V-functor : /~P÷If , then ~-colim L = 0 ® L

where

L

functor

11 ---÷IA , since [~ ® L, X]

On the other hand, if

Y

= Nat(0, [L(-),X])

of

~-limF--if-Nat(0,F)

If. L

is seen as a pro-

for each object

is the Yoneda embedding : h °p ÷ V A

then

X of

V.

0-colim Y = 0

28

since

~(q,

L) = Nat(q, L)

by Yoneda ! , L

and Nat(q, ~ ( Y ( ) ,

is isomorphic to

Nat(Y(),

L)) = Nat(q, Nat(Y(), L)). Now,

L).

We shall study more precisely two particular important examples for our purpose. The Bousfield-Kan homotopy limits. Let

A be the well known category of ordered sets [n] = (0, I, 2, ..., n} , n • NN

and non decreasing maps. Let

S = set A°p

be the category of simplicial sets, which

is cartesian closed as is•very presheaf category. Let I be an ordinary small category, X

and

Y

two functors : I ÷ S , then Bousfield-Kan

[7]

define

Hom(X, Y)

as the kernel of : ]-[[X~,Y~] • | ] [Xi,Yi] iel ~ ~ i+i'eI J where the set

I(i, i')

. But

~ [Xi:.Yi,] = --[-[- [X i x l(i,i'), Yi' ] i-~i•I i,i'el J

is considered as a trivial (constant) simplicial set.

Therefore this Hom(X, Y) is nothing but vially enriched in

There is a canonical functor over

i

Nat(X, Y), where

I, X

and

Y

are tri-

S .

to each object

i

I/of

: I ÷ Cat

which associates the category

I . A category

C

I/i

being a particular simplicial set

(often noted

Ner C , but here we shall forget, as in

the functor

I/-

[7 ] , the notation Ner),

can be regarded as I ~ S .

Definition (Bousfield-Kan) : Let

X : I ÷S

be a functor, then

holim X = 14orn (I / -, X) . So, with the previous remark, holim X

is the

I/-

indexed limit of

X .

The lax limits of Gray. There are several ways to describe lax limits either from lax transformations [13] , [4 ]

which are generalized natural transformations, or as indexed limits

[ I 7] .

We shall give a new description of the indexation, which links the two points of view and is more appropriate to our aim. The Total category. Let

%

be the full subcategory of

i < 2 . Each functor

~

to

co < ]

A , formed by all the objects

[i]

÷ Cat determines a diagram : q0

~ t]

with : j.t 0 = j.t I = I,

cI

) C2 ql

q1"t0 = q0"tl '

n't0 = q0"t0 '

n'tl = q1"tl

such that

27 Let us define the total category Tot of this diagram

as the category whose objects

are the pairs (C, f : t0(C ) ÷ t I (C)) consisting of an object C of C0and a morphismf of

C1

such that j(f) = IC

and whose morphisms f'

and

q1(f) . q0(f) = n(f)

(C, f)~(C',

f')

are the morphisms

C g C'

such that

t0(g) = t 1(g) • f .

The total category has a clear universal property. Precisely let the inclusion Tot F

(In]

is the

Y2 : b2 ÷ Cat

can be seen as a category). Then if F: ~ + Cat , the category

Y2-indexed limit of

F, since clearly :

Tot F -~ Nat(Y2, F). The laxcones. Let

B

& ÷B

and

B

be 2-categories (i.e. categories enriched in Cat)land F a2-functor

. A lax cone from

an object

X

to

F

is the following data :

for each

A e ~

:

a morphism

X

for each

f : A+ A'

:

a 2-morphism

X

TA

> F(A) F(f) F

satisfying obvious coherences,

for

compositions

and

')

2-morphisms. These cohe-

rences are such" that a lax cone is exactly an object of the total category of the following diagram : ~B(X,FA) A with

t~ 'A,A'

[~(A,A'),B(X,FA') ] -- ~ ~ ~ [A(A,A') x~(A' ,A"),B(X,FA") ] ~--~-~,A,A' ,A"

t 0((TA)Ae A) = F(f) . TA ,

[qO(OA,A ')] A,A' ,A''(f'g) = F(g).eA, A,(f)

t I((~A)AeA ) = TA, ,

((n(@A,A,)~A,AtA,,(f,g)

= @A,A,,(~.f)

[J (@A,A ') ] A = @A,A(IA )'

[(qI(@A,A ')] A,A',A ''(f'g) = @At, A"(g)

Actually, this diagramm is determined by the right K~n extension of /~ / - : & - - - ÷ A

2 , where

l~/~

~ J_~ ~(A,~) ~(A,A')x~(A', A ~---A,A' wbere

d0CA f A' g ~3

=

A' g

d I(A f A' ~ ~) = A g÷f

~ F

along

is the internal category in Cat : ~ ~o ~)(** [ I A(A,A') x•(A' ,A") xA(A", ~) ~ A,A' ,A"

28

i(A ~f ~) =(A = A +f ~) P0(A+f A' ~ A" h ~) = (A f A' h~g c0 P I ( A I'+-A

aA"h-~) h ~)

m ( A f A' ~ A "

Whence t h e l a x l i m i t s We s h a l l d e n o t e

= (A'

~A" ~ ~)

A g-~f A" h ~)

=

are the

Y2 ® ~ / - l i m i t s .

Y2 ® A / -

by

Remark. T h i s new d e s c r i p t i o n

L(A) .

of lax limit leads to a generalization,

[6 ] , o f t h e B o u s f i e l d - K a n homotopy l i m i t s &/~

is an internal

simplicial

category in

: if

~

S and N e r ( A / ~ )

is a simplicial

studied in category,

is a cosimplicial

then

s p a c e . The

i n d e x a t i o n f o r t h e s e homotopy l i m i t s i s t h e p r o f u n c t o r

H(A) = A - - - * ! to generalize

, defined by the

H@A) (~) = D i a g ( N e r ( ~ / ~ ) )

replacement

scheme o f

[71

. This indexation allows us

to simplicial

categories.

The monads and t h e s e m i a d s . Let

~

be the 2-catego~

2-cells

~ : ~ + t ~.

with a single object

and

k t

~ : t2 ÷ t

=~.tk=t,

~.~

Then it is clear that a monad on a category such that its value at

~

limit of this 2-functor

is

C

Let

D

t

=~.

t~

is a 2-functor

C . It is well known too

:

from

[13] , [4]

~

to Cat

that the lax

is the category of algebras of the monad and that there

is a cotriple on this category of algebras, of

~ , g e n e r a t e d by a 1 - c e l l , and two

satisfying the well-known relations

that is an action of

D c°

(the dual

for the 2-cells). ~

be the sub 2-category of

semiad a 2-functor from

~

and a natural transformation Then the lax limit of are the pairs

a

D

generated by the 2-cell

to Cat, that is a category ~ : T2 ~ T

C

such that

(c, h • c ÷ T c )

such that

are the morphisms

The universal lax

cone

is

U(c, b) = c , and the 2-cell

b . ~(c) = b . T b

f: c ÷ c'

given by the

such that

forgetful

B : T . U + U

T

~ . ~ T = ~ . t ~.

semiad is the category of algebras

(c, b) ÷ (c', b')

~ . Let us call a

with an endo-functor

whose objects

and whose morphisms

f . b = b'

functor

given by

CT

. T f .

U : CT + C

6(c, b) = b : T c + c .

There is no longer an adjunction between C and C T, but it is clear too, that we have a functor

F : C ÷ CT

with

F(c) = (T(c), ~(c))

, such that

U . F = T . Fur-

ther more there is a natural transformation

q " F . U + I CT

Indeed

defines a natural transformation. Let

dl

q(c, b) = b : (T(c), ~(c)) + ( %

b)

be the 2-category with only one object

a 2-cell Proposition.

q : t-4~

~ , generated by a 1-morphism

. We can sum up this result in the following

There is a canonical action of

t

and

:

d7 on the category of algebras of a

29

semiad. III. The c~nonical action on G-indexed limits. These two last examples raise the question: is this fact general, is there always an action on the category of algebras! Let

~

be a

~g-category, A --~-~ ]I an

indexation. The profunctor ~ can be viewed as a functor Proposition. The

t-indexed limit of

nical action on each Proof.

~

A ÷ V •

is a ~-mono~d and this mono~d has a cano-

~-indexed limit,

~-lim t = U ~, ~]] = Nat(~, @) has an obvious structure of

F : ~ +~g t-lim ~

be a M-functor. So

~-lim F = ~ ,

acts on the ~-indexed limit of

~-functor and

L

M-mono~d. Let

and it is clear that

F . More generally, let

the t-indexed limit of

and the canonical action

F]] = Nat(t, F)

F :~ + B

be a

F . So we have ~(B,L) =Nat(t, ~(B,F -))

~-lim @ x ~(-, L) ÷ B(-, L)

which is equivalent by

the Yoneda le~raa to a morphism : t-lim ~ ÷ B(L, L) . It is easy to see that it is a morphism of V-monoids. Examples. The monad case. We have seen that the indexation limit of

y : ~op ÷ Cat~

tegory of the monad

k L~_~ I

is the lax

so that we can exhibit it as a monad on the Kleisli ca-

D(-, ~)

on

k(~, ~) . A simple but tedious computation of

its category of algebras shows us that this category is exactly mono~d structure is that of

~co(~, ~) and the

~co . The canonical action on the category of algebras

of a monad is the usual comonad. In the same way, we can study the semiad case. We must calculate the category of algebras of a semiad on the lax colimit of the semiad ~(-, ~) which is just action is

J/ (~, ~)

the

and the monoid structure is that of

on

~(~, ~) ,

J/ . The canonical

one described by the former proposition. More details will be

given in the proof of the next proposition. Remark. I choose this proof for sake of simplicity and quickness. But it is a very general result that (as in

the

case

of

V-functors [9] ) right Kan extensions

of profunctors are equipped with an action of the codensity monad (which always exists since we deal with profunctors) so that they can be factorized through the Kleisli category

[ ]6 ] of that codensity monad of profunctors, which in our case

has only one object and so is a mono~d. This general result is used in

[ 5 ] to

show in a very simple and categorical way that Kan extensions are shape invariant, so we could say that

~-lim t

is the "shape" mono~d of ~ .

IV. An appT,icatio~ to coherent homotopy. Following Dwyer-Kan gory

[10] , the standard resolution

C (a single object

t 2 = t)

•

F~ C

of the idempotent cate-

with a single non trivial morphism

is a simplicial category.

t

such that

30

But a category

is a particular simplicial set (via its nerve), so a 2-category is

a particular simplicial category. Now forthis category C , no composite of non identity maps is an identity, so F~ C see that this 2-category is just

is actually a 2-category and it is not hard to ~

.

We are now going to study the consequences of the higher homotopy coherences involved in the data of a simplicial functor from

B

to a simplicial category

which I keep on calling a semiad. In the special case

B

B = Top , I shall speak of

a coherent homotopy idempotent. The 2-pro-functor

L~)

simplicial profunctor. joint

K

: B ---+ |

indexing

lax limits can be considered as a

The simplicial embedding Cat L

preserving products, preserves

X

of

F

F : ~] ~ B

[14] ), being

the case of a 2-functor

ends and along

in a 2-category

B

F : [4 + IB

F

L(~), is still the 2-mono~d

we studied previously the action of

Y

F

has an L~H)-indexed limit L,

such that, if

c~(~)

a(~) . v = F(t) , and a 2-cell between

and the constant 2-functor on

iH(~, -)

then there

is the canonical projection v . ~(~) ~

given by

9(~) =~(t,-)

and

L(~) O

-)

~(t,-)j

:

~

"

~(~, -)

since we can verify that : @(t)

. @(~) I[-I(~, - )

= IH(U, - )

and

O ( t 2) = @(t)

. (9(t) ~ I ( t ,

. E(t, -)

- ) IH(!J, - )

= tI(p,

-)

= M(]~ . t 1~, - )

. I~(1~ t ,

-)

= tt(p

. ]] t ,

Whence a natural transformation : L(I0 ÷ H(~, -)

such that

~ T(~) = ~](t, -)

is the

between

: I](~,-) ÷~(~,-)

JH(~, -) = IH(ll,

IL .

on the li-

Y : B °p ÷ Cat~ . On the other hand, we have a lax cone

@(t)

In

a simplicial

L . Firstly let us consider the 2-enriched situation. We saw that

lax colimit of

~.

JJ. We have the B

Proof. The proof will be given by a careful study of the action of mit

. Thus

a simplicial functor.

v : F(~) ÷ L

L ÷ F(~) , we have

L(~)

L(l~)-indexed limits (homotopy limits in

L~H)-lim

Proposition. If the simplicial semiad exists a map

is also the

is the simplicial

following result about this action in the general situation of category and

L(~) -indexed

L(~)

considered as a simplicial functor. Therefore the sim-

plicial mono~d acting on the simplicial the sense of Gray

X

considered as a simplicial profunctor along

the lax limit of a 2-functor L(~)-indexed limit

has a simplicial ad-

exponentiations,

limits. So the right Kan extension of a 2-profunctor right Kan extension of

S

-)

31 if

• is the universal lax cone associated to the lax colimit of

there is a 2-1ax cone

6 between ~(~)O

and

Y . Furthermore,

T, given by

s i n c e t h e second members o f t h e f o l l o w i n g e q u a l i t i e s ~(~).T(N)o(t) are equal,

=

T(t).T(~)H(~,--)

,

b e c a u s e o f t h e c o h e r e n c e o f t h e l a x cone

~. Whence a 2 - n a t u r a l

transformation

L = Nat(LOt),F)

. The u n i v e r s a l

=

T(t).T(t)

S . Let L be t h e L (H) - i n d e x e d l i m i t o f F , t h a t L (N) - i n d e x e d cone o a s s o c i a t e d

We have a map o f s i m p l i c i a l Nat(~,F) :

and we v e r i f y

~(~).~

= F(t)

to the 2-cell

.

o(~) = Nat (T (~) , F) : Nat(L(tt) ,F) ---, N a t ( I ~ ( ~ , - ) ,F) = F(~) , and so on.

[q(t,-)

T with respect

d: T ( ~ ) a ~ I L ( H )

Now l e t us b e g i n w i t h a semiad F:14 ~ is

~(t).~(~)~(t,-)

s e t s w: F ( ~ ) - ~ L

F(e) = N a t ( I t ( ~ , - ) , F )

to o(t)

L

i s g i v e n by

= Nat(T(t),F)

, that is: , Nat(L(V),F) = L

, since

~(~) .~ = N a t ( T ( ~ ) , F ) , N a t ( ~ , F )

= Nat(T(~)~,F)

~ Nat(H(t,-),F)

= F(t).

F u r t h e r m o r e we have a 2 - c e l l b e t w e e n ~ . o ( ~ ) and 1 L g i v e n by N a t ( d , F ) and so F "splits"

at

L .

More g e n e r a l l y

l e t F : [ t * A be a semiad i n a s i m p l i c i a l

i n d e x e d l i m i t o f F and ¢ t h e u n i v e r s a l commutative d i a g r a m w i t h n a t u r a l

isomorphisms:

N a t ( T ( ~ ) , / A ( X , F - ) ) : N a t ( L 0 t )/A(X,F-)) A(X,o~) F u r t b e r m o r e we g e t a n a t u r a l A(X,F(~)) m

c a t e g o r y A. Let L be t h e L ( ~ ; -

L O t ) - i n d e x e d c o n e . Thus we have t h e f o l l o w i n g

* NatOH(~,-),IA(X,F-))

: /A(X,L)

, A(X,F(~))

f i n X) t r a n s f o r m a t i o n :

Nat(H(~,-)),/A(X,F-))

mat(~JA(X,F-))

Nat(L(H),N(X,F-))

~-~ N(X,L)

and so by t h e Yoneda len~na a morphism

v: F(~)--,I,

such that this natural

t i o n i s j u s t /A(-,v)

that o(~).v

is F(t), since

. Then i t

And now t h e 2 - ( n a t u r a l )

is clear

~.r(~)

cell

dJ atd

A(X,L) ~ Nat(L(i~) , A ( X , F - ) )

N a t ( r (~) ~,N(X,F-) )

A(X,L) ~ Nat(L(tl) ,N(X,F-})

transorma= ~(t,-).

32 determines, by Yoneda ! , a 2 - c e l l i n

N(L, L)

between

~ . ~(x)

and

1L •

Corollary. The homotopyidempotent ( i . e . idempotent i n the homotopy category H0-Top) a s s o c i a t e d to a coherent homotopy idempotent, s p l i t s . References. I. C. Auderset, Adjonctions et monades au niveau des 2-categories, Cahiers de Top. et G6om. Diff., XV (1974), 3-20. 2 J. B~nabou,

les distributeurs, Inst. Math. Pures et Appl. Univ. Louvain la Neuve

Rapport n ° 33 (1973). 3

F. Borceux and G.M.KelIy, A notion of limit for enriched categories, Bull. of

the AustralianMath. 4

Soc., 12 (1975) 45-72.

D. Bourn, Natural anadeses and catadeses, Cahiers de Top. et G~om. Diff. XIV

(1974) 371-480. 5

D. Bourn and J.M. Cordier, Distributeurs et th~orie de la forme, Cahiers de Top.

et G~om. Diff., XXI (1980) 161-189. 6

D. Bourn and J.M. Cordier, Une formulation g~n~rale des limites homotopiques,

Notes, Univ. Amiens (1980). 7 A.K. Bousfield and D.M. Kan, Homotopylimits, completions and localizations, Springer Lecture Notes in Math., 304 (1972). 8

J.M. Cordier, Sur la notion de diagran~e homotopiquement coherent , Proceedings

3~me colloque sur les categories Amiens 1980 (~ para~tre). 9

E.J. Dubuc, Kan extensions in enriched category, Springer Lecture Notes in

Math., 106 (1969). 10

W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, Journal of

P.A. Algebra, 17 (1980) 267-284 . 11

J. Dydak, A simple proof that pointed FANR-spaces are regular fundamental re-

tracts of ANR's, Bull. Acad. Polon. Sci. Math., 25(1977) 55-62. 12

P. Freyd and A. Heller

(in preparation).

13

J.W. Gray, Formal category theory, Springer Lecture Notes in Math., 391 (1974).

14

J.W. Gray, Closed categories, lax limits, homotopylimits, Journal of P.A. Alge-

bra, 19 (1980) 127-158.

15

F.W. Lawvere, Teoria d e l l e c a t e g o r i e sopra un topos di base, Mimeographed notes,

Perugia (1973).

Topology,13,(1974),

16

G.B. Segal, Categories and cohomology t h e o r i e s ,

17

R. S t r e e t , Limits indexed by category valued 2 - f u n c t o r s , Journal of P.A. Alge-

293-312.

bra, 8 (1976), 149-181. 18 M. Thiebaud, S e l f - d u a l s t r u c t u r e semantics and a l g e b r a i c c a t e g o r i e s , Dalhousie Univ., Halifax, N.S., (1971).

A REMARK ON C A R T E S I A N

H. B r a n d e n b u r g

I. A c a t e g o r y

A with

if for every A - o b j e c t For b a c k g r o u n d for c e r t a i n

and M.

finite p r o d u c t s

X the functor

concerning

aspects

cartesian

of a l g e b r a i c

c l o s e d n e s s and its

to

cited

there.

the c a t e g o r i e s

they c o n t a i n (for UNIF or UNIF,

spaces

[12]).

gorical

Aspects

tively

of T o p o l o g y

cartesian

respect

in TOP or UNIF.

(see r e m a r k

general

problem

category

of TOP or UNIF w h i c h

problem

is included

gorical

Topology

tain the

THEOREM

there

following

theorem,

point discrete space, As a c o n s e q u e n c e subcategory str o n g l y

of TOP m u s t

easily

and nega-

in the m o r e

reflective

closed.

recent

Note

survey

that

article

subthis

on Cate-

For the case of TOP we obbe proved

subcategory

in section

of TO__~P contains

I every c a r t e s i a n

consist

of c o n n e c t e d

space

of c a r d i n a l i t y

continuous

of X form an example contains

subspaces

here

car-

space w h i c h

can be a n s w e r e d

interested

w h i c h will

of T h e o r e m

ty is the only n o n - c o n s t a n t

TOP w h i c h

of usual

have

i.e.

2:

the two-

then it is not cartesian closed.

rigid H a u s d o r f f

all powers

TO_~P or UNIF

a non-trivial

11).

F.

on Cate-

subcategories,

is c a r t e s i a n

[7], P r o b l e m

If a reflective

I.

we are exists

in TOP are

fact,

Conference

whether

his q u e s t i o n

in H. H e r r l i c h ' s

(see

by this

a non-indiscrete

formation

Since

(c) below),

whether

Motivated

that

subcategories

that their p r o d u c t s

epireflective

to the

closed

1980 O t t a w a

containing

it is known

are c o r e f l e c t i v e

and Analysis)

closed

subcategories

are c l o s e d with products

(at the

importance and topolo-

and the l i t e r a t u r e

closed,

cartesian

products.

asked

[3].

TOP of t o p o l o g i c a l spaces and

are not c a r t e s i a n

from the usual

has r e c e n t l y

closed

[15],

All these c a t e g o r i e s

Schwarz

non-trivial

analysis,

[14],

and they have the d i s a d v a n t a g e

different

tesian

[5],

some nice n o n - t r i v i a l

see

cartesian closed

is called

topology,

algebra we refer Although

[2],

Hu~ek

X x - has a right adjoint

gical

UNIF of u n i f o r m

CLOSEDNESS

show that all c a t e g o r i e s

spaces. ~2

mapping

of a r e f l e c t i v e

only c o n n e c t e d

spaces

[4].

_AX o b t a i n e d

closed r e f l e c t i v e If X is a

(i.e.

the

identi-

from X into X) , then subcategory However,

~X of

one can

in this w a y are not

34

cartesian

closed.

cartesian

closed

We c o n j e c t u r e

rem shows

the v a l i d i t y

reflective

that

there exists

subcategory

of TOP.

of the c o r r e s p o n d i n g

no n o n - t r i v i a l

Our

second theo-

statement

for u n i f o r m

spaces. THEOREM

If a reflective subcategory

2.

indiscrete

space,

2. T h r o u g h o u t

this note all

and i s o m o r p h i s m - c l o s e d .

information

about

is i n t e r e s t i n g

PROPOSITION.

functor

I we will

Since A c o n t a i n s closed,

exists

T on the

a topology

(ii)

• is admissible,

Hence

where

Y is A - p r o p e r ,

i.e.

set C(Y,Z)

of c o n t i n u o u s

the e v a l u a t i o n

map

for every A - o b j e c t

is continuous,

topology

from

e:Y ×

(C(Y,Z),T)

~ Z

X and

for every contin-

w h e r e ~(f) (x) (y)=f(x,y).

exist

on C(Y,Z)

on C(Y,Z).

to A.

In order

X in A and a c o n t i n u o u s (C(Y,Z),~)

two spaces Y,Z

in A such

is not A-proper.

with

the usual

To this

topology

and

space Y × Yo' w h e r e Yo is the s u b s p a c e

e , both Y and Z b e l o n g

~

mappings

there

e(y,g)=g(y).

space of i r r a t i o n a l s

let Z be the p r o d u c t

~(f):X

that

of A - o b j e c t s

properties:

Y U {o} of the reals w i t h the usual metric.

topology

[8].

the countable

it is easy to v e r i f y

for each pair Y,Z

to show that there

admissible

end let Y be the

to

proposition

f:X × Y ~ Z the m a p p i n g

(C(Y,Z),T)

it suffices

that every

in TOP and contains

then

i.e.

is continuous,

~(f) :X ~

following

Then A is not cartesian closed.

the f o l l o w i n g

uous m a p p i n g

For addi-

we refer

of TOP which is closed with

a singleton,

if A is c a r t e s i a n

(i)

use the

Let A be a subcategory

respect to countable products

Y into Z w i t h

subcategories

to be full

C is reflec-

for itself.

infinite discrete space ~. Proof:

are a s s u m e d

A of a c a t e g o r y

has a left adjoint.

reflective

In the p r o o f of T h e o r e m which

subcategories

A subcategory

tive in ~ if the e m b e d d i n g tional

of UNIF contains a non-

then it is not c a r t e s i a n closed.

Being

homeomorphic

Let T be an a r b i t r a r y to p r o v e

mapping

that there

admissible

exists

a space

f:X × Y ~ Z such that

is not c o n t i n u o u s

we

start w i t h an a r b i t r a r y

to

35

continuous

mapping

g:Y ~ Z satisfying

for e a c h

y 6 Y, w h e r e

pl:Z

Consider

an a r b i t r a r y

point

hood.

By the

borhoods (yo,g)

continuity

V and V'

6 V'

cl V is not

tinct

points

X be the

Yo

~ Yo are

in Y w h i c h

of the

where

compact,

of Y

no c o m p a c t

map

that

that

exists

{Ynln

consisting

e w e can cl V c V'

a sequence

6 ~}

neighborfind

for

i=I,2}. of dis-

in Y.

non-negative

neigh-

and

(yn)n6~

is c l o s e d

of all

= 0

the p r o j e c t i o n s .

W = {z 6 Z I IPi(Z) I < I

there

in cl V such

subspace

has

evaluation

of Yo and a U 6 T s u c h

x U c e-l[W],

Since

Ip1(g(y)) I < I and p2(g(y))

~ Y and p2:Z

Now

let

numbers.

O

Being

homeomorphic

subspace retract

to Y, t h e

of X x y the of X × Y

continuous

(see

mapping

space

X belongs

space A =

({0}

[11],

II,

§26

f r o m A into

× Y)

has

a continuous

[

Z defined

that

an o p e n

neighborhood

there

exists

hand we conclude

from and

diction

shows

that

6 ~})

is a

Consequently

the

is c o n t i n u o u s .

an m 6 ~

that

Y=Yn

Then

there

~(f) [B] c U.

satisfying

e(Ym,~(f) (x))

is not

and

x y ~ Z.

e ( Y m , ~ ( f ) (x))

~(f)

2).

if x > 0

B of 0 in X such

=

(pl(g(ym)),mx)

f:X

(C(Y,Z),T)

an x 6 B and

(X x {Ynln

by

(pl(g(yn)),nx)

~(f) :X ~

As a c l o s e d

if x = O

extension

Suppose

U

Corollary

S g(Y)

(x,y)

to A.

mx

> I.

exists

Moreover On the o t h e r

= ~(f) (x) (y m) = f ( x , y m) 6 W that m x

continuous,

< I.

which

This

completes

contrathe

proof. Proof

of T h e o r e m

taining

the

countable

1:

L e t A be a r e f l e c t i v e

two-point

discrete

discrete

space

tue of t h e p r o p o s i t i o n .

space

e, t h e n A Hence

subcategory

{0,1}.

is not

we assume

of T O P

If A c o n t a i n s

cartesian that

closed

e does

con-

the by v i r -

not b e l o n g

to

A. Suppose is g i v e n

t h a t A is c a r t e s i a n by r e × r e : e x e

reflection

of e.

the ~ - r e f l e c t i o n and

that

of exe

Then

~ re × re, w h e r e

To v e r i f y

l_~In£er~ x {n}

closed.

this

always = re x

fact

has

the

I Jn6e{n}

one

the A - r e f l e c t i o n re:e only

form

~ re d e n o t e s has to n o t e

r(exe)

= re x r~

=

of e × e the Athat

l~In6ere x {n},

by t h e c a r t e s i a n

36

closedness Now

of A

consider

, where

the

I_~i d e n o t e s

continuous

f(n,m)

mapping

tinuous that

re-re[e]

fact,

n = m

and

m • 1

if

n • m

and

m = I

{O,1}

To prove Xo

% @ and

re = re[e]

(since

the

second

£ re-re[e]

such

h : r e ~ re d e f i n e d

~

observation {0,1}

[

re(1)

Applying

essentially

of r e x r e

we can show that

{x}×re

6 rear

if

g(x,x)

= 0

the

same argument

We conclude (a) T h e o r e m taining

- by

te space.

to our assumption. exists

if

n % I n

this

section

I implies

an mapping

that

In fact,

[9], T h e o r e m

1.1

rex{re(n)}

I

g(x,y)

g(x,x)

A cannot

every

it h a s

that

is i m p o s s i b l e .

subspaces

the argument

that

with

space X which

=

, which

to the

if

Repeating

Consequently

a finite

sian closed. which

[e].

in p a r t i c u l a r

a contradiction!

In

is a h o m e o m o r p h i s m

there

= I

of r e x r e w e o b t a i n

[e],

con-

We claim

Then the continuous

g(x,x)

I

spaces

that

= O.

if

g'x're'n~'~ ~ ,; = If 0

x,y

~ re

, contrary

assume

a unique = f"

every x 6 re-re[e].

h o r e = r e = i d r e o r e a n d h • idre

each x 6 rear

exists g o r xre

by f x

for

re:e

e 6 A

assertion

there that

= I for

that

that

that g(Xo,Xo)

h(x)

satisfies

such

g(x,x)

imply

i.e.

by

otherwise

that

would

6 A),

defined

if

g:rexre

in A.

~ {0,1}

I

to t h e p r e c e d i n g

mapping

f:e×e

= ~ I 0

According

the coproduct

again

for t h e

sub-

= 0 for e a c h p a i r

= 0 for

each x 6 rear

be cartesian

closed.

[e]-

[]

some remarks. reflective

is n o t

to c o n t a i n

- always

subcategory

indiscrete

cannot

the reflective

contains

of TOP

con-

be cartehull

the two-point

o f X, discre-

37

(b) If a r e f l e c t i v e s u b c a t e g o r y A of TOP contains a n o n - i n d i s c r e t e space X w h i c h is not TI, then it is not c a r t e s i a n closed.

In case

that X is not T this follows from the fact that A m u s t contain the o b i r e f l e c t i v e hull of X (e.g. see [10]) and hence a n o n - i n d i s c r e t e finite space.

If X is a To-Space,

spaces is c o n t a i n e d in A

then the c a t e g o r y of sober

([10], T h e o r e m

1.3).

In particular,

the

t w o - p o i n t d i s c r e t e space belongs to A. (c) No

non-trivial epireflective

sian closed,

s u b c a t e g o r y of TOP can be carte-

since it has to contain the t w o - p o i n t d i s c r e t e space.

This answers S c h w a r z ' s q u e s t i o n m e n t i o n e d However,

in the introduction.

a simpler proof of this fact results

from the o b s e r v a t i o n

that there exist z e r o - d i m e n s i o n a l T 1 - s p a c e s X,Y,Z and a c o e q u a l i z e r q:Y ~ Z such that i d x X q : X x y ~ XxZ is not a q u o t i e n t m a p p i n g in TOP (e.g. see

[I], Exa/nple 4.3.4).

E s s e n t i a l l y the same a r g ~ e n t

that there is no n o n - t r i v i a l e p i r e f l e c t i v e tegory of p s e u d o t o p o l o g i c a l

spaces

shows

s u b c a t e g o r y of the ca-

[13].

(d) Every e p i r e f l e c t i v e s u b c a t e g o r y of an e p i r e f l e c t i v e subcategory of TOP is r e f l e c t i v e in TOP. a/nple, to e p i r e f l e c t i v e

Hence T h e o r e m

I applies,

for ex-

s u b c a t e g o r i e s of the c a t e g o r i e s of Haus-

dorff spaces or c o m p l e t e l y regular spaces. 3. In order to prove T h e o r e m 2 let A be a r e f l e c t i v e s u b c a t e g o r y of UNIF c o n t a i n i n g a n o n - i n d i s c r e t e space X.

If A is c a r t e s i a n

closed,

l_~ldenotes the copro-

then X~xl In6 {n} = l_~In6 X~×{n}, w h e r e

duct in _A" jection,

For each n 6 ~ let in :X~x{n} ~ X~xi---in6~{n} be the in-

and let Pn:Xex{n}

Pn is u n i f o r m l y continuous, mapping

f:X~xi_~in6~{n}

~ X be defined by

((xi),n) ~ x n.

Since

there exists a u n i f o r m l y c o n t i n u o u s

~ X s a t i s f y i n g foi n = Pn for each n 6 ~.

M o r e o v e r there are two points x,y 6 X and a u n i f o r m cover U of X such that no element of U c o n t a i n s both x and y. c o n t i n u i t y of f there exist u n i f o r m covers

By the u n i f o r m

V of X and W of

{~in£~{n}

such that V×W refines f-1(U). It follows that V refines every -I Pn (U), c o n t r a d i c t i n g the fact that the subspace {x,y} e of X e is not t o p o l o g i c a l l y discrete.

Hence A cannot be c a r t e s i a n closed

w h i c h c o m p l e t e s the proof of T h e o r e m 2. It is w o r t h m e n t i o n i n g that our proof of T h e o r e m 2 m a k e s no use of star-refineraents of u n i f o r m covers,

hence T h e o r e m 2 is v a l i d

38

also for the category N E A R of nearness follows that no non-trivial reflective

subcategory

tion applies,

[6].

Moreover

subcategory

of NEAR is cartesian closed.

for example,

to the category

spaces

epireflective

This observa-

to the category of proximity

of contiguity

spaces

it

of an epispaces or

[6].

REFERENCES

Elements of Modern Topology, McGraw-Hill,

[I]

R. Brown, (1968).

[2]

E.J. Dubuc and H. Porta, Convenient categories of topological algebras and their duality theory, J. pure appl. Algebra I (1971)

New York

281-316.

[3]

S. Eilenberg and G.M. Kelly, Closed categories, in: Proc. of the Conference on Categorical Algebra, La Jolla 1965, ed. by S. Eilenberg et.al., Springer-Verlag, Berlin-New York (1966).

[4]

H. Herrlich, On the concept of reflections in general topology, in: C o n t r i b u t i o n s to E x t e n s i o n Theory of T o p o l o g i c a l Structures , (Proc. Sympos., Berlin, 1967), 105-114, Deutscher Verlag d. Wissensch., Berlin (1969).

[5]

H. Herrlich, Cartesian closed topological categories, C o l l o q u i u m Univ. Cape Town 9 (1974) 1-16.

[6]

H. Herrlich, A concept of nearness, (1974) 191-212.

[7]

H. Herrlich, Categorical Topology 1971-1981, in: General Topology and its Relations to M o d e r n A n a l y s i s and Algebra V (Proc. of the Fifth Prague T o p o l o g i c a l Symposium, Prague 1981), H e l d e r m a n n Verlag, Berlin, (to appear).

[8]

H. Herrlich and G. Strecker, Category Theory, H e l d e r m a n n Verlag, Berlin, (1979).

[9]

R.-E. Arch.

Hoffmann, der Math.

33

(1979)

[11]

K. Kuratowski, (1966).

[12]

M.D.

ed.,

n~chterner Rdume, Manus-;

Topology, Vol. I, Academic Press, New York,

Rice and G.J. Tashian, Cartesian closed coreflective subcategories of uniform spaces, (preprint). F. Schwarz, Cartesian closednes8, exponentiality, and final hulls in pseudotopological spaces, (preprint). U. Seip, Kompakt erzeugte Vektorrdume und Analysis, Springer Lecture Notes

[15]

4

258-262.

R.-E. Hoffmann, Charakterisierung cripta Math. 15 (1975) 185-191.

[14]

sec.

Appl.

Reflective hulls of finite topological spaces,

[10]

[13]

General Topol.

Math.

in Math.

273

N.E. Steenrod, A convenient M i c h i g a n Math. J. 14 (1967)

(1972).

category of topological spaces, 133-152.

CROSSED COMPLEXESAND NON-ABELIAN EXTENSIONS Ronald Brown School of Mathematics and Computer Science, University College of North Wales.

Philip J. Higgins Oepartment of Mathematics, Science Laboratories, Durham University.

and

Durham, U.K.

Bangor, Gwynedd, U.K.

Introduction

Crossed complexes may

be thought of as chain complexes with operators from a

group (or groupoid) but with non-abelian features in dimensions one and two.

We start

by surveying briefly their use. The definition of crossed complex is motivated by the standard example, the

otopy crossed conrplex ~ Here

~]~

of a filtered space

is the fundamental groupoid

hom-

~ : X 0 c X] c ... c X n c Xn+ ! c ... c X.

~I(X|, X0)

of

homotopy classes

rel i

of maps

(I, i) ÷ (X;, X O) , with the usual groupoid structure induced by composition of paths. For

n a 2 , ~n_X is the family of relative homotopy groups

p E X0 . map

For

n ~ 2 , there is an action of

~]~

on

~n(Xn, Xn_ ] , p)

6 : ~n~ + ~n_]~ ; there are also the initial and final maps

The rules which are satisfied by all such crossed complex (§]). complexes.

~

for all

~n~ , and there is a boundary 60

6 ] : ~|X + X 0

are taken as the defining rules for a

In particular, the rule

66 = 0

shows the analogy with chain

Of course the individual rules are connnonly used in homotopy theory, with-

out necessarily considering the total structure. By a

reduced crossed complex

C

we mean one in which

have been considered for some 35 years. [2].

CO

is a point.

These

They were called "group systems" by Blakers

He writes that he follows a suggestion of Eilenberg in using these group systems

to apply the homotopy addition lemma in his investigation of the relationship between the homology and homotopy groups of pairs.

His proofs involve a functor from reduced

crossed complexes to simplicial sets; the values of this functor have been shown recently by Ashley [I] to be

simplicial T-complexes, and Ashley has proved the hard theo-

rem that this functor gives an equivalence T-complexes.

N

between crossed complexes and simplicial

This equivalence generalises the well known equivalence of chain com-

plexes and simplicial abelian groups, due to Dold and Kan [27; Theorem 22.4], and the functor

N

generalises also the nerve of a groupoid, which we use in §3.

Reduced crossed complexes satisfying in each dimension a freeness condition were called "homotopy systems" by Whitehead [3], 32], and his main example was is the filtration of a CW-complex

K

by its skeletons.

~

where

The paper [3]] gives inter-

esting relations between homotopy systems and chain complexes with operators: we shall generalise these results to crossed complexes in [10]. the papers [30, 31, 32] is

reallsability.

In §]7 of [32] Whitehead sketches a proof

of a theorem announced in §7 of [3]], that if of finite dimensional homotopy systems, and

An overall consideration in

~ : C ÷ C' is a homotopy equivalence C

is realisable as

~

for some

40

CW-complex K + K' .

K , then

C'

is also realisable as

~'

and

~

is realisable by a map

The approach to simple homotopy theory in this section of [32] seems to have

Deen ignored and indeed its predecessor [31] is not widely read. Huebschmann, Holt and others (cf. [20, 17] and the historical note [26]) have shown how crossed complexes may be used to give an interpretation of all the cohomology groups

Hn(G; A)

of a group

G

with coefficients in a G-module

A .

Lue has explain-

ed in [24] how related ideas had been developed earlier for varieties of algebras, rather than just for groups.

However, the tie-up with classical cohomology was not

made explicit (cf. p.172 of [24]). We have given in [6, 7] a colimit theorem for the homotopy crossed complex of a union of filtered spaces.

This theorem includes the usual Seifert-van Kampen theorem

on the fundamental groupoid of a union of spaces; it also includes the Brouwer degree theorem

(~n Sn = ~ ) ,

the relative Hurewicz theorem, and a subtle theorem of J.H.C.

Whitehead on free crossed modules [31; §16].

The proof of the colimit theorem in [7]

involves in an essential way two other categories equivalent to crossed complexes, namely m-groupoids and cubical T-complexes [6, 8].

With simplicial T-complexes [l],

~-groupoids [9] and poly-T-complexes [22], there are now five categories known to be equivalent to crossed complexes, the proofs in each case being highly non-trivial. The papers [16, 18] give other work on crossed complexes. One of our aims here is to show how the homotopy addition lemma (which plays a key rSle in the work of Blakers [2] and of the authors [6, 7]) is also important in the cohomology of a group G . We do this by showing that the standard crossed resolution CG , which is constructed algebraically in [20] and applied further in [2]], in fact arises as

~BG , the homotopy crossed complex of the classifying space of

The boundary maps in

CG

G .

are determined by the homotopy addition lenmla.

Our further aim is an exposition of the Schreier theory of non-abelian extensions. Much

has been written on non-abelian extensions and cohomology,

(cf. [5, 12, 13

23] and the further references there), but it is notable that, while tnereare accounts in several books on group theory, texts on homological algebra remain largely silent on the subject, presumably because there is no known exposition using chain complexes, on which expositions of the abelian case are rightly based.

Here we show that the non-

abelian features of crossed complexes allow an exposition closer to the abelian case, involving morphisms and homotopies.

We strengthen the theory, by presenting an equiv-

alence of groupoids which on components induces the usual one-one correspondence of sets.

We also generalise the theory, to extensions of groupoids rather than just

groups, and to "free" equivalences of extensions. l.

Crossed Complexes We recall from [6] the definition of the category (here denoted

complexes. A crossed complex

C

(over a groupoid) is a sequence

XC) of crossed

41 60 "'" ---+ Cn satisfying (].I)

the following

C1

"'6> Cn-I

J> "'" ---÷ C2 ~

C! ~ 61

C0

axioms:

is a groupoid with

CO

as its set of vertices

and

60 , 6 !

as its initial

and final maps. We write

Cl(p, q)

for the group (1.2)

For

n e 2 , Cn

disconnected (1.3)

for the set of arrows

is a family of groups

groupoid

The groupoid

C1

over C 0) operates

(x,a)~-+ x a

Here if

Cn(p) ~ Cn(q)

if

We use additive

p

For

p

to

q

notation

n e 3

(p,q E C O )

and

C](p)

q

and

a £ C1(p,q)

Cn(P)(n

C! , where

, then

e 2)

for all their identity is a morphism

CI

Cn(P)

Cn(n e 2)

lie in the same component

for all groups

0

(equivalently, C n i s a

the groups

on the right of each

n ~ 2 , 6 : C n ÷ Cn_ l

the action of a =-a+x+a.

{Cn(P)}PEC0

and for

x E Cn(p) and

and we use the same symbol (l .4)

from

C](p,p). totally

are abelian.

by an action denoted

x a c Cn(q)

.

(Thus

of the groupoid

CI .)

and for the groupoid

CI ,

elements.

of groupoids

acts on the groups

C|(p)

over

CO

and preserves

by conjugation:

x (1.5)

= 0 : C n ÷ Cn_ 2

~

for

n _> 3 (and

~0~ = 616 : C2 ÷ CO

as follows

from

(1.4)). (1.6)

If C2

c e C 2 , then

as conjugation

6c

by x

In the case when We observe

operates

trivially

CO

= -c + x + c

~roupoid

as a

, or, simply, C](p)

n ~ 3

and operates

on

.

C

C2(p)

two as defining

over

for

(x, c e C2(P))

is a single point, we call

that the above laws make each

(Cl, CO)

Cn

c , that is 6c

we take the laws up to dimension

is a module

on

a

a crossed module

C2

over

CI(p)

;

crossed module over the

as a

crossed Cl-mOdule.

, and we take the laws

reduced crossed complex.

Let

n ~ 3 .

Then

(1.1) - (1.3) as defining

Cn(p)

Cn

as

module over the groupoid (Cl, c O ) , or, simply, as a Cl-module. A morphism f : C + D of crossed complexes is a family of morphisms of groupoids fn : Cn ÷ On , compatible of

C 1 , D!

on

with

Cn , D n .

the boundary

We denote by

maps

XC

Cn ÷ Cn-!

the resulting

' Dn ÷ Dn-l category

and the actions

of crossed

com-

plexes. By restriction (over groupoids). identity fl

of structure, Let

(as happens

Suppose union of the

erators

f : C ÷ D

throughout

are the identity,

we call

be a morphism

§5) we write f

Cl(P)

, p ~ CO .

[x] £ C 2 with

and of crossed

of crossed modules.

f

C , a set

Then we say

6[x] = hx

of modules,

as a pair

If

(f1' f2 ) "

f0 If

modules

is the f0

and

morphism of crossed Cl-mOdules.

a

given a crossed module

and for any other crossed for all

we have categories

.for all

Cl-module

C'

C

X

and function

is

the free crossed cl-module on gen-

h

from

x E X , if such elements and elements

x ~ X , there is a unique morphism

f : C ÷ C'

[x]' c C 2' of crossed

X

to the

[x] are given,

with

6[x]'

= hx

Cl-mOdules

such

42

that

f[x] = [x]'

for all

uced case [31]. [11]),

C2

is constructed,

x a E C2(q) +yb+x

for all

This d e f i n i t i o n becomes

to the group case, given

C1 , X

x ~ X , a ¢ Cl(60Xx ' q), and

q ~ C 0 , wita the usual relations

w h e r e these make sense, and witll C1

is

free on generators

free crossed C1-module on these generators be a crossed complex.

[15] of the groupoid

C1

A crossed complex C2

C

A crossed complex

C

If

C

is exact and

(or, equivalently,

G C

~i C

Px ' x E X.

is the quotient

subgroupoid

6C 2 .

The

(for some

(on some graph

% : X 2 + CI) , and for

XI),

n e 3 , Cn

Xn).

exact if for

is

is a groupoid,

n e 2 =

Im

then

C

(6

G .

Cn+ 1 --+ C n)

:

•

together with an isomorphism

with a quotient m o r p h i s m

crossed resolution of

called a

equal to the zero at

is a free groupoid

(6 : C n --+ Cn_l)

Ker

6 : C2 ÷ C I

n e 3 , the induced structure of #iC-module.

CI

is a free crossed Cl-mOdule (on some

6[x]

with

, x E X , if it is a

totally disconnected

C n , for

free if

is

C

[x] e C2(Px)

with

-x a

~ x a) = - a + % x + a

fundamental groupoid

Its

by the normal,

rules for a crossed complex give

is a free ~iC-module

Ix] = x 0

can be regarded as a crossed Cl-mOdule

Such a Cl-module

C

is given in

% , as t h e g r o u p o i d w i t h g e n e r a t o r s

A module over

Let

the usual one in the red-

(an exposition of w h i c h

and

a=yb-a+%x+a

trivial.

x ~ X .

Analogously

C1 + G

w h o s e kernel

is

Wl C ÷ G

6C2) is

free crossed resolution if also

It is a

C

is

free. Let

G

follows. groupoid w

be a groupoid.

Let on

X

generating

C2

÷ C I) is the G-resolution

w

is

G

~ : C1 ÷ G .

be a function to the union of the

closure of the image of Let

G

X , with quotient m o r p h i s m

: R ÷ C1

G .)

A free crossed resolution of

be any subgraph of

Ker ~ .

CI(p)

may be constructed

Let

R

(CI, CO)

be any set and let

(X; R, w)

determined by

w

.

is a Then

G-module of identities for the presentation (cf. [II]). + Cn ÷

... ÷ C 3 ÷ K

of

K

by G-modules;

: C 2 + C 1 to give a free crossed resolution

of

as

be the free

, p E C O , such that the normal

(The triple

be the free crossed Cl-mOdule

G

and let

G .

presentation of K = Ker(6

: C2

Choose any free

this may be spliced (Such a construction

into for groups

is used in [20, 21].) As explained

in the introduction,

homotopy crossed complex defined by the skeletons (This is due to Whitehead case follows. 71~ is

w~

7n X = 0

for

~ l ( X , x o) •

p

example of a crossed complex

space

~ .

X ; then

~

Let

, and the homology of Hn(Xp)

(cf. [32; Footnote n e 2) , then

w~

in dimension ~

(i.e.

is exact,

is the

b e the filtered space

from w h i c h

the more general

two is given in [II].)

Ker 6/Im 6) is for

, p E X 0 , where

41]).

~

is a free crossed complex.

[31; §16] in the reduced case,

A simple proof of freeness

~I(X, X0)

based at

key

of a CW-complex

phic to the family of groups X

a

of a filtered

Xp

In particular,

n e 2

is the universal if

X

Further isomor-

cover of

is aspherical

(i.e.

and so it is a free crossed r e s o l u t i o n

of

43

2.

The homotopy addition lemma This is a basic,

it expresses

but not so easy to prove,

the idea that "the boundary

Its formulation

involves

all the structural

and so for completeness Let and let n > I

An An

from

then determine vI

to

n-simplex

have its filtration

~I(A l, A~)

of the homotopy

with ordered

set of vertices

by skeletons

A rn "

vI

Then

o , say.

Intuitively,

of its faces".

crossed

complex,

is also written

a .

determines

~n(A n, A nn-l' Vn)

The face maps

3io ~ ~n_l ~n , and the map

respectively,

{Vo,Vl,...,Vn} , is for

The unique arrow of 3 i : A n-I + A n

u : A 1 ÷ A n , which

sends

v0 ,

uo ¢ Zl ~n .

(The homotopy addition le,~na) The elements

1.

Proposition

to

elements

Vn_ l , v n

theory.

is the composite

elements

cyclic group with generator v0

in homotopy

we state it here.

be the standard

an infinite

lemma

of a simplex

a may be chosen so that

the boundary : ~n(A n, A nn - l '

Vn) ___+ ~ n - l ( A ~ - I ' Ann-2' Vn)

is given by -81o + 800 + 320

if

n = 2 ,

300 - (830)u° - 31o + 820

if

n = 3 ,

n-I E (-l)lSi O + (-l)n(3no)U° i=O

if

n > 4

For a proof of this result, homotopy

3.

[29].

lemma is given for m-groupoids

A corresponding

cubical

form of the

as Lemma 7.1 of [6].

The standard crossed r e s o l u t i o n Let

of

addition

see for example

o

G

be a groupoid.

G , in which

of n-tuples

NnG

of elements

ui

The geometric

realisation

The simplicial

structure

homotopy write

crossed

CG

There

on

complex

2.

Let

G

of

G

X = INGI

~

for this crossed

Proposition

is a well-known

is the set of composable

NG

such that

simplicial

set

(ul,...,Un)

u i + ui+ !

is defined

is known as the classifying

induces

a structure

(for the skeletal complex and call

be a groupoid.

NG

elements

on

for

space BG

of CW-complex

filtration

[28], of

X)

on

the nerve

G n , i.e. 1N

i < n .

of

G .

X , and so the

is defined.

We

it the standard crossed resolution of

Then

is a free crossed resolution of

CG

G .

G

and has the following structure. (i)

CoG = G O ; CIG

is the free groupoid on the sub-graph

vertices and all the non-identity arrows of

G .

The basis element of

CIG

u E G*

ation is extended to

G

(ii)

C2G

corresponding to by setting

is written

(u,v) ¢ N2G*

consisting of all the [u] , and this not-

[0p] = 0p

is the free crossed CiG-module on generators ~[u,v]

for all

G*

: -[u + v| + [u] + Iv|

(the composab~e pairs of

G*)

.

[u,v]

E C2G(~Iv)

with

44

(iii) For

is the free G-module on generators

n >- 3 , CnG

for all (u l,...,un) ~ NnG* . We also let [Ul,...,u n] ~ CnG some

ui : 0 .

(iv)

6 : C3G ÷ C2G

be the identity at

~lun

[Ul, .... u n] E CnG(~lUn )

if

(Ul,...,Un) ~ NnG

and

i8 given by

6[u,v,w] = [v,w] - [u,v] [w] - [u + v,w] + [u,v + w] ,

for al~

(u,v,w) E N3G .

For

(v)

n >- 4 , 6 : CnG ÷ Cn_IG

is giVen by n-I i Z (-1) [Ul,...,u i + ui+i, .... u n] i=l

~[Ul, .... u n] = [u 2 ..... u n] +

+ (-l)n[ul, . . . ,Un_l ]

[u n]

.

D

This proposition follows from the homotopy addition lemma, the standard description of the face operators in if

G

NG , and the fact that

is a group, then Proposition 2 shows

homogeneous) crossed resolution of how

CG

G

CG

BG

as defined in §9 of [20].

6 : C3G ÷ C2G

We have now shown

should be noted; the values of this

are in a family of (generally) non-abelian groups. a crossed complex

A

Note that

arises geometrically.

The curious formula for

functor

is aspherical [281.

to be the same as the standard (in-

A(CG)

abelianises

C

a chain complex

AC

is the bar resolution of C2

There is a funetor assigning to

with operators from G

TIC

[I0]; for this

(cf. [25]), for the group case).

However,

and so loses information.

The 3-simplices of

NG

may be pictured as 3

u

+

0

(cf. p.12 of [25]). n > I .

Now

NG

v

~

u

I

is a T-complex in which every n-simplex is thin for

Every T-complex has a groupoid structure in dimension I, and the above picture

illustrates the 3-simplex used to prove associativity [I] of this groupoid. suggests the link between

4.

~ : C3G ÷ C2G

This

and associativity in extension theory.

Homotopies The notion of homotopy has a similar importance for crossed complexes to that for

chain complexes.

However, because of the more complicated structure of crossed com-

plexes, there are several possible conventions for the definition of homotopy, and there are also two levels of generality (corresponding to free and based homotopy in

45

the topological

case).

Our definition

lemma in the algebra of m-groupoids m-groupoids, Let

If If

(4.2)

be morphisms

and

Cop e Dl(fp,gp)

x c Cn(q)

: CI ÷ D2

of crossed complexes.

e n : C n ÷ Dn+l(n >- 0)

p ~ C O , then

n ~ 2 81

from the cubical homotopy addition

a topic w h i c h we hope to develop elsewhere.

f , g : C ÷ D

is a family of functions (4.1)

follows

[6], applied to a natural notion of homotopy for

, then

.

A ~omotopy

0

: f : g

with the following properties.

If

x e Cl(p,q)

, then

Ol x E D2(g q) .

0nX e Dn+l(g q) .

is a derivation

over

gl

, that is if

x + y

is defined

in

Cl

then O1(x + y) = (01x)gY + ely (4.3)

For

n e 2 , e n : C n ÷ Dn+ l

a ~ Cl(P, q) , x ~ Cn(p)

,

, y c Cn(q)

If

x e C1(p, q)

then

(4.5)

If

n ~ 2 , and

x E Cn(q)

gy = gl y .

where

ga = gla .

gx = -e0p + fx + 00q - (6elx)

(A similar definition, [23].

A homotopy p ~ CO

but with different

For further comments,

0 : f = g

conventions,

eq = e0q . is given in the reduced case

see Remark 4 at the end of the paper.)

which are used by H u e b s c h m a n n

if

00p

is an identity for all

(It is these homotopies, [20].)

For emphasis,

with different

the more general

called ~re~ homotopies.

e : f = g , e' : g = h

is defined by then

conventions,

f0 = go ) "

kinds of homotopy are sometimes If

where

is said to be tel C O

(so that in consequence

.

then

gx = (fx) eq - 8n_l~X - ~0nX ,

by Whitehead

gl , that is, if

, then

en(xa + y) = (enx)ga + 8ny (4.4)

where

is an operator m o r p h i s m over

are (free) homotopies,

their composite

~ = e + 8'

~0 p = Cop + 8~p , p ~ C O , and if n ~ I and x ~ C1(p, q) or x ~ C (q), e' n q . It is easily checked that # is a homotopy f = h .

~n x = e~x + (0nX)

In the next section we will be considering which are the identity on

CO = D O .

only crossed complex morphisms

C + D

Therefore we w r i t e

(C, D)f

and

for the groupoids w h i c h have such morphisms vely the free, and the

rel C O , homotopies.

are thus the respective

sets of homotopy

(C, D) as objects,

and w h o s e arrows

The sets of components

classes of morphisms

over

are respecti-

of these groupoids C O = D O , and they

are w r i t t e n respectively [C, D]f

5.

[C, D]

.

Non-abelian extensions Throughout

A

and

this section,

is totally disconnected

~t~nsion

of

A

by

G

G

(i.e.

and A

A

will be groupoids

is a family

is a pair A ~ E ~ G

A(p)

such that

GO = A 0

, p e A 0 , of groups).

and An

46

of morphisms

of groupoids

(5.2)

p

is a quotient m o r p h i s m of groupoids.

(5.3)

i

maps

p

A

i

p

are the identity on objects.

isomorphieally

onto

is a quotient m o r p h i s m means

for more details action of

E

see [15].)

on

A

~

A

A free equivalence

a crossed

is an isomorphism;

i

(large)

free equivalences

A .

~ E

if ~

Extf(G,

A) of

E

induces an

This can be extended

trivially

is (with the quotient m o r p h i s m

A

by

P

> G

G

is a commutative

diagram

G that

~

also is an isomorphism.

is the identity.

Here

Act A

is an isomorphism

on

A

and A

and the equivalences

Under our assumption Act A

of

this implies

the extensions

For any groupoid

q

in

E/Ker p ÷ G ;

Such

W e can thus form two

groupoids

both having objects

of

conjugation

which

i' ~ E' ~

equivalence

is an

an isomorphism

G .

A ~

induces

E-module.

of such extensions A

such that

p

: ... ÷ 0 ÷ 0 ÷ A ÷ E

a crossed resolution of

a free equivalence

Ker p . that

For such an extension,

making

to a crossed complex p)

and

the following properties.

E0 = GO

(That

and

satisfying

(5.1)

A

there is a groupoid

A(p) ÷ A(q) that

A

A)

,

G , but having arrows respectively

the

of extensions.

has the same objects

determine

Ext(g,

by

Act A

as

of actions on the vertex groups

A , and an arrow in

of groups.

Act A

There is a conjugation map

is totally disconnected,

from

p

to

~ : A ÷ Act A.

this map and the action of

a crossed complex ---+ 0 ---+ ... ---+ 0 ----+ A--~+~ Act A

w h i c h we w r i t e

XA .

olution

G , then the action of

~

of

(o, 1) : E ÷ X A isomorphism further

If

(where

A i

E

R> G

o : E ÷ Act A)

(~, ~) : X A ÷ X A

where

is an extension with associated E

on

.

A free equivalence

~ : Act A ÷ Act A

, Extf(G,

e : (CG, X A) The m o r p h i s m

e

since this result

details.

is given by

is the restriction

our m o r p h i s m

H, N, G

of

ef .

of standard

a $~ = ~((~-1a)8) ;

, .

W e give the proof only for

read our

for the group case, and

but with differences

G, A, E ; his factor set

k : C2G ÷ A ; his a u t o m o r p h i s m

a ~-+ a u

ef .

theory, w e do not give full

are given in [14],§|5.1

rather than free equivalence,

: for Hall's

A)

* Ext(G, A)

is a reformulation

[Some of the calculations

for equivalence follows

induces a m o r p h i s m

as in (*) induces an

There are canonical equivalences of groupoid8 ef : (CG, xA)f

Also,

by conjugation

o'n = ~o : E ÷ Act A .

T h e o r e m 3.

Proof.

A

crossed res-

of

N

for

in notation as (u,v) e N u ~ G

becomes

becomes

our

47

morphism

h : CIG ÷ Act A ; his choice

morphism

£ : CIG ÷ E ; his f u n c t i o n

A morphism

CG ÷ x A

+ Act A , k : C2G ÷ A

over

u ~-+ ~

GO = A0

such that

of coset r e p r e s e n t a t i v e s becomes our

~ : H ÷ N

k

becomes our d e r i v a t i o n

~ : CIG ÷ A . ]

is d e t e r m i n e d by a pair of m o r p h i s m s

is an operator m o r p h i s m over

h : CIG

h , and such that

the equations h~ = ~k , k~ = 0 hold.

(These equations are equivalent to the first two equations

of [14], and indeed

k~ = 0

in T h e o r e m 15.1.1

is, b y P r o p o s i t i o n 2, equivalent to the "factor set" con-

dition k[u + v,w] + k[u,v] h[w] = k [ u , v + w] + k[v,w] for all by

G

(u,v,w) ~ N3G*

.)

G i v e n such a m o r p h i s m

is defined by setting

set of pairs

(u,a)

E0 = GO

such that

and for

u E G(p,q)

CG ÷ X A , a n e x t e n s i o n

p, q e G O , letting

, a e A(q)

v e G(q,r)

, b ~ A(r)

.

The v e r i f i c a t i o n that

reader (cf. p.220 of [14]).

We write

Suppose now given two morphisms (h,k)

, (h',k')

as above.

B = 80 ' ~ = @I "

Then

Let

~

E

E(p,q)

of

A

be the

, with addition

(u,a) + (v,b) = (u + v, k[u,v] + a h[v] + b) for

,

E

,

is a groupoid is left to the

E = e(h,k) CG ÷ x A

over

G O , w h i c h w e w r i t e as pairs

@ : (h,k) = (h',k')

is a d e r i v a t i o n over

h'

b e a (free) homotopy, and if

and w r i t e

u e G(p,q) , v ~ G(q,r) ,

w e have h'[v] = -Bq + h[v] + Br - ~ [ v ] k'[u,v] = k[u,v] Br - a~[u,v]

,

.

A s t r a i g h t f o r w a r d c a l c u l a t i o n shows that k'[u,v] + ~ [ u , v ]

= -~[u + v] + k'[u,v] + (~[u]) h'[v] + ~[v]

(and this verifies that our d e f i n i t i o n of e q u i v a l e n c e agrees w i t h that on p.22! of [14]).

Define ef(8)

: e(h,k) ---+ e(h',k') (u,a) ~

Then by

ef(8)

(u, ~[u] + a Bq)

, u ~ G(p,q)

, a E A(q)

is an i s o m o r p h i s m of groupoids which, w i t h the a u t o m o r p h i s m

a ~-+ a Bq , a ~ A(q)

, defines a free equivalence of extensions.

. A ÷ A

Conversely,

given any

free e q u i v a l e n c e

arises in the above w a y if ÷ A

A

~ e(h,k)

> G

A

~ e(h',k')

> G

B : G O ÷ Act A

ined b y

n(u,0) = (u,~'u)

e(h,k)

complex

E

.

Let

h'

B(q) = ~]A(q)

, and : G ÷ A

~ : C|G

the f u n c t i o n

~'

def-

of

is equivalent to

.

Finally, w e show that any e x t e n s i o n some

is d e f i n e d by

is defined b y extending to a d e r i v a t i o n over

~ : CIG ÷ G

A _~i E -P+ G

A

by

G

b e the quotient m o r p h i s m and consider the crossed

obtained by trivial extension of the crossed E - m o d u l e

A .

Consider

48

the d i a g r a m C3G l I + 0

6 ~ C2G i Ik + ~ A

6 .> CIG i I~ + ~ E

A ~ A c t The crossed complex groupoid. (h,k)

CG

is free, while

So the identity on

is a m o r p h i s m

is defined by

G

CG ÷ X A

over

GO

A

is exact,

has a lift

G I I= % G

(~,k)

and both have

: CG -> E .

and an equivalence

G

Let

as fundamental

h = o~ .

of extensions

Then

e(h,k) -> E

(u,a) ~-+ ~[u] + ia .

Thus the crossed complex approach in non-abelian resolution

E

~

is successful

extension theory are so-to-speak

(a kind of universal

example)

because

compressed

some of the difficulties

into the standard crossed

and in particular

into the formula for

: C3G ÷ C2G • By standard homotopy Corollary

Let

4.

C

arguments,

we obtain from T h e o r e m 3;

be any free crossed resolution of the groupoid

G .

Then there

are equivalences of groupoids

Corollary

Let

Let

5.

e~ : (C, xA)f

> Extf(G, A),

e'

~ ext(G,

: (c, ×A)

N i--~ F -P-+ G

F-module

N .

G

Let

C

C ÷ ~

induces

n : F ÷ Act A

is injective. of groupoids

a set normally generating

Then a m o r p h i s m such that

h(r)

such that

C! = F

and

N = 6(C2)

5 is when P ÷ XA

A

. is centreless,

i.e. w h e n

is determined by a m o r p h i s m

is a conjugation

of

A

for each

r

in

N .

isation of Dedecker's work on non-abelian

"

.

also enable one to give a crossed complex version of a general-

be as above and suppose given a crossed ~0 = GO)

G

,

, (~, X A) --+ (C, X A)

special case of Corollary

The above methods

is free.

isomorphisms

(~, xA)f--+ (C, xA)f

: A ÷ Act A

A)

~ Ext(G, A)

be a free crossed resolution of

An interesting

F

obtained by trivial extension of the

~f(G,

e" : (~, xA)

Then the projection

D

Then there are equivalences of groupoids e~ : (~, xA)f ~

Proof.

.

be an extension of groupoids such that

P denote the crossed resolution of

crossed

A)

A H-~tension of

A

by

G

ether with a m o r p h i s m of crossed modules

cohomology ~-module

and extensions

A

is an extension

(where A-~

~ E ~

[12].

Let

G , A

is a groupoid with G

as above tog-

49

i

A

~E

I In fact if, by extending exes

~

and

of crossed

xHA

trivially,

respectively,

we regard

a function

then the above diagram

a co~ugation $

from

HO

xHA ÷ x~A

alence of

H-extensions

and a conjugation

in which

(~,~)

and

: x~A ÷ x~A

a groupoid

' q ~ ~0

(~,~)

such that

Ext,(G,

' such that Define a free equiV-

.

: ~ ÷ ~'

o'n = ~

generalisation

e is similar

on components

over the identity

•

: (CG, ×~A)

G , A , ~

~ ~(G,

A)

,

~ Ext'(G,

A)

.

(Dedecker's

result

is the bijection

results

that given a morphism

~ * xHA

to Oedecker's

A theory of extensions

internal

category

includes

the above equivalence

of groupoids,

to define similar

for the group case. ~ ÷ x~A

(where

e

If

X

is a CW-complex,

the aohomology of

ants of function

spaces

Remark 4. such that

A homotopy ft(Xn)

are discrete,

c Zn+ 1

for

complexes.

C

maps

fo' f|

We will

induces

do not include

and also extensions

are not used.

complex,

it seems reasonable

simply as

[~,

C] .

(A

to Postnikov

invari-

to have applications

invariant.

: ~ ~ ~

is a homotopy

prove elsewhere

a homotopy

the non-abelian

is a homotopy

theory,

in

cohomology.

n ~ 0 .

Consequently,

of

[19] he rel-

is given in [23], using

in E2] and applied

It would be interesting

of filtered

then such a homotopy

above for CW-complexes,

is a crossed

is developed

in [3].)

ft

C

complexes

X with coefficients in

theory of such a non-abelian

to Dedecker

Dedecker's

The results

and crossed and

On p.309 of is as in Coroll-

to be the coequaliser

of T-algebras

This generalises

of groupoids.

nor free equivalences,

~

2-cocycles.

and cohomology

for T-algebras.

idea for chain complexes

homotopy

crossed

objects

induced

are groups.)

has proved related

(for groups)

ates such morphisms

3.

G ,

Such free equivalences

ary 5), one can define an extension A ÷ E ÷ G by taking E __+ two ma~9 A--+ F m A (the semi direct-product). In a letter

Remark

on

of Theorem 3.

to that of Theorem 3. when

Huebschmann

[20] he shows

2.

for which there is

The (strict) equivalences are those

A) .

ef : (CG, x~A)f

Remark

(o,l) : ~ ÷ xHA

There are equivalences of groupoids

Theorem 6.

1.

compl-

is the identity.

We have the following

Remark

H(q)

(~,~)

~(a) = a Bq , a ~ A(q)

to be an isomorphism

(~,~)

form under composition

e

as crossed

is a morphism

to be an isomorphism

to the union of the

~(x) = -Bp + x + Bq , x e N(p,q)

by

these crossed modules

complexes.

Define

The proof

I

~fo = ~f|

cohomology

that if

X0

ft : X ÷ Z and

of morphisms

suggested

Z0

of

in Remark

3

50

R E F E R E N C E S O.

H. ANDO*, A note on the Eilenberg-MaeLane invariant, Tohoku

I.

N.K. A q~iLEY, Crossed oompl~e8 and T-~omplexe8, Ph.D. Thesis, University of Wales, (197fl).

Math. J. 9 (1957), 96-104.

2.

A.L. BLAKEKS, So~e relations between homology and homotopy groups, Ann. of Math., (49) 2 (1948), 428-46].

3.

R. ~ROWN, Cohomology with chains as coefficients, Proc. Lond. Math. Soc., (3) 14 (1964), 545-565.

4.

R. BROWN, On K~nneth suspensions, Proc. Camb. Phil. Soc., (1964), 60, 713-720.

5.

R. BROWN, Groupoids as coefficients, Proc. Loud. Math. Soc., (3) 25 (1072), 413-426.

6.

R. B~OWN and P.J. HIGGINS, The algebra of cubes, J. Pure Appl. A1 E. 21 (1981), 233-260.

7.

R. BI~OWN and P.J. HIGGINS, Colimit theorems for relative homotopy groups, J. Pure Appl. AlE. 22 ( 1 9 8 1 ) , 11-41.

8.

R. BROWN and P.J. HIGGINS~ The equivalence of w-groupoids and cubical T-complexes, Can. Top. G~om. uiff., (3e Coll. sur les cat4gories, d~di4 a Charles Ehresmann), 22 (1981), 349-370.

9.

R. BROWN and P.J. HIGGINS, The equivalence of crossed complexes and ~-groupoids, CaLl. Top. G~om. Diff. , (3e Coll. sur les categories, d4di~ a Charles Ehresmann), 22 ( 1981), 371-386.

10.

R. BROWN and P.J. HIGGINS, On the relation between crossed complexes and chain complexes with operators, (in preparation).

1|.

K, BROWN and J. HUEBSCHMANN, Identities among relations, in L~Dimenaiona~ and T.L. Thiekstun, Lond. Math. Soc. Lecture Note Series 48 (1982).

12.

P. OEOECKER, Les foncteurs Ext~ , H 2 4891-4894.

13.

P, OEDECKER and A. FREI, Gdn4ralisation de la suite exacte de cohomologie non ab~lienne, C.R. Acad. Sci. Paris, 263 (1966), 203-206.

14.

M. HALL, JR., Tile gheo~d of groups., MacMillan (]959).

]5.

P.J. HIGGINS, Ca~egorv~es c ~

16.

P.J. HIGGINS and J. TAYLOR, The fundamental groupoid and homotopy crossed complex of an orbit space, (these proceedings).

17.

O.F. HOLT, An interpretation of the cohomology group~

]8.

J. HOWIE, Pullback functors and crossed complexes, Cah. Top. G~om. Diff., 20 (1979), 281-295.

et

H2

TapoLogy, Ed. R. Brown

non ab~liens, C.R. Acad. Sci. Paris, 258 (1964),

~l~oupo~ds, van Nostrand Math. Studies, 32 (1971).

Un(G, M) , J. Alg., 60 (1979), 307-318.

19.

J. HUEflSCHMANN, Letter to P. Dedeeker, (4th June, 1977).

20.

J. NUEBSCHMANN, Crossed N-fold extensions of groups and cohomelogy, Comm. Math. Helv., 55 (1980), 302-314.

2].

J. HUEBSCtD4ANN, Automerphisms of group extensions and differentials in the Lyndon-Hoehsehild-Serre spectral sequence, J. Algebra, 72 (1981), 296-334.

22.

D.W. JONES, Po~l-T-comp~exe8,

23.

R. LAVEND~OMME and' J.R. ROISIN, Cohomologie non-ab~lienne de structures alg4briques, J. Algebra, 67 ( 1 9 8 0 ) , 385-414.

24.

A.S-T. LUE, Cobomology o f groups r e l a t i v e

25.

S. MAGLANE, Topology and l o g i c as a s o u r c e o f a l g e b r a ,

26.

S. MACLANE, H i s t o r i c a l

27.

J.P. MAY, S~npl~cia~ objects in a~gebralc topologyj van Nos~rand Math. Studies II (1967).

Ph.D. Thesis, University of Wales, (in preparation).

to a v a r i e t y ,

n o t e , J . A l g e b r a , 60 ( 1 9 7 9 ) ,

J . A l g e b r a , 69 ( 1 9 8 ] ) ,

155-174.

B u l l . Amer. Math. S o c . , 82 ( 1 9 7 6 ) ,

1-40.

319-320.

28.

G. SEGAL, Classifying spaces and spectral sequences, Publ. Math. I.H.E.S., 34 (1968), 105-112.

29.

G.W. WHITEHEAD, Elemen~a of /~omo~op~ ~heoz,~, Graduate texts in Maths. No. 61, Springer, Berlin~eidelberg-New York, (1978).

30.

J.H.C. WHITEHEAD, Combinatorial hometopy I, Bull. Amer. Math. Soc., (55) 3 (1949), 213-245.

31.

J.H.C. W~ilTEHEAD, Combinatorial hometopy II, Bull. Amer. Math. Soc., 55 (1949), 453-496.

32.

J.R.C. WHITEHEAD, Simple homotopy type, Amer. J. Math., 72 (1950), 1-57.

* NO~.

Reference [0] continues work of [2J.

UN CRITERE DE REPRESENTABILITE PAR SECTIONS CONTINUES DE FAISCEAUX Yves DIERS D~partement de Math~matiques, U.E.R. des Sciences Universit~ de Valenciennes, 59326 VALENCIENNES O. Introduction. Etant donn~ un foncteur tions chaque objet

B

de

B

d'un faisceau ~ valeurs dans

~

et fibres dans

globales d~fini sur la catggorie dans

~. La cat~gorie

~

flexive dans la cat~gorie

U : $ ÷ ~, on d~termine dans quelles condi-

est isomorphe g l'objet des sections globales continues

~ais~A

~, universel pour le foncteur sections

des faisceaux g valeurs dans

~

et fibres

peut alors ~tre plong~e d'une fa~on pleinement fiddle cor~~ais~A

si bien que chaque objet de

~

peut s'identifier

son faisceau repr~sentant. On utilise la construction universelle des spectres, topologies sepctrales et faisceaux structuraux donn~e dans [6] et on est rameng ~ d~terminer dans quelle condition le morphisme canonique de chaque objet de

~

vers l'objet des sections globales con-

tinues de son faisceau structural, est un isomorphisme. On montre qu'une condition n~cessaire et suffisante est que le foncteur

U

soit cog~n~rateur finiment r~gulier.

Cette notion, plus forte que celle de foncteur cog~n~rateur prDpre [7] et plus faible qne celle de foncteur codense [12], est obtenue ~ partir des notions de famille monomorphique stricte on effective

E8] on r~guligre

cog~n~ratrice par monomorphismes stricts

~]

[5] de morphismes, de famille d'objets

et de morphismes de presentation finie

relative [7~, et est d~crite de plusieurs fa~ons diffgrentes. Dans certaines conditions, un foncteur est cog~n~rateur finiment r~gulier si et seulement si il est cog~n~rateur. Ainsi si et si

~

est une sous-cat~gorie de

~

B

est une cat~gorie arithm~tique

[5] et [15],

ferm~e pour les ultraproduits et dont les mor-

phismes sont exactement les monomorphismes de

~

dont le but est dans

$, alors

est une sous-cat~gorie cog~n~ratrice finiment r~guli~re si et seulement si une sous-cat~gorie cog~n~ratrice, d'un produit d'objets de

c'est-g-dire si tout objet de

~

~

est

est sous-objet

&. II s'en suit un th~orgme de representations par sections

continues de faisceaux qui contient t o u s l e s

thgor~mes de reprgsentations qui utili-

sent habituellement des versions g~n~ralis~es du th~or~me chinois sur les syst~mes de congruences. En appliquant les r~sultats ~ des foncteurs

U

oubli de structure ad~quats entre ca-

tegories d'ensembles munis de structures alg~briques, on obtient d'une part, de tr~s nombreux th~or~mes connus de representation par sections continues de faisceaux dont quelques uns sont d~taill~s ici, et d'autre part, des nouveaut~s parmi lesquelles la representation des anneaux commutatifs r~guliers formellement r~els par des faisceaux de corps ordonn~s, celle des groupes ab~liens sans torsion par des faisceaux de groupes abgliens totalement ordonn~s, celle des espaces veetoriels r~els par des faisceaux d'espaces vectoriels euclidiens ou par des faisceaux d'espaces vectoriels norm~s, celle des ensembles par des faisceaux d'ordinaux finis. Une originalit~ de ces derni~res

52

representations

est que les faisceaux repr~sentsntsont

ces topologiques

non "spectraux"

au sens de Hochster

en g~n~ral pour bases des espa[9] car non To-s~pargs

et ~ven-

tuellem~nt non quasi-compacts. On utilise les notations et les rgsultats de [5], ]. Foncteurs

cog~n~rateurs

de presentation IB

finie

~

tout morphisme

B/~

U-injective

des objets de de

B

si tout morphismes

core appel~es

U : /A ÷ ~. Un morphisme

au-dessous U

de

B. ll est dit

de

finie

[7]

U-injectif

si

se factorise ~ travers lui. Plus ggngrale-

de morphismes

g : B -~ UA

de

B

Les families monomorphiques

families monomorphiques

f : B -> C

s'il est un objet de presentation

~

vers

(fi : B -> Ci)ic I

l'un de ses membres.

dams

finie relative

g : B -~ UA

ment, une famille

On consid~re une cat~gorie localement

[7] et un foncteur

est dit de presentation

dams la catggorie

finiment rgguliers.

[6].

de m~me source de vers

U

IB, est dite

se factorise g travers

r~guli~res

de morphismes

strietes ou effectives

dans

de

~, en-

[81, sont ~tudi~es

[5].

|.0. D~finition. famille

Le foncteur

U-injective

U : ~A ÷ ~

de morphismes

est coggn~rateur

de prgsentation

finiment r~gulier si toute

finie relative de

~

est monomor-

phique r~guli~re. Rappelons

qu'un foncteur

au sens de Grothendieck tion

HO~B(f,UA)

U : /A ÷ ~

est dit coggn~rateur

[|4] si tout morphisme

: HO~B(C,UA)

-~ HO~B(B,UA)

est n~cessairement

isomorphique.

|.]. Proposition.

Si le foneteur

U

f : B ÷ C

propre de

~

[7] ou cog~n~rateur tel que l'applica-

soit bijeetive pour tout objet

est cog~ngrateur

A

de

/A,

il est cogg-

finiment rggulier,

ngrateur propre doric coggn~rateu_r. Preuve

: Soit

f : B ÷ C

un morphisme de

bijective pour tout objet finie,

la categoric

objets de

B/B

de presentation monomorphique

B/~

A

de

Soit

tel que l'application ~

l'est aussi et le morphisme finie au-dessus de lui

finie relative.

Chaque morphisme

rggulier.

Le morphisme

m,n : C ~ D

r~guliers.

deux morphismes

g : D + UA

f

f

est colimite filtrante des i.e. f = lim f. i-~+ i

Ii reste g montrer que mf = nf

gmf = gnf

donc

et soit gm = gn

f

k : D + K puisque

; il se factorise done ~ travers

k ; ce qui implique que

est bijective

et donc que

et par suite

k

est monomorphique

dams les representations

jamais d'adjoint ~ gauche, mais ils ont ngcessairement donn~ par les fibres des faisceaux repr~sentants. a un multiadjoint

g gauche et pour chaque objet

une famille universelle

de morphismes

de

avec

f. :B+C. l i

est ~pimorphique.

est bijective

Les foncteurs qui interviennent

soit

f. : B ÷ C. est U-injectif donc i I est alors monomorphique r~gulier comme colimi-

v~rifiant

vgrifie

HomB(f,UA)

~tant localement de presentation

de prgsentatiOn

te filtrante de monomorphismes

Tout morphisme

~

~. La cat~gorie

B

vers

leur COnoyau~

Hom~(f,UA) HO~B(k,UA)

m = n.

par faisceaux ne poss~dent

un multiadjoint

~ gauche

[4]

On suppose donc que le foncteur B

de U.

U

~, on note (~i :B÷UAi)icSpecu(~

53 1.2. Proposition. Si le foncteur

a un multiadjoint ~ gauche, il est cog~-

U :~ ÷ 8

n~rateur finiment r~gulier si et seulement s i i l de

~

existe une classe

de morphismes

telle que

(I) tout morphisme diagonalement universel de morphismes de

~

de source

B

B

vers

U

est colimite filtrante de

et

(2) toute famille U-injective de morphismes de

~ , est monomorphique r~guli~re.

Preuve : La condition n~cessaire est satisfaite en prenant pour phismes de presentation finie relative de 0

~

de morphismes de

universelle

B

0

la classe des mor-

~. R~ciproquement supposons qu'une classe

satisfasse (I) et (2). Pour chaque objet

B

de

~, la famille

(Ni : B ÷ UA.) est monomorphique. En effet, si T est un objet de pr~i ~ et m,n : T ~ B sont deux morphismes vgrifiant D.m = N.n i i i c Specu(B), alors d'apr~s (|), pour chaque i, il existe un morphisme

sentation finie de pour tout d. : B ÷ D. i i

de

~

au-dessus de

N. i

tel que

d.m = d.n. La famille l i

est U-injective, donc monomorphique r~guli~re d'apr~s (2). Par suite tat est aussi vrai pour un objet quelconque d'objets de presentation finie de

(Ni). Soit

morphisme

fk(i) : B + Ck(i)

d.1 : B ~ D.I

de

~

)

m = n. Le =~sul-

~, puisque celui-ci est colimite (Ni)

~tant monomorphiques,

route

est aussi monomorphique puisque plus fine une famille U-injective de morphismes

~. Chaque morphisme

~. se factorise ~ travers un i k(i) c K. D'apr~s (I), il existe un morphisme

avec

au-dessus de

(d i : B + Di)icSpecu(B)

~

(fk : B ÷ Ck)kc K

de presentation finie relative de

de

~. Les families

famille U-injective de morphismes de [5] qu'une famille

T

(di)icSpecu( B

N.,I qui se factorise ~ travers

fk(i)" La famille

ainsi obtenue est U-injective donc monomorphique r~guli~re.

elle est moins fine que la famille

(fk)kcK

et m~me r~guli~rement moins fine [5]

puisque route image directe de la famille Ii s'en suit que la famille Lorsque les objets de

A

(fk)kc K

(d.) est U-injective donc monomorphique. i est monomorphique r~guli~re (prop. 2.1 [5]).

sont des ensembles munis d'une structure alg~brique d~finis-

sable par une th~orie logique du premier ordre, la cat~gorie

$

est ~ ultraproduits.

Nous allons montrer que, dans ce cas, il suffit de consid~rer les families finies de morphismes. 1.3. Proposition. Si le foncteur ultraproduits d'objets de (I)

U

U :A ÷ ~

a un multiadjoint g gauche et rel~ve les

& ([6] 4.1), les assertions suivantes sont gquiValentes

:

est cog~n~rateur finiment r~gulier,

(2) toute famille finie U-injective de morphismes de (3) il existe une classe

D

~, est monomorphique r~guli~re,

de morphismes de presentation finie relative de

~

telle

que a) tout morphisme diagonalement universel de de morphismes de

0

B

vers

b) toute famille finie U-injective de morphismes de Preuve :

U

est colimite filtrante

et

(I) => (2):Soit

(fi : B ~ Ci)ic[],n]

est monomorphique rgguli~re

une famille finie U-injective de

54

morphismes de

~. La cat~gorie

(B/~) n

existe une petite cat~gorie filtrante

~tant localement de presentation finie, il ~

et un diagramme

d'objets de presentation finie de

(B/B) n

que les morphismes

sont de presentation

tout

fik : B + Cik

((fik:B + Cik)icE1,n~kcK

dont la colimite est

(fi). C'est-g-dire

finie relative et que pour

i £ El,n],

fi = ~ fik" Pour ehaque k ¢ ~, Is famille (fik:B ÷ Cik)ie[l,n] keK est U-injective done monomorphique rfiguli~re. Notons (fijk : Cik + Cijk' f~jk: Cjk ÷ Cij k)

la somme amalgam~e de

(fik : C + Cik, fjk : C + Cjk)

et

n

n

Pik :

R Cik + Cik la projection canonique. Le morphisme (fik) : B + ~ Cik i=I i=l est noyau des deux morphismes (f~jkPik)(i,j)e[l,n]2 et (f~jkPjk)(i,j)e[1,n]2 de f,. (f~j:C. " i ÷ C.., xj xj :C.j -~ n Cij) la somme amalgam~e de (fi : B ÷ Ci, f. : B -+ Cj) et Pi : ~ C. ÷ C. la J i=l i i projection d'indice i, alors par passage ~ la colimite filtrante suivant ~, le morN phisme (fi) : B ~i=IH C.x est noyau des deux morphismes (f~:pi)J (i,j)cE1,n]2 et source

n i~l Cik

(f'.'.p.) 2 lj j (i,j)e[l,n] la famille

(~Tn) (i,j)=(l,l)Cijk.

et de but

n ~ C. i=l i

de source

(fi)ieEl,n ]

est monomorphique

morphismes

de

r~guli~re. ~

la classe des morphismes de presentation

~.

(3) => (1) :avec la proposition de

(n,n) ~ C .. Cela implique que (i,j)=(],l) ij

et de but

(2) => (3) : est satisfait en prenant pour finie relative de

Si l'on note

~

].2, il suffit de montrer que toute famille U-injective

est monomorphique

r~guli~re.

Soit

(fk : B + Ck)kc K

une telle

famille. Supposons qu'il n'existe aucune sous-famille finie U-injective de Pour chaque partie finie que

K de K, notons D(K o) o se factorise g travers l'un des morphismes

Hi

D(Ko) # Specu(B) ,

D(~) = ~

et

D(KoO

(fk)kcK .

l'ensemble des i ¢ Specu(B) fk

avec

K I) = D(Ko ) 0 D(K l)

tel

k e Ko. Les relations montrent que les parties

compl~mentaires finies de

des parties D(K ) dans Specu(B) quand K parcourt les parties o ~ o K, forment une base de filtre sur Specu(B ). Soit F un ultrafiltre plus

fin. Ii existe un objet (UAi)icSpecu(B)

AF

suivant

de

$

tel que

torise ~ travers un morphisme

fk

avec

colimite filtrante et que le morphisme existe

I ¢ F

fk" L'inclusion

tel que le morphisme I C D({k})

r~sulte que la famille

k c K. Puisque fk

D({k})

de

NF : B + UA F

UA F = I ~

se fac-

icl~ UA.I est une

finie relative,

il

H UA i se factorise ~ travers iel D({k}) e F, ce qui est en contradiction dans

Specu(B)

appartient ~

poss~de une sous-famille finie U-injective

est monomorphique

r~guli~re. La famille

est done monomorphique

injectives. La sous-famille

d~fini

est de presentation

implique alors

(fk)kcK

(fk)keKo

(H i : B * UAi)icSpecu(B)

soit l'ultraproduit

(Ni)icl : B ÷

avec le fair que le compl~mentaire de

La sous-famille

UA F

F. Le morphisme canoniquement

(fk)kcK

F. Ii en (fk)kcK . o

de m~me que toutes les families U-

est r~guligrement plus fine que la famille o

55

(fk)keK

(2 [5])

puisque toute image directe de

(fk)k~K

est U-injective donc mono-

morphi~ue. De la proposition 2.1 [5], il r6sulte que la f~mille

(fk)keK

est monomor-

phique rgguli~re. ].4. Proposition. Si le foncteur

U

est codense []2], il est cog~ngrateur finiment

r~gulier. Preuve : Si &

U

est codense, tout objet

B

de

B

est limite de t o u s l e s objets de

au-dessous de lui, ce qui implique que la famille de t o u s l e s morphismes de

vers

U

B

est monomorphique r~guli~re (prop. 5.4 F5]). Toute famille U-injective de

morphismes de source

B

est plus fine que la famille pr6c6dente donc est monomorphi-

que ; elle 8st m~me rgguli~rement plus fine puisque toute image directe d'une famille U-injective est U-injective donc monomorphique (prop• 2.1

; elle est donc monomorphique r6guligre

~]).

1.5. Proposition.

Le foncteur

existe un foncteur

V : K ÷ ~

U :~ ÷ B

est cog6n~rateur finiment r~gulier s'il

tel que le foncteur

UV

soit cog~n~rateur finiment

r~gulier. Preuve : Toute famille U-injective de morphismes de presentation finie relative de m~me source de

~

est UV-injective donc monomorphique r~guli~re.

2. Le crit~re de repr6sentabilit~. 2.0. Th~or~me. Soit

U :~ ÷ ~

un foncteur tel que : I)

ment de presentation finie, 2) U diagonalement universel d'un objet

~

est une cat~gorie locale-

admet un multiadjoint ~ gauche, 3) tout morphisme B

pr6sentation finie relative de source

de B

~

vers

U

est colimite de morphismes de

diagonalement universels pour

U, 4)

U est

cog~n~rateur finiment r~gulier. Alors tout objet dans

~

B

de

et fibres dans

~

d~termine un faisceau

FB

Spe_cu(B)

~ valeurs

$, dont l'objet des sections globales est isomorphe ~

qui est universel pour le foncteur sections globales que le foncteur

de base

Bet

F : ~ais ~A ÷ ~ ; c'est-g-dire

F admet un adjoint g gauche pleinement fiddle. Si les conditions

I), 2), 3) sont satisfaites, la condition 4) est en fait n~cessaire et suffisante pour obtenir la conclusion. Preuve : Les conditions ]), 2), 3) sont les conditions d'applications du th6or~me 3.1 de E6] dont on utilise ici les notations et les r~sultats (cf. 3.0, 3.], 3.3, 3.4, 3.5). Soit

B

un object de

a) la famille universelle foncteur

U

est monomorphique puisque le

est coggn~rateur. Cela implique que les morphismes de

universels pour f, g : C ~ D

~. (H i : B + UAi)ieSpecu(B)

U

sont ~pimorphiques. En effet si

sont deux morphismes v~rifiant

on a ~3f6 = ~jg6

ce qui implique

(nj• : D ÷ UAj)jaSpecu(D)

~jf

=rljg

6 : B + C

~

est l'un d'eux et si

f6 = g6, alors pour tout donc

f = g

diagonalement

j e Specu(D),

car la famille

est monomorphique.

b) Montrons que le foncteur

D : ~(B) ÷ D(Specu(B)) °p

est une 6quivalence de cat6-

56

gories. II est surjectif sur les objets d'apr~s la construction de est fiddle puisque les morphismes de morphisme entre deux objets de ~(B) de

tels que

A'(B)

~(B). Soit

D(6) C D(6'). Notons

(6,6'). La relation

6tant 6pimorphiques, 6 : B ÷ C, 6' : B ~ C'

D(6|6) = D(6) N D(@') = D(6)

gonalement universel de presentation finie relative

@|

est U-injectif.

6

est plein.

(dk : (C,@) ÷ (Ck,@k))keK

D(@) = keK ~J D(6k)" La famille relative est U-injective.

6 : B ÷ C

PB(Specu(B))

dans

~(B), est un faisceau. On en d~duit

canoniques

B ÷ FB(Specu(B))

F

F : ~ais ~A ÷ ~

FB(D(6)) ÷ Fc(Specu(C))

F : ~ais ~

FB(Specu(B))

on d~duit

est pleinement

fiddle

est un isomorphisme.

que l'application

Specu(~ ) : Specu(C) ÷ Specu(B)

~.j : A.1 ÷ A.j

ouvert (prop. 3.3.6 [6]) la fibre de

l'isomorphisme

le morphisme canoniquement ~ ~. Alors

Soit

et

j. Le morphisme

Fc(Specu(@)) ~

A'(B)

est un faisceau sur

est monomorphique

au point

= B

et du

(U~j) ~i = ~.6. Puisj i

fibre de

est la fibre de F6

en

i

(F~)D(@)

r~guli~re.

Or on a

(6k : B ÷ Ck)ke K

1.2, il

de morphismes

ke~K D(6 k) = Specu(B) , doric, puisque (FB(Specu(B)) ÷

r~guli~re. Compte tenu des isomorphismes

e), la famille pr~c~dente est isomorphe g la famille

3. Un crit~re de reprgsentabilit~

FC

est doric

: FB(D(6)) ÷

~'(B). D'apr~s la proposition

Specu(B), la famille de morphismes

est monomorphique

:

est cog~n~rateur finiment r~gulier. Tout morphisme

est colimite filtrante de morphismes de

FB

(F6)D(6)

est un plongement hom~omorphique

suffit donc de montrer que toute famille U-injective de

d~fi-

j e Specu(C ). Posons

U~. : UA. ~ UA.. On en d~duit que le morphisme i i j est un isomorphisme. U

On

F~ : F B ÷ Fc(Specu(~)) ~

l'isomorphisme d6fini par

(F@)i : (FB)i ÷ (Fc(Specu ( 6 )))i

f) Montrons que le foncteur

@ : B ÷ C e &'(B),

est un isomorphisme.

Specu(B). Montrons que le morphisme

i = (Specu(6))(j)

FB(Specu(B))

et

I), 2), 3) que le foncteur adjoint ~ gauche

(Specu(6),F 6) : (Specu(B),F B) ÷ (Specu(C),F C)

FB(D(~k)))ke K

F B ~ PB

PB(D(6)) = C

~tant des isomorphismes,

FB(D(~)) ÷ Fc(Specu(C))

est un morphisme de faisceaux sur

Ni

Cela exprime pr~ci-

dgfini par

soit pleinement fiddle. Montrons d'abord que pour

ni par le foncteur adjoint ~ gauche ~

Fc(Specu(C))

telle que

~12])

le morphisme canoniquement d6fini

en

r~guligre.

PB : V(Specu(B)°P ÷ ~

e) Supposons maintenant avec les conditions

note

vers

de morphismes de presentation finie

Elle est donc monomorphique

que le foncteur adjoint ~ gauche au foncteur

au foncteur

~(B)

6'

= PB(D(IB)) = B.

d) Les morphismes

(Th. 1, p.88,

C'est donc

est un morphisme de

une famille de morphismes de

(dk : C ÷ Ck)ke K

s~ment que le pr~faisceau structural pour

D

la somme amalgam~e

implique que le morphisme dia-

6~1@~

dans la cat~gorie ~(B). Ainsi le foncteur

et il

deux objets de

(61 : C ÷ CI, 61 : C' ÷ C|)

un isomorphisme d'aprgs 4). Ce qui implique que

c) Soit

N(Specu(B))

il y a au plus un

(6k).

special pour les categories arithm~tiques.

Le th6or~me suivant contient les th~orgmes de repr~sentabilit~ de faisceaux qui utilisent habituellement

par sections continues

une version g~n~ralis6e du th~or~me chinois

57

sur les syst~mes de congruences. arithm~tiques, 3.0. Th~orgme. trice de

6

exactement Alors

A

So it

6

une cat~gorie arithm~tique

les monomorphismes

de

FB

de base

teur sections globales

: Soit

valences

R

on note

B

B

de

B

g

et

un objet de

vers

~

et

~. Notons

phismes diagonalement

B

de

L'ensemble ~

nalement universels pour

de

~

d~ter-

$, dont

F

admet un

R

~. Pour chaque

(DR : B + B/R)RcSpecu(B )

En effet, si

g : B ÷ X

la relation d'~quivalence de

h : B/R + X

B

est dans

est un m~rphis-

sur

tel que

B

~

~, donc

B/R

multir~flexive ~R : B + B/R

les morphismes

$. Ils sont aussi de presentation B

par un

de

o0

~. Notons R

est une

et o0

~R

est

est fermg pour les colimites finies dans

~tant monomorphiques,

de

g

est

II est imm~diat qu'une telle fac-

de la forme

A'(B)

engen-

h D R = g ; puis-

engendr~e par un nombre fini d'~Igments

universel

R e Specu(B) ,

vers

~

de

A'(B)

sont diago-

finie relative.

~tant colimite

Tout mor-

filtrante de morphismes

A'(B), la condition 3) du th~or~me 2.0 est satisfaite.

II reste g montrer que 6. Pour chaque objet phique puisque de morphismes

~ de

$ B

est une sous-cat~gorie de

¢og~nfiratrice dans ~

est monomorphique.

B, les morphismes

De la proposition

de morphismes

4. Applications. de representation foncteurs

Le th~or~me 2.0

de

A'(B)

7.11

[5], il r~sulte que

est monomorphique

r~guli~re.

1.3 en prenant comme morphismes

permet de retrouver de tr~s nombreux

oubli de structure ad~quats.

de classe

th~or~mes

II suffit de l'appliquer

connus g des

De nombreux exemples de foncteurs

I), 2), 3) du th~or~me

sont satisfaites

[6]. II reste au lecteur ~ d~terminer dans quels cas, le foncteur

rateur flniment r~gulier.

est monomor-

A'(B).

par sections continues de faisceaux.

U :$ + 6

pour lesquels les hypotheses

dans

de

finiment r~guli~re de

(NR : B + B/R)

~. Par suite toute famille U-injective

Le r~sultat d~coule alors de la proposition de source

coggn~ratrice

6, la famille universelle

toute famille finie U-injective

U

B

et fibres dans

que le foncteur

La famille

est donc une sous-cat~gorie

le morphisme

de

B

; on obtient ainsi une factorisation de

sur

de

et tout objet

l'ensemble des relations d'~qui-

(qR : B ÷ B/R)RcSpecu(B).

relation d'~quivalence quotient.

sont

~.

est dans

l'unique monomorphisme

l'ensemble des morphismes

B/~. Les morphismes

B/R

quotient.

$, on note

R e Specu(B)

de la famille

cog~n~ra-

&

et qui est universel pour le fonc-

Specu(B )

~, le morphisme

torisation est unique. ~ A'(B)

B

est universelle.

h : B/R ÷ X

~

est isomorphe g

le morphisme

est un objet de

morphisme

~

de

fiddle.

vers un objet de

dr~e par

de

g valeurs dans

dont l'objet quotient

NR : B + B/R

de morphismes

X

B

une sous-cat~gorie

dont le but est dans

Specu(B)

un objet de

sur

~

F : ~ais ~A + B ; c'est-~-dire

adjoint g gauche pleinement

que

6

et

et telle que les morphismes

est une sous-cat~gorie multir~flexive

l'objet des sections globales

me de

et El5].

ferm~e pour les ultraproduits

mine un faisceau

Preuve

Pour la d~finition et des exemples de categories

on peut se reporter ~ ~]

sont dorm's U

est cog~n~-

Nous en ~tudions quelques uns. Pour les categories

arithm~-

88

tiques (4.3 ~ 4.7), on utilise plutSt le th6or~me 3.0. avec lequel on est ramen6 montrer que le foncteur inclusion objet de

~

U :& ÷ ~

est cog6n6rateur, c'est-~-dire que tout

est sous-objet d'un produit d'objets de

&. Or c'est une propri6t6 souvent

bien connue dont la preuve repose essentiellement sur le lemme de Zorn. Les representations (4.8 g 4.12) sont nouvelles. Les faisceaux repr6sentants poss~dent l'originalit6 d'avoir pour bases des espaces topologiques non "spectraux" au sens de Hochster [9] car non To-s6par6s et 6ventuellement non quasi-compacts. 4.0. Repr6sentation d'un anneau cormnutatif par un faisceau d'anneaux locaux [2], [3]. Compte tenu de 7.0 [6], il suffit de montrer que le foncteur

U : Gocc ÷ S n e

est

cog6n6rateur finiment r6gulier. Un ultraproduit d'anneaux locaux 6tant un anneau local, le foncteur classe

~

U

relgve les ultraproduits. Utilisons la proposition 1.3 avec la

des morphismes de la forme

A ÷ A[a-l]. Soit

(A ÷ A~aTl])i~[l,n~ ~

mille U-injective. Pour chaque P c Specu(A), le morphisme -I A ÷ A[ai(p) ] avec i(P) c If,hi ; alors

vers un morphisme

A ÷ Apse

une fa-

factorise

~ tra-

ai(p) ~ P. L'id6al de

A

engendr6 par l'ensemble des 616ments id6al premier de ea famille

A, est ~gal g

(A ÷ A[a71])icD,n]a

a. pour i ~ [l,n] n'6tant eontenu dans aucun l A. La suite a I , ...,an engendre done le A-module A. est donc monomorphique r6guli~re (8.O~5]).

ainsi la repr6sentation classique de

A

par son faisceau structural

On obtient

A.

4.1. Repr6sentation d'un anneau commutatif par un faisceau d'anneaux ind6composables [3], El 3]. Soit

U : /Anclnd ÷/Anc

le foncteur inclusion (7.5 [6]). Pour un anneau

famille des anneaux quotients de l'anneau des idempotents de vers

U. Chaque morphisme

(A ÷ A/PA)

o3

P

A, est une famille universelle de morphismes de

A ÷ A/PA

A. Montrons que le foncteur

U

A ÷ A/Ae

o7

est coggn~rateur finiment rggulier. Un ultra-

ultraproduits. Utilisons la proposition 1.3 avec la elasse A ÷ A/Ae. Soit

est un idempo-

e

produits d'anneaux indgcomposables ~tant indgcomposable, le foncteur

forme

A

est colimite filtrante de morphismes diagonalement

universels de presentation finie relative de la forme tent de

A c ~nc, la

dgcrit l'ensemble des id~aux premiers

0

U

relgve les

des morphismes de la

P c Specu(A), il existe

(A ÷ A/Ae.). [',n] une famille finie U-injective. Pour chaque llXC I i(p) c [ ,3 tel que le morphisme A ÷ A/PA se factorise

travers

i.e tel que

A ÷ A/Aei(p)

tient ~ t o u s l e s

ei(p) c P. Par suite l'idempotent

id6aux premiers d'idempotents de

(A + A/Aei)ic[l,n ]

~ e appari=I i A ; il est donc nul. La famille

est done monomorphique r6guli~re (8.2 [5]).

4.2. Repr6sentation d'un treillis distributif par un faiseeau de treillis locaux Compte tenu de 7.9 [6], il suffit de montrer que le foncteur

U : ~rDLoc ÷ ~rD

~],~] est

cog~n~rateur finiment r~gulier. Ce foncteur rel~ve les ultraproduits puisqu'un ultraproduit de treillis locaux est un treillis local. Utilisons la proposition 1.3 avec la classe

~

principal de

des morphismes quotients de la forme E

engendr6 par

jective. Pour chaque

a. Soit

E ÷ E/(a)

(E ÷ E/(ai))iEEl,n~

~ c Specu(E) , le morphisme

E + E/~

o3

(a)

est le filtre

une famille finie U-infactorise g travers un

59

morphisme

E ÷ E/(ai(~))

a I V ... V an Donc

avec

i(~) e If,n], et par suite

appartient alors g t o u s l e s

ai(~) e ~. L'61~ment

filtres premiers de

E ; il est ~gal ~

I.

E/(a I) ~ ... ~ (an ) = E/(a I V ... V an ) = E. La famille (E ÷ E/(ai))ie[1,n] est

monomorphique rgguli~re d'apr~s 7.O.1, 8.5.1, 8.5.2 [5]. 4.3. Representation d'un anneau com~utatif r~gulier par un faisceau de corps commuta-

[3],

tifs

[10].

Le foncteur inclusion

U : ~c ÷ &ncReg (7.3 [6~) satisfait les hypotheses du th~or~me

3.0. En effet la cat~gorie gorie cog6n~ratrice de r6gulier

A ¢ ~ncReg

(A ÷ A/P)PeSpec(A )

SncReg

est arithm6tique (8.5 [5]), ~c

est une sous-cat6-

/AncReg puisque l'intersection des id~aux maximaux d'un anneau est r~duite ~ z6ro et donc la famille des anneaux quotients

est monomorphique,

~c

est ferm6e pour les ultraproduits et tout

sous-anneau r~gulier d'un corps cormnutatif est un corps. 4.4. Repr6sentation d'un anneau fortement r~gulier par un faisceau de corps [0_]. Le foncteur inclusion 3.0, la cat~gorie

U : ~ ÷ ~nForReg (7.4 [6]) satisfait les hypotheses du th~or~me

/AnForReg

6tant arithm~tique (8.5 [5]) et tout anneau A c /AnForReg

6tant un sous-anneau d'un produit de corps. 4.5. Representation d'un groupe ab~lien r~ticul~ par un faisceau de groupes abgliens totalement ordonn~s ~I]. Le foncteur inclusion

U : ~bTotOrd ÷ ~bRet (7.10 [6]) satisfait les hypotheses du

th~or~me 3.0, la cat~gorie

AbRet

~tant arithm~tique (8.5 ~]).

4.6. Representation d'un anneau commutatif fortement r~ticul~ par un faisceau d'anneaux totalement ordonn~s [II]. Le foncteur inclusion

U : IAncTotOrd ÷ ~ncForR~t

du th~or~me 3.0, la cat~gorie

(7.11

[4)

satisfait les hypotheses

AncForRet ~tant arithm~tique (8.5 [5]).

4.7. Representation d'un anneau commutatif rggulier fortement r~ticul~ par un faisceau de corps commutatifs ordonn~s Le foncteur inclusion

[11].

~cOrd ÷ SncRegForRet

(7.12 [6]) satisfait les hypotheses du

th~or~me 3.0. 4.8. Representation d'un anneau commutatif r6gulier formellement r~el par l'anneau des sections globales d'un faisceau de corps commutatifs ordonngs. On considgre la cat~gorie

~ncRegFormRl des anneaux commutatifs unitaires r~guliers

formellement r~els i.e. qui v~rifient l'axiome :

~Xl, ...,Xn,

1+x~ + ... + x n2

ble, et des homomorphismes d'anneaux et le foncteur oubli de structure AncRegFormRl. Le foncteur A £ SncRegFormRl (O) - I ~ P (x ~ P

ou

U

admet un multiadjoint ~ gauche. Le spectre de

relativement g

(1) P + P C P

vers

U

est l'ensemble des parties

(2) PP C P

(3) P O (-P) = A

P

de

o7

telles que

(A ÷ A/p N (=P))

P, est une famille universelle de morphismes de

U. La topologie spectrale est engendr~e par les parties

{P : -(a~ + ... + a~) e P}

A

(4) Vx C A, ~y c A(xy c - P=>

y c P)) (cf. 7.27 [6]). La famille des anneaux quotients

munis des ordres quotients de A

inversi-

U : ~cOrd ÷

a I ..... a n e A. Elle n'est pas

D(al,...,a n) =

To-s~par6e car toutes

60

les parties

P

qui d~finissent un ordre sur

(-P) = O, sont des points denses de

r~me 2.0 sont satisfaites. Montrons que id~aux maximaux

M

de

A

A i.e. qui v~rifient en plus (5) P

Specu(A). Les hypotheses (1), (2), (3) du th~oU

est coggn~rateur liniment r~gulier. Les

sont r~els, donc les corps quotients

A/M

rgels et par suite ordonnables. La famille des anneaux quotients de maximaux est donc une famille de morphismes de morphique, donc le foncteur duits. Chaque morphisme

U

A

vers

par ses id~aux

U. Or c'est une famille mono-

est cog~n~rateur. Le foncteur

A ~ A/p ~ (_p)

sont formellement A

U

rel~ve les ultrapro-

est colimite filtrante de morphismes quotients

A ÷ A/I

o~ I est un ideal de type fini de A. Si (A ÷ A/I ,...,A ÷ A/in) est une 1 famille finie U-injective, elle est monomorphique donc monomorphique r~guligre puisque

la cat~gorie foneteur

U

AncRegFormReel

est arithm~tique (prop. 7.O.1.

[~).

Ii s'ensuit que le

est cog~n~rateur finiment r~gulier.

4.9. Representation d'un groupe abglien sans torsion par le groupe des sections globales d'un faisceau de groupes ab~liens totalement ordonn~s. Le foncteur

U : &bTotOrd ÷ SbSTor (7.30 et 7.31 E ~ )

est cog~n~rateur finiment r~gu-

lier car il est surjectif sur les objets puisque tout groupe ab61ien sans torsion est totalement ordonnable. 4.10. Representation d'un espace vectoriel r~el par l'espace vectoriel des sections globales d'un faisceau d'espaces vectoriels euclidiens. Montrons que le foncteur

U : ~ucl ÷ ~ec(~)

est coggn~rateur finiment r~gulier (7.32

[6]), en utilisant la proposition 1.2 avec la classe E ÷ E/X E/Xi)ic I

o~

X

e

Xi(l)+...+Xi(n)

et un suppl~mentaire

P la forme

X'

quadratique positive

q

q e Specu(E). II existe

sur

9. Soit

(E ÷

est de dimension finie done poss~de une base

avec E

E. Soit

i(1),...,i(n) c I. L'es-

de codimension finie. Tout ~Igment

x = Xlel+...+Xpep+X'

factorise

des morphismes de la forme

est un sous-espace vectoriel de dimension finie de

une famille U-injeetive de morphismes de

pace vectoriel

D

par

i(n+l) c I

Xl,...,~

c ~

et

x

de

E

el,...,

~tant de

x' c X', on d~finit la forme

q(x) x~+...+x 2. Alors ISO(q) = X', donc P tel que le morphisme quotient E + E/Xi(n+l )

E + E/ISO(q) , c'est-~-dire tel que

Xi(n+l)C

ISO(q)

donc tel que

(Xi(1)+...+Xi(n) N Xi(n+l) = {0}. II s'ensuit que la famille des morphismes quotients (E ~ E / X (• 1) , . . .,E ÷ E/Xi(n+l) ) prop. 6.0 [5], la famille

est monomorphique r~guli~re (8.4 [5~). D'apr~s la

(E ÷ E/Xi)ic I

est monomorphique r~guli~re.

4.11. Representation d'un espace vectoriel rgel par l'espace vectoriel des sections globales d'un faisceau d'espaces vectoriels norm~s. Le foncteur

U : ~orm(~) ÷ ~ect(~)

(7.34 [6])

est eog~n~rateur finiment r~gulier

puisqu'il factorise le foncteur cog~n~rateur liniment r~gulier

Eucl ÷ ~ect(~) (prop.l~)

4.12. Representation d'un ensemble par l'ensemble des sections globales d'un faisceau d'ordinaux finis. Montrons que le foncteur

U : @rdfin ~ ~ns (7.36 ~])

des morphismes de la forme

lier en utilisant la proposition 1.2 avec la classe E ÷ E/R

o~

R

est une relation d'~quivalence sur

est coggn~rateur finiment r~gu-

E

engendr~e par un ensemble fini.

61

Soit

(E * E/Ri)ie I

Soit

R

une famille U-injective de morphismes de

la relation d'gquivalence sur

E engendr~e par

finiment engendrge. La relation d'~quivalence seulement si (x = y o u

%

sur

(lea classes d'~quivalences de

9. Soit i(I) ..... i(n)el.

Ri(1) U ... U Ri(n). Elle eat E x

d~finie par et de

y

x ~ y

suivant

si et R

sont

des singletons)) poss~de un ensemble fini de classes d'~quivalence. L'ensemble quotient E/~

peut ~tre muni d'une structure d'ordre total et eat donc en bijection avec un

ordinal fini. II existe alors

i(n+l) e I

factorise l'application quotient

tel que l'application quotient

E ~ E/b, c'est-~-dire tel que

deux relations d'~quivalence

R

lea relations d'~quivalences

Ri(1),...,Ri(n)

suit que la famille

et

~

Ri(n+l ) C

E + E/Ri(n+1) ~. Or lea

sont premieres entre-elles (8.3 [5]). Donc sont premieres avec

(E ÷ E/Ri(1),...,E ÷ E/Ri(n+l))

Ri(n+l). II s'en-

eat monomorphique rgguli~re

(8.3. [5]). D'apr~s Is proposition 6.0 de ~] is famille

(E ÷ E/Ri)iE I e s t

monomor-

phique r~guli~re. 4.13. Quelques contre-exemples. On montre facilement que lea foncteurs ~c ~ Snc, &ncDifLoc ÷ &ncDif, ~oc ÷ An

~om ~ ~nc,

E6~ ne sont pas cog~n~rateurs propres, donc

ne sont pas cog~n~rateurs finiment r~guliers et par suite ne donnent pas de th6or~mes de representations. REFERENCES [O] R.F. ARENS et J. KAPLANSKY. Topological representatio n of algebra_~s, Trans. Amer. Math. Soc. 63, pp. 457-481, |948. [I] A. BREZULEANU et R. DIACONESCU. Sur la duale de la cat~gorie des treillis, Rev. Roumaine. Math. Pures et Appl. 14, pp. 331-323, 1969. I~1 J.C. COLE. The bicategory of topo~ and Spectra, preprint. M. COSTE. Localisation, spectra and sheaf representation, Lecture Notes in Math. 753, Springer-Verlag. Berlin-New-York, 1979. ~] Y. DIERS. Familles universelles de morphismes, Ann. Soc. Sci. Bruxelles, 93, III, pp. 175-195, ;979. [5] Y. DIERS. Sur lea familles monomorphiques rgguli~res de morphismes, Cahiers de Top Geom Diff, XXI-4, pp. 44"I-425, 1980. [6] Y. DIERS. Une construction universelle des spectres, topologies spectrales et faisceaux structuraux, Archiv der Math, ~ paraltre. [7] P. GABRIEL et F. ULMER. Lokal pr~sentierbare Kategorien, Lecture Notes in Math. 221, Springer-Verlag, Berlin-New-York, 1971. [8] A. GROTHENDIECK, M. ARTIN, J.L. VERDIER. Th~orie des topos et cohomologie ~tale des schemas, Lecture Notes in Math 269, Springer-Verlag, Berlin Heideberg New-York, 1972. [9] M. HOCHSTER. Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, pp. 43-60, 1969. IOl P.T. JOHNSTONE. Rings, Fields, and spectra. J°ur" AIm" 49, PP" 238-260' 1977" ; K. KEIMEL. The representation of lattice-ordered groups and rings by sections in sheaves. Lecture Notes in Math. 248, Springer-Verlag, Berlin-New-York, 1971. 2] S. MACLANE. Categories for the working Mathematician, Springer-Verlag, New-YorkHeidelberg-Berlin, 1971. I I R.S. PIERCE. Modules over commutative regular rings. Mem. Amer. Math. Soc. 70, 1967. 14 H. SCHUBERT. categories, Springer-verlag, Berlin-Heidelberg-New-York, 1972. 5 A. WOLF. Sheaf representations of Arithmetical Algebras. Mem. Amer. Math. Soc. 148, pp. 87-93, 1974.

Kan

extensions

and

systems

Armin

Given

a diagram

problems: M:

P

When

K

P is M'

~ A? C a n

a similar

way

that

it

in

the paper

we

functors

systems

[F,K]~

fits

k A of

a right Kan

the

Frei

functors

extension

the

of

sole

in the d i s c u s s i o n

apply

general

the

theory

of

some

a given

The

first

for t r e a t i n g

naturally

consider

the

following

functor

M' ~ R a n K M be c l a s s i f i e d

inducing

imprimitivity? reason

we

RanKM

M satisfying

as the r e p r e s e n t a t i o n s

fied by Mackey's answered

M'

> T

of i m p r i m i t i v i t y .

one

of t h e

second.

to a s p e c i a l

are c l a s s i -

question

it a g a i n

in

has been

briefly A t the

situation

is

e n d of

in m o d u l e

theory. All

concepts

gory

and all

mula

as

in

u s e d are

V-concepts,

right Kan

where

extensions

V is a b i c o m p l e t e

are p o i n t w i s e ,

given

closed

cate-

b y the K a n

for-

(2).

In the d i a g r a m

(i)

p

K

P denotes of K,

> T

a small

given

by

} SK

E

category,

ISKI

D is the o b v i o u s E is the

D

=

K any

functor

ITI a n d b y S K ( X , Y )

extension

embedding,

m [p,v]OP

of t h e

identifying

a n d S K the = Nat

identiy

an o b j e c t

shape

category

(T(Y,K.),T(X,K.)).

on objects

to a f u n c t o r

X in S K w i t h

the

and

functor

T(X,K-). Let A be a c o m p l e t e RanKF

for all F in

of t h e

formal

called

indexed

FK = RanKF *)

category

Hom-functor limit

is g i v e n

Supported

[P,A])

by

(it a c t u a l l y a n d F: (see

P

[A],

suffices

) A a functor. or

) a n d the Y o n e d a

[B,K] lemma,

by

the F o n d s

National

that

Suisse

where

A contain

B y the d e f i n i t i o n that

the r i g h t

notion Kan

is

extension

83 (2)

FK(-)

FK admits

= HOmp(T(-,K),F).

a canonical F(~)

We r e c a l l be shape

extension

F K = FD w h e r e

= H O m p ( E ;;,F).

that a f u n c t o r w i t h d o m a i n invariant.

In turn,

F admits

T which

factors

a canonical

over D is said to

extension

F = FE

where A

which

F(*)

= Homp(,,F)

is c l e a r l y

continuous.

The o p e r a t i o n s and Y:

( )K,

(^) is just the P

> [P,V] °p.

RanyF(,)

(-) and functor

( ) extend Rany

faithful

= Homp(*,F)

one has that Y*(^)

embedding

for any c o n t i n u o u s

f u n c t o r M"' : [P,V] Op

~ M"' R a n y Y Z R a n y ( M "

(^)Y* ~

Id(Cont[[P,v]OP,A],where C o n t [ [ P , v ] ° P , A ] [[P,v]°P,A]

Y) as Y is codense;

consisting

= ~(*).

~ Id[P,A].

has M "

of

ways

Indeed

On the o t h e r hand,

category

in o b v i o u s

, w h e r e Y is the Y o n e d a

= HOmp([P,v]°P(*,Y),F)

As Y is f u l l y

to f u n c t o r s

of c o n t i n u o u s

~ A one

thus denotes

the

full sub-

functors.

A

We also o b s e r v e

= Homp((-,K)

that

((ED)*~) (-) = FED(-)

,F) = FK(-).

Summarizing

we have

Theorem

Let A be c o m p l e t e .

gramm

i.

= HOmp(ED(-),F)

Then,

with

the n o t a t i o n s

above,

the dia

64 A

( )

[P,A]

[P,A] commutes

up to n a t u r a l

Corollary M:

2.

> A if and o n l y

tinuous~

furthermore

functor

M = M"' Y is,

M K ~ M'

and M"' ~ ~.

Remark.

From

If a f u n c t o r M'(*)

that

isomorphisms.

M':

> A is of the

if it is of the

for a g l v e n

is,

the p r o o f

of the

M"i: [P,V] °p

exist

in

Hence

the

According candidates

a left

[P,V] °p. functors

When

theorem

Y))

~

adjoint, A :

1 the

M satisfying where

the

M'

unique

M'" R a n y Y of H O m p

and h e n c e

~ M K. As

some

M'"

con-

continuous,

one

the

satisfying

~ RanyM"' Y t h e n

we have

preserves

preserve

all

extensions

all

the

limits

that

is r e p r e s e n t a b l e .

limits.

M'" of M'

[P,V] °p is a r a t h e r

to c l a s s i f y

an i s o m o r p h i s m

[P,v]°P(A(A,~'Y), *)

~

M'"(*) ~ N a t ( M ''iY,*)

continuous

possible,

M'"

[P,V] (*,A(A,M'" Y))

V, then

~ M K for

we have:

definition

in C o n t [ [ P , v ] ° P , A ]

to T h e o r e m

it is p r e f e r a b l e ,

= M'" ED w i t h

> A satisfies

~ A(A,HOmp(*,M"'

M'" has

M'

form M'

form M' ~ M'" ED w i t h

up to i s o m o r p h i s m ,

~ Homp(,,M'" Y) . By the

A(A,M'"(*)

Cont[[P,v]°P,A]

y*

A functor

P

> Cont[[P,v]°P,A]

classify

large

candidates

the

category M by

func-

65

tors h a v i n g d o m a i n S

K

We call a f u n c t o r M": w i t h M'" c o n t i n u o u s [SK,A]

consisting

takes v a l u e s ponding

• We n e x t i n v e s t i g a t e

SK

) A a s y s t e m of i m p r i m i t i v i t y

and d e n o t e of s y s t e m s

in I m p s [ S K , A ] ;

corresponding

by I m p s [ S K , A ]

takes values

functor

The

we use the same symbol Imps[SK,A].

The

functor (

( ) clearly

restricted

and we use E*

~ Imps[SK,A].

of

for the c o r r e s -

f u n c t o r E*,

in I m p s [ S K , A ] ,

Cont[[P,v]°P,A]

if M" = M ' E

the full s u b c a t e g o r y

of i m p r i m i t i v i t y .

functor with codomain

to C o n t [ [ P , v ] ° P , A ]

that p o s s i b i l i t y .

With

for the

this n o t a -

t i o n we have:

Theorem

3.

In the s i t u a t i o n

of d i a g r a m

(1) a s s u m e

be small.

T h e n the f o l l o w i n g

statements

are e q u i v a l e n t :

(i)

E*:

Cont[[P,v]°P,v]

Imps[SK,V]

is an e q u i v a l e n c e .

(ii)

E*:

Cont[[P,v]°P,A]

Imps[SK,A]

is an e q u i v a l e n c e

complete

( ):

[P,V]

Imps[SK,V]

is an e q u i v a l e n c e .

(iv)

( ):

[P,A]

Imps[SK,A]

is an e q u i v a l e n c e

(v)

E is codens e .

(±i) i m p l i e s

We have that

S K to

for all

A.

(iii)

Proof•

T, and h e n c e

(i) and

(E*o ( ? ) ) ( ~ )

(iv) i m p l i e s

= Homp(E~,?)

=

(iii)

for all c o m p l e t e

A.

trivially•

( 3 ) ( ~ ), that is, E*(^)

=

(-).

A

Since

( ) is an e q u i v a l e n c e ,

this e n t a i l s

(iii)

and

to

(v) :

(ii) is e q u i v a l e n t

that

(i) is e q u i v a l e n t

(iv). N e x t we show that

(iii)

to

implies

66

For 9:

any F,G:

P

[P,V](F,G)

> V we have

an i s o m o r p h i s m

) [P,V] (RanEE(F),G)

given

[P,V](F

G)

Imps[S K

V] (F,G)

Imps[S K

V] ( H o m p ( E ~ , F ) , H o m p ( E # , G ) ) ,

RanE([P

v]°P(G,E~))

via

(F),

(-), by the d e f i n i t i o n

by the K a n

[P,v]°P(G,RanEE(F))

by

of

(-),

formula,

as r e p r e s e n t a b l e s

preserve

RanEE

,

[p,v] ( R a n E E ( F ) , G ) , natural

in F and G, which,

isomorphism morphism,

~:

RanEE

hence

to show

the

of R a n E E

with

counit

I:

• Id.

that

we h a v e

jects

of I m p s [ S K , A ] , h e n c e

M'

ciated From

It is c l e a r

> A is of the is i m p r i m i t i v e

with

and,

from

again

and do c h o o s e

1 and

B(1)

is the u n i t

an iso-

that

is,

RanEE

= Id

for B = RanE(BE)

property as the

of RanE(BE) . This

second

arrow that

holds

is an i s o m o r -

E* hits

all ob-

it is an e q u i v a l e n c e .

f o r m M'

and

3 we

> Nat(AE,BE)

the d e f i n i t i o n s

that

= M"D with M"

M'.

Theorems

we m a y

Nat(AE,B(1))

if A is c o n t i n u o u s ,

so is E*.

that

that

of the u n i v e r s a l

phism

T

is ~ E ,

> A

E* ) N a t ( A E , B E )

isomorphism

If M':

of R a n E E

by a n a t u r a l

(ii) . If E is codense,

is an i s o m o r p h i s m ,

[P,V] °p

Nat(A,B)

afortiori

is i n d u c e d

Id E .... ~ E.

for any A:

is the

counit

(v) i m p l i e s

For B in C o n t [ [ P , v ] ° P , A ] and

The

lemma

E is codense.

It r e m a i n s counit

by the Y o n e d a

then

have

M"

is a s y s t e m

in I m p s [ S K , A ]

we

of i m p r i m i t i v i t y

say asso-

67

Corollary M:

P

4.

A functor

) A if a n d o n l y

Corollary dense}

5.

The

in t h i s

M K ~ M'

The

M"

• A is of the

(-)

is an e q u i v a l e n c e

for a g i v e n there

imprimitive

is an M,

used comes

Taking

for K:

P

finite

groups

HCG

V = k-Mod,

the

to the

Mackey.

For

Theorem

3 has

from the

considered

functor as

( )

M' w i t h

some

if E is c o -

associated

the

see

kH

of s y s t e m s [F],

following

system

with

again

taken

as a o n e - o b j e c t

which

takes

t in T to t h e

P-modules

as

functors

(P-Mod) ° p a n d the

functor

o f S ° p to the P - m o d u l e (Z) = H o m p ( T , - ) .

in P - M o d ,

then

codense,

T. T h e

(-)

hence

S °p

between

sense

isoof

s ° P = ( E n d p T ) °p

ringhomomorphism

the u n i q u e

~ s°P-Mod

instance

generated By Theorem

T is d e n s e

ca-

the

[PV] ° p b e c o m e s

(-) : P - M o d

is an e q u i v a l e n c e .

P ---mT

one-object

Interpreting

m (P-Mod) ° p t a k e s

(see for

L e t K:

ring

the

b y t.

the c a t e g o r y

(that is a f i n i t e l y

the P - m o d u l e

theory.

and D becomes

functor theorem

is t h e n

in the

of K is the

left multiplication

A Morita

if T is a p r o g e n e r a t o r

in m o d u l e

category

category

E:

taking

representations

Imps[SK,A]

as a f u n c t o r

f r o m P to Ab,

of

[M].

application

shape

and

to c o i n d u c t i n g Our

of g r o u p s :

group-algebras

categories,

of i m p r i m i t i v i t y

[K] a n d

The

V = Ab.

-> k G of the

~ k-Mod.

tegories.

Here

kH

of r e p r e s e n t a t i o n s

as o n e - o b j e c t

interpreted

is t h e n

if a n d o n l y

u p to i s o m o r p h i s m ,

theory

corresponds

functors

category

details

K

be a r i n g h o m o m o r p h i s m ,

tor)

unique

~ T the embedding

considered

morphic

by

f o r m M' ~ M K for

a n d M ~ M".

terminology

of H,

T

if it is i m p r i m i t i v e .

functor

case

of i m p r i m i t i v i t y

M':

[P])

is g i v e n says

projective 3 the

in P - M o d .

We

object

that genera-

functor thus

E

have

68

Theorem

6.

it is dense

Remark. ( ):

Let P be a ring and T a P-algebra.

If T is a p r o g e n e r a t o r

in P-Mod.

The d i s c u s s i o n

[P,A]

leading

~ Imps[SK,A]

to T h e o r e m

3 points

is an e q u i v a l e n c e

out that when

it g e n e r a l i z e s

a Morita

equivalence.

Bibliography

[A]

.C. Auderset, ries,

[B,K]

Adjonctions

Cahiers

F. B o r c e u x

[F]

A. Frei,

Shape

A. Frei

and induced

sion?, H. Kleisli,

[M]

G.W.

when

Mackey,

B. Pareigis, Teubner,

Soc.

2-categoXV,I(1974).

for e n r i c h e d

12

representations,

A question

L.N.

in Math.

Coshape-invariant

XXII-I

vol.

of limit

Math.

(1975).

to appear

in c a t e g o r i c a l

is a s h a p e - i n v a r i a n t

Springer

theorem,

in

functor

719

functors

shape

a Kan exten-

(1979). and M a c k e y ' s

induced

Cahiers

de Topol.

et G~om. Diff.,

representations

of groups

and q u a n t u m

(1981).

Induced

mechanics, [P]

A notion

Diff.,

des

Mathematicae.

representation Vol.

au niveau

et G~om.

Austral.

and H. Kleisli,

theory:

[K]

Kelly,

Bull.

Quaestiones [F,K]

de Topol.

and G.M.

categories,

et m o n a d e s

Benjamin-Boringhieri

Kategorien (1969).

und Funktoren,

(1968). Math.

Leitf~den,

SMOOTH STRUCTURES

by Alfred Fr~licher

A smooth

structure

on a set S consists

a set F c ]RS of f u n c t i o n s

of a set C c S

such that C and F d e t e r m i n e

tion t h a t FoC c C°°(~R, IR). The sets w i t h smooth ~if

we take as m o r p h i s m s

curves or,

equivalently,

with respect

The m a i n results the a t t e n t i o n

presented

to the e x c e l l e n t

show h o w c l a s s i c a l

calculus

finite dimensional[) singularities,

elementary

and are v a l i d

e.g.

cf

[3] or

properties

for each c a t e g o r y

For c e r t a i n m o n o i d s

[12].

explicitly

condition

Siciak

by d e s c r i b i n g

structure

result

(not~ecessarily

finds also o b j e c t s

with

in this direction. are those w h i c h we call the

is g e n e r a t e d

T h e y are easily o b t a i n e d

in a similar way, r e p l a c i n g B of B . For other examples,

the s m o o t h

structure

[7]. The p r o o f

yields

notion of C -maps.

between

the l i n e a r i t y the usual

by L a w v e r e ~ S c h a n u e l

for and

will be c o n s t r u c t e d

and for m a n y

w i t h the smooth the

theorems

of Boman

is f a c i l i t a t e d convex

of c a l c u l u s

functions

~-morphisms.

is a c a r a c t e r i z a t i o n

form

The

of smooth m a p s [2], of B o c h n a k

and

by u s i n g a m i n i m a l

spaces w h i c h does for

of the d i f f e r e n t i a l

results

and

of the function-spaces.

together

locally

A necessary

of this c o n d i t i o n

for any F r 4 c h e t - s p a c e

spaces w h i c h g e n e r a l i z e s

of c l - m a p s

closed.

first p r o v e d closedness

and that the C°°-maps are e x a c t l y

not r e q u i r e

nevertheless

and was

cartesian

the smooth curves

[I] and of H a i n

caracterization

is c a r t e s i a n

in [3]. The v e r i f i c a t i o n

in order to o b t a i n t h i s

Fr4chet

instance

: first we d r a w a n d then we will

manifolds

and cocompleteness.

which

In §3 it will be shown t h a t

between

there

the c a t e g o r y

was g~ven

The functor y i e l d i n g

C -Fr4chet-manifolds

basic

of~

within~one

(JR, IR) by any s u b m o n o i d

the m o n o i d C (JR, i~] is d i f f i c u l t

a smooth

to the

[4].

sufficient

Zame

set-up,

completeness

by any fixed set B and C

good w i t h respect

properties

but we shall not at all go here

ones,

yield a c a t e g o r y

here go in two d i r e c t i o n s

of C - d i f f e r e n t i a b l e

A m o n g the c a t e g o r i c a l

and

to the functions.

categorical

fits in this

of curves

each other by the condi-

structures

those m a p s w h i c h b e h a v e

~9

at a p o i n t but

and in p a r t i c u l a r

the usual

70

In r e c e n t y e a r s categories

several a u t h o r s u s e d w i t h a d v a n t a g e c a r t e s i a n c l o s e d

in o r d e r to d e v e l o p c a l c u l u s

for n o n - n o r m e d v e c t o r

c o n v e r g e n c e s t r u c t u r e s were u s e d in [6], c o m p a c t l y g e n e r a t e d

spaces

:

spaces in [ ~ ] ,

a r c - d e t e r m i n e d s p a c e s in [ii]. W h i c h c a r t e s i a n c l o s e d c a t e g o r y is the n a t u r a l one for c a l c u l u s ? If one w a n t s to study C~-maps,

the a n s w e r

vector spaces with a compatible

Some ideas and r e s u l t s in

s m o o t h structure.

seems clear

:

this d i r e c t i o n are g i v e n in the last section. The r e s u l t s understand~

p r e s e n t e d here are v e r y easy to f o r m u l a t e and to

h o w e v e r s e v e r a l p r o o f s r e q u i r e h a r d a n a l y s i s and thus can o n l y be

i n d i c a t e d in this e x p o s i t o r y article.

All p r o o f s w e r e c a r r i e d out in d e t a i l

in a s e m i n a r on s m o o t h f u n c t i o n s a t the U n i v e r s i t y of Geneva~ another

p r o o f of the t h e o r e m of Lawvere,

and the a u t h o r was p r e s e n t e d .

in p a r t i c u l a r

S c h a n u e l and Zame due to H. J o r i s

I w i s h to e x p r e s s m y g r a t i t u d e

for a v e r y

a c t i v e p a r t i c i p a t i o n and s u b s t a n t i a l c o n t r i b u t i o n s at this s e m i n a r in p a r t i c u l a r to G o n z a l o Arzabe,

Henri J o r i s and O s c a r P i n o - O r t i z .

§l T H E C A T E G O R Y ~ O F

S M O O T H S P A C E S A N D ITS E L E M E N T A R Y P R O P E R T I E S IR

A s m o o t h s t r u c t u r e on a set S is a c o u p l e F c I R S such that the

(C,F)

where C c S

"duality" C = D,F and F = D*C holds,

with

D.F = {c

: ]R ~ S~ foc 6 C°°(]R, ~R)

for all f £ F}~

D*C = {f

: S ~]R~

for all c 6 C}.

foc 6 C~( IR, JR)

A s m o o t h space is a t r i p l e

and

(S,C,F) w h e r e S is a set and

(C,F) is a

s m o o t h s t r u c t u r e on it. The morphisms

s m o o t h spaces form a c a t e g o r y ~ from

~.(C) c C' or,

(S,C,F)

so

(S',C',F')

for which,

are t h o s e m a p s d

by d e f i n i t i o n ,

: S ~ S' w h i c h s a t i s f y

e q u i v a l e n t l y ~*(F') c F.

The set of s m o o t h s t r u c t u r e s on a fixed set S is o r d e r e d manner

: (C,F)

m o r p h i s m from structure by C O

is c a l l e d finer than (S,C,F) to

(S,C',F').

(C',F')

and is o b t a i n e d as f o l l o w s (C,F)

IR

t h e r e is a f i n e s t

it is c a l l e d the s t r u c t u r e g e n e r a t e d

: F = D*C 0 ; C = D,F. S i m i l a r l y one has S : it is the c o a r s e s t

g e n e r a t e d by any set F 0 c ~ R

s t r u c t u r e w i t h F 0 c F and is o b t a i n e d as C = D , F 0 ; F = D*C. the s m o o t h s t r u c t u r e s of a fixed set S form a c o m p l e t e forgetful

in the u s u a l

if the i d e n t i t y m a p of S is a

For any set C O c S

(C,F) on S such that C O c C~

the s t r u c t u r e

the

f u n c t o r from ~ t o

It f o l l o w s t h a t

lattice,

that the

sets has a left and a r i g h t adjoint,

and that

71

~is

c o m p l e t e and cocomplete,

limits or colimits b e i n g o b t a i n e d

(as in the

c a t e g o r y of t o p o l o g i c a l spaces) by taking them in the c a t e g o r y of sets and then p u t t i n g the initial resp.

final structure on them• We note in p a r t i c u l a r

that the p r o d u c t of smooth spaces

(Si,Ci,Fi), i 6 I, is the object

(S,C,F}

w i t h S = × S. and C c o n s i s t i n g of those curves c : ~R ~ S w h o s e component i6I 1 c i : IR ~ S. b e l o n g 1 An o b j e c t

for all i 6 I, to C i.

(S,C,F) of ~ i s

called separated if for all a # b 6 S

there

exists f 6 F with f(a) ~ f(b).

~ denotes the r e s p e c t i v e full subcategory sep The inclusion functor has an obvious left adjoint and it follows

of ~ . that

~

is also c o m p l e t e and cocomplete. The forgetful functor from sop to sets still commutes w i t h limits, but not with all colimits. The one

sep point set w i t h its u n i q u e smooth structure is o b v i o u s l y

a final object of ~ ,

and it also yields a r e p r e s e n t a t i o n of the forgetful functor from ~ t o Another

important object is the triple

(~R, C~(]R, ~ )

, C°°(IR, ~R} ).

It will be d e n o t e d simply by IR and is generator and c o g e n e r a t o r of For any o b j e c t X = cx =

(SX, C x, F x) of ~ w e JcP(m,

x)

;

have

sets•

sep

:

~x : ~ C ~ ( x

,m)

.

The results of this section do not depend on the nature of the monoid C~(IR, IR) . As it was shown in [3], t h e y hold for the c a t e g o r y ~ g e n e r a t e d a n a l o g o u s l y by any m o n o i d M of maps of any set B to itself of BB). C a r t e s i a n c l o s e d n e s s of ~ h o w e v e r and sufficient condition was given.

(i.e. M a s u b m o n o i d

d e p e n d s on M; in [3] a n e c e s s a r y

In the following section we discuss this

c o n d i t i o n and its v e r i f i c a t i o n for the case B = R, M = C°°(IR, Z{) .

~2 CARTESI~ C~OSEON~SS OF dTPAND

~ep

F r o m the m e n t i o n e d p r o p e r t i e s of the o n e - p o i n t object it follows that if there is a functor

H :~ep

×~yielding

cartesian closedness,

it can be chosen such that the u n d e r l y i n g set of H(Y,Z) ~(Y,Z).

In p a r t i c u l a r one must get on ~ ( ~ ,

structure

(F,~)

such that, w i t h ~(x,y)

F = { ~ : ]R ~ C~°(]R, ~ )

; ~

JR)

then

is the function space

= C°°(]R, JR)

a smooth

: = 7(x) (y),

: m~IR

and } = D*F. Since t r i v i a l l y F c D,(D*F)

~]R a morphism }

the couple

(F,#) will be a smooth

structure iff D,(D*F) c F. Let us d i s c u s s the m e a n i n g of this condition. A c c o r d i n g to the d i s c r i p t i o n of p r o d u c t s i n ~ ,

a map G : 2

~ ] R is a m o r p h i s m

72

~

~

iff for all o,T 6 C°°( ~ , JR)

G is smooth along all smooth curves

one has Go(U,T) 6 C°°(~R, JR) , i.e. iff 2 . A c c o r d i n g to a remarkable

(O,T) of ~

t h e o r e m of Boman [2] this is equivalent to G 6 C ~ ( ~ 2, JR) yields a b i j e c t i o n F ~ C°° (

2, ~ )

and hence 7 ~

. From this and the d e f i n i t i o n of ~ as

D*F we get = {~ : C=°(]R, ~)---~IR; x ~ ~(G(x,-))

is in C ~ ( m ,

We call the elements of ~ smooth functionals explicitly;

for all G 6 C ° ° ( ~ m ) }

. One does not k n o w all of them

h o w e v e r for the linear ones one has

P r o p o s i t i o n 1.

The linear smooth functionals [9 have compact support

for each (p there exists a compact K fl I K

JR)

= f2 ~ K

( i.e.

of IR w i t h the p r o p e r t y

~ t0(f1) = [p(f2 )) and satisfy i9(lim f ) = lim ~ ( f ) n n n-~o n-~o

if fl,f2,..,

is a sequence in C°°(~R, JR)

such that for all k -> 0 the deriva-

tives

f(k)converge locally u n i f o r m l y for n ~ oo . This m e a n s that the linear n smooth functionals are exactly the d i s t r i b u t i o n s of compact support. This result is due to van Que and Reyes [13]

c o n s t r u c t i n g for c o n v e n i e n t subsequences f

, f nI

such that G(i/k,y)

P r o p o s i t i o n 2. I II

= fnk(Y) and G(0,y)

2 If G : IR ~

For all x 6JR,

G(x,-)

; it can be proved by

.... a function G 6 C~ (IR2 JR) n2

= lim fn(y)n-mo

satisfies 6 C~( ~ , ~ )

,

For all linear smooth functionals ~ • x ~ ~ G(x,-)

is in C°°(IR, IR)

then G £ C°°( m 2, JR) . This important result was proved by Lawvere,

Schanuel and Zame [12].

It can be p r o v e d by showing first the c o n t i n u i t y of G and its first o r d e r partial d e r i v a t i v e s ~1 G and D2G. This is quite d e l i c a t e for DIG and we found it useful to show first that DIG is p a r t i a l l y continuous in the second v a r i a b l e and ~IDIG is locally bounded. Once one has o b t a i n e d the continuous different i a b i l i t y of G, the p r o o f is completed by showing that DiG and ~2 G satisfy the same conditions I and II ; for D2G this is easy, and for DIG one m a k e s use of p r o p o s i t i o n i. F r o m p r o p o s i t i o n 2 is follows i m m e d i a t e l y that structure on C°°(~, ~ )

(F,~) is a smooth

. W e remark that for this it w o u l d be enough to prove

73

p r o p o s i t i o n 2 under the a s u m p t i o n that I holds and II holds for all smooth functionals q). However,

if one w o u l d a l l o w n o n - l i n e a r ones in the p r o o f that

G or ~l G or ~)2G are c o n t i n u o u s it w o u l d be

hard to get f u r t h e r , b e c a u s e one

does not haw~ the analogue of p r o p o s i t i o n 1 for n o n - l i n e a r smooth functionals

Theorem. for

The c a t e g o r y ~ o f

smooth spaces is cartesian closed. The same holds

sep The f u n c t i o n - s p a c e structure can be d e s c r i b e d e x p l i c i t l y

objects Y,Z of ~ o n e

d e f i n e s on the f u n c t i o n - s p a c e ~ ( Y , Z )

In fact, for

a structure

(C,F) by C = {d : ]R ~

~(Y,Z);

~

: ]RZ Y ~ Z a morphism}

F = D*C. Using that

(I',~) is a smooth structure on C°°(IR, JR) it is easy to show that

(C,F) is a c t u a l l y a smooth structure. D e n o t i n g the smooth space formed by the set ~ ( Y , Z )

with this structure by H(Y,Z)

it is s t r a i g h t f o r w a r d to show that

one has the universal p r o p e r t y : X -~ H(Y,Z) and

a morphism ~=~

: X Z Y ~ Z a morphism

this yields f u n c t o r i a l i t y of H and cartesian closedness of ~ . Since Z separated implies H(Y,Z) sep

separated, cartesian

closedness of

is o b t a i n e d by r e s t r i c t i o n of the functor H.

B e c a u s e in p r o p o s i t i o n 2 o n l y linear smooth functionals are used it follows easily that the structure of H(Y,Z) ~(Y,Z) support,

~[

of the form ~ ~ ~(f°~oc)

c 6 Cy =

9(IR,Y)

is g e n e r a t e d by the functions

w h e r e ~ is a d i s t r i b u t i o n of compact

and f 6 m Z = ~ ( Z

, m)

.

§3 THE S M O O T H STRUCTURE OF FRECHET SPACES AND MANIFOLDS. F o r ~ n the couple j OO~C(~, n )

, cOO( ~ n , ~ ) )

is a smooth structure. This

is not at all trivial, but it is equivalent to Bomans t h e o r e m [2], w h i c h says that a function o n ~ n

a l r e a d y quoted

is smooth if it is smooth along all

smooth curves. Using p a r t i t i o n s of u n i t y one gets a more general result for any finite d i m e n s i o n a l p a r a c o m p a c t C -manifold V, (c ~ ( ~ , V )

, C

:

(V, ~ ) )

is a smooth structure on V. So every such m a n i f o l d can be c o n s i d e r e d as a smooth space, and the C°°-maps b e t w e e n them are exactly the

~O-morphisms.

In order to get the same results for a greater class of v e c t o r spaces and m a n i f o l d s we need first of all a t h e o r e m w h i c h g e n e r a l i z e s B o m a n ' s result

;

74

this

theorem

and Hain

The useful

will

at t h e

same

time

aeneralize

results

of Bochnak-Siciak

[i]

[7]. following

(E,F w i l l

set-up

in t h i s

for c a l c u l u s

section

always

between denote

locally

convex

separated

spaces

locally

is

convex

spaces).

Definition. class

A map

f : E ~ F between

locally

convex

spaces

is c a l l e d

of

C 1 if for a l l x , h 6 E. df(x,h)

exists

: = w-lim i/~. I~0 a c o n t i n u o u s m a p df

and yields

By w-lim

we mean

Hence

the

df(x,h) 1 lim ~.

for all

1 [ F', We

Proposition

the

i.

limit

((lof) (x+lh)

-

: E × E ~ F.

in F w i t h

is c a r a c t e r i z e d

F' b e i n g

require

in p a r t i c u l a r

(unique)

(f(x+lh)-f(x))

to t h e w e a k

topology.

by

(lof)(x))

the topological

so l i t t l e

because

linearity

of d f ( x , - ) .

= l(df(x,h))

dual

o f F.

it is e n o u g h

to g e t the u s u a l

In f a c t o n e h a s

If f : E ~ F is of c l a s s lim l/l-(f(x+Xh] i~0

respect

- f(x))

C

1

properties,

:

, then

= d f ( x 0 , h 01

X~X 0 h~h 0 We

remark

(and n o t o n l y simultaneous strict

that

here

with

respect

limit

exists

differentiability.

Corollary.

to p r o v e

limit

is w i t h

to t h e w e a k shows

that

As e a s y

If f is o f c l a s s

In o r d e r useful

the

C

1

respect

topology!),

we a r e c l o s e

consequences

, then

proposition

to t h e t o p o l o g y

a n d the to w h a t

we have

f is c o n t i n u o u s 1 the

following

fact

that

of F this

is s o m e t i m e s

called

:

a n d df(x,-) "mean value

is linear, theorem"

is

:

Proposition

2.

interval

I c]R

(special

case

L e t A c E be c o n v e x into E related : d : c').

d(1) The proof For

by

application

f : EI×...xE n

(loc)'(~)

c,d

= l(d(~))

: I ~ E maps for all

for a n y ~ < ~ of I o n e h a s

6 A for ~ < ~ < ~ ~

is a s i m p l e a map

Then

and closed;

c(~)

- c(~)

6

~ 6 I and

1 6 E'

:

(~-~)-A .

of the H a h n - B a n a c h F the n o t i o n

of an o p e n

theorem.

"partially

of c l a s s

C I'' is

75

defined

in t h e u s u a l way;

i.e.

di~.=- Elx...XEnXE.l ~ F h a v e It f o l l o w s 1 C .

as u s u a l

that

the partial

to e x i s t

and

differentials

have

to be c o n t i n u o u s

f is o f c l a s s C 1 if a n d o n l y

in all v a r i a b l e s

if it is p a r t i a l l y

of

class

n now maps of class C : n+l f : E ~ F is o f c l a s s C if it is of c l a s s

Inductively Definition. c l a s s C n.

we d e f i n e

f is of c l a s s

For d n there operator

T behaves

C

1

a n d df is o f

C°~ if it is o f c l a s s C n for all n 6 N.

is a c h a i n

rule.

much better; Tf

It is c o m p l i c a t e d

it is d e f i n e d

and

for t h i s t h e

as

: E×E ~ FXF (x,h) ~ ( f ( x ) , df(x,h))

The

chain

rule then

says

a l s o gof, a n d T n ( g o f ) Since

: if f : E ~ F a n d a

: F ~ G are of class C

n

, then

= TncroTnf.

for f of c l a s s

C

1

t h e m a p df is l i n e a r

(and c o n t i n u o u s )

in t h e

s e c o n d v a r i a b l e , o n l y t h e f i r s t p a r t i a l d i f f e r e n t i a l is of i n t e r e s t a n d y i e l d s 2 2 2 a m a p D f : E x E X E ~ F. D f e x i s t s a n d is c o n t i n u o u s iff f is of c l a s s C , a n d t h e n D 2 f (x,_,_)

is b i l i n e a r

a n d D n+l as t h e

first partial

class

C n if a n d o n l y

Dnf(x,-,...,-)

if D l f ..... D n f

to s h o w t h a t

existence

D n-1 f. If we

suppose

symmetric.

differential

is n - m u l t i l i n e a r

If we w a n t must verify

and

and

a map

continuity

'

In ) ~

on d e f i n e s

D 1 f as df

of Dnf.

shows that

f is o f

One

a n d are c o n t i n u o u s ,

f of c l a s s C n-1 of the g

admits

first partial

basis

L e t E be m e t r i z a b l e

convergent

sequences

U

a sequence

: IN ~ ] N ,

is c o n t i n u o u s .

suppose

that

following

and a n ~ = lima n n-~o

i n E. T h e n t h e r e of r e a l s

of

form

the e x i s t e n c e It is t h e n Under

for the z e r o - n e i g h b o r h o o d s

w e can s h o w t h i s c o n t i n u i t y if we 2 g :~ ~ E of class C , using the

Lemma.

differential

a + l ~ h l + . . . + A n-n h

(x,h) ~ D n f ( x , h , . . . , h )

a denumerable

is e v e n of c l a s s C n, we

: IRn ~ E o f t h e

n t h e c o m p o s i t e m a p fog is o f c l a s s C , we g e t e a s i l y n D f ( x , - , . . . , - ) a n d its m u l t i l i n e a r i t y a n d s y m m e t r y . that the map

and then

symmetric.

for all m a D_s

that

(11'''"

and

exist

Recursively

I

n

exist

(cf.

to s h o w that E

( i . e . i f E is m e t r i z a b l e )

[7])

b 0 = limb n n-~o

a strictly

with a limit

enouqh

the asumption

fog is of c l a s s C n for all

lemma

'

of

:

be l i m i t s

increasing

of

function

10 = l i m ~ ,and a f u n c t i o n n n-x=

78

g

: ]R

2

~ E of c l a s s C°O s u c h t h a t

g(ln,~) This fact

it is f a l s e

3.

E metrizable :n

supposes

that

show that

n~2

. Then

could not

There

is

better

get t h e a b o v e

result

" , which

we

is o n l y a s l i g h t n

f has t h e p r o p e r t y

the norms.

However,

and this

that

but

between

if

of H a i n

[7].

to B a n a c h

classical

spaces)

.

notion

of

"

our notion

: E ~ L

n

C n a n d F r 4 c h e t - C n.

(E;F)

continuity

C n and

is in p a r t i c u l a r n+l

Proposition

Let

g

:

spaces

of

]Rn

(F' t h e t o p o l o g i c a l to a s s u m e

that

still weaker,

f(n)

true

~ Fr~chet-C n ~ C

the

~ F be s u c h t h a t d u a l of F) the

and

with respect

is n o r m - c o n t i n u o u s ,

if f is of c l a s s

coincides

spaces many

them yield

(n)

of f

then

C n+~.

to f

Hence

with the

different

same notion

"of

notions

class

log is of c l a s s C

suppose

Mackey-topology

t h a t F is l o c a l l y

classical

n+l

"of c l a s s

C~ ' '

as

f o r all

t h a t F is c o m p l e t e

(in f a c t

o f F is s e q u e n t i a l l y

complete,(cf

[9]

). T h e n

g is

C n.

Using and Siciak

our who

set-up

the proof

is a l m o s t

gave this proposition

in t h e

:

n

our C -notion

. For not-normable

[i0]).

or,

(in

spaces;

E of c l a s s C

"Fr@chet-C n

imply the

(cf.

complete,

difference

he u s e s t h e here

does not

the one we use

of c l a s s

call

~ F

all

it is e n o u g h

because

result

: IRn+l

: E×...xE

almost

4.

nA2

C~

(who r e s t r i c t s

for all g

shall

convex

C n if a n d o n l y

= Dnf

for B a n a c h

n o t i o n of F r ~ c h e t - C

exist;

locally

: ]Rn ~ E of c l a s s

C n, H a i n

if f is o f c l a s s

C shows

we a s s u m e

that

and this

is F r ~ c h e t - c n ;

proposition

, then the map

(n)

f(n) is c o n t i n u o u s ,

U {0}.

than the respective

f is of c l a s s

: E ~ F is of c l a s s C

1 6 F'

following

f is of c l a s s

f o g is of c l a s s C n+l

" m a p of c l a s s C n

Cn "

in t h e

F is o f c l a s s C n for all g

to

This

for all n £ ~

f : E ~ F be a m a p b e t w e e n

n 6~,

This proposition

If f

why

(n)

for n : i).

Let

;

In o r d e r

He

+ p. 5

also explains

Proposition

fog

= ao(n)

the

s a m e as t h a t g i v e n

c a s e n = i , cf.

[i].

by Bochnak

77

Combining

Boman's

following

theorem,

Theorem

1 .

theorem with propositions announced

in [5]

Let f : E ~ F be a m a p b e t w e e n

that E is m e t r i z a b l e are e q u i v a l e n t

and F

3 and 4 one gets easily the

:

(locally)

locally convex

complete.

T h e n the

spaces

following

and suppose conditions

:

i)

f is of class Cco

2)

f. (C~(IR,E))

3)

f*(C°°{F, IR)) C Cco(E, ]IR)

4)

f*(F')

= Cco( JR, F)

co

Corollary. smooth

For any F r ~ c h e t

structure

and the

m C

(E, IR) space E, the couple

on E. Hence F r @ c h e t

~-morphisms

between

Theorem

2.

of u n i t y

of~from

co

, C

can be c o n s i d e r e d

t h e m are e x a c t l y

If we w a n t to get o b j e c t s sure that p a r t i t i o n s

spaces

(~ (JR,E) the m a p s

(E, IR))

is a

as smooth

spaces

of class C°°.

Fr4chet manifolds

we m u s t m a k e

exist.

Let V be a p a r a c o m p a c t

space E w h i c h has the p r o p e r t y

Fr4chet manifold modelled

t h a t to each n e i g h b o r h o o d

over a F r ~ c h e t

V of zero there

co

exists

a C -function

Then

(JR,V),

between

C

(V, IR)) is a smooth

such spaces are exactly

Remark.

According

Fr4chet-manifolds C

f : E ~IR with

(V,W).

This

the u n i v e r s a l

V , W the n a t u r a l

property

structure

closedness

smooth

can be d e s c r i b e d that

= i and f(x]

= 0 for x ~ V.

on V. The

morphisms

the m a p s of class Cco.

to the. c a r t e s i a n

structure

f(0)

of ~ w e

structure

on the function

explicitly

for any such m a n i f o l d

get for any such space

in a simple way and has

X a Rap

f : X ~ C

(VrW]

is

co

of class C ~ C

(V,W)

iff ~

: X ~ V ~ W is of class C

again a m a n i f o l d

t h i n g s to look at in this vector

spaces

will give

set-up.

More natural

equipped with a compatible

some ideas

. One can then ask

? Of c o u r s e F r 4 c h e t m a n i f o l d s

in this direction.

smooth

: when

is

are not the n a t u r a l

are m a n i f o l d s structure~

modelled

The last

over

section

78

§4 C A L C U L U S

A smooth v e c t o r structure,

i.e.

an a r b i t r a r y

F O R S M O O T H V E C T O R SPACES

space

is a v e c t o r

such that the v e c t o r

object

X :

(S,C,F)

of ~ t h e

w i t h H(X, JR), is a smooth v e c t o r w a y an e q u i v a l e n c e f ~

relation

g ~=~ (f°e)" (0) =

space w i t h a c o m p a t i b l e

space o p e r a t i o n s function

space.

are

set F, being

If we define,

smooth

~-morphisms.

For

identified

for p 6 S, in the usual

"~ " on F by P (goc)" (0) for all c 6 C w i t h c(0)

= p

P then the q u o t i e n t space, tangent not,

is, due to c a r t e s i a n

c a l l e d the c o t a n g e n t

space of X at p as a q u o t i e n t

in general,

smooth v e c t o r evaluation remarks

a vector

space

seem v e r y useful

to o b t a i n

such a result,

E'

= p};

as a subspace

of F w i t h r e s p e c t

in this d i r e c t i o n

is the f o l l o w i n g

indefinitely

Let E be a smooth v e c t o r

{c 6 C; c(0)

here; way.

the

it is

into the

to the

but these It does not

on them.

we want to study spaces

also a smooth v e c t o r

we can introduce

spaces come in a natural

to put a t o p o l o g y

smooth v e c t o r

In o r d e r

of the space

but can be imbedded

formed by the d e r i v a t i o n s

smooth v e c t o r

The question between

space,

at p. We do not go further

show that

c l o s e d n e s s of ~ ,

space of X at p. S i m i l a r l y

differentiable

some r e s t r i c t i o n s

space,

: are the

(CE,F E)

~-morphisms

in the usual

on the spaces

its smooth

sense

seem useful.

structure.

We put

: = E* N F E

where E* notes the a l g e b r a i c real-valued

linear

Definition. points, cf

smooth

dual of E. So E'

functions

The smooth vector

generates

the smooth

is the v e c t o r

space of the

on E.

space E is c a l l e d c o n v e n i e n t

structure

and y i e l d s

if E' s e p a r a t e s

a comolete

bornology

on E;

[8]. This completeness

locally

convex t o p o l o g y

complete; Fr~chet

cf

condition

is e q u i v a l e n t

on E y i e l d i n g

[9]. A c c o r d i n g

space E the natural

E' as t o p o l o g i c a l

to the results smooth

to the c o n d i t i o n

of

structure

dual

t h a t any

is locally

§3 we see that for any (C

(JR,E)

, C

(E~ IR)) is

convenient. If c : IR ~ E is a space E

(i.e. c 6 CE)

hypothesis that

that there

~-morphism

one d e d u c e s exists

from LR to a c o n v e n i e n t

from the s e p a r a t i o n

a u n i q u e map,

denoted

smooth vector

and the c o m p l e t e n e s s

by c',

?

from]]{ to E such

79

loc"= F r o m the o t h e r asumption

(loc)"

(that E' generates the smooth structure of E) it

follows then i m m e d i a t e l y that e" we obtain

~O-morphisms

for all 1 6 E'

: ~ ~ E is also a

c (n) : ]R ~ E for n 6 ~

~morphism.

Inductively

and we see that c is indefini-

tely d i f f e r e n t i a b l e in the usual sense with respect to any locally

convex

t o p o l o g y on E y i e l d i n g E' as t o p o l o g i c a l dual. M o r e o v e r one v e r i f i e s that the (linear) map H(IR,E) ~ H(IR,E)

sending c into c" is a ~ - m o r p h i s m .

U s i n g this we get similar results for the general case T h e o r e m i.

Let d : E 1 ~ E 2 be a ~ Y - m o r p h i s m

:

b e t w e e n c o n v e n i e n t smooth vector

spaces. T h e n the map d~ defined by d~(a,h) ~-morphism.

=

: E1 ~ EI ~ E2

(~OCa,h)" (0) where Ca,h(l) = a + l h

For any a 6 E 1 the map d~(a,-)

is also a

is linear. The

H(E1,E2) ~ H(E 1 rl El, E2) sending ~ into d~ is also a If E~ separates points of E2, then o b v i o u s l y

(linear) map

~-morphism.

(H(E1,E2))'

separates

points of H(E1,E2). And if E~ g e n e r a t e s the smooth structure of E2~ then the remark at the end ~ ~(io~oc)

generate

' certainly

addition

that

H(EI,E2)

one sees

that

for

Theorem

2.

the

of the form

for 1 6 E~, c 6 C and ~0 a d i s t r i b u t i o n of compact support 2 E 1 ' smooth structure of H(E I, E2); since these functions are linear

the

(H(E1,E2))

of §2) the functions H(E1,E 2) ~

(cf

The

~-morphisms

generates satisfies

E 2 convenient

category is,

formed by

the the also

by

restriction

structure

of

H(E1,E2).

is

convenient.

completeness H(E1,

the of

E 2)

condition

convenient the

By

functor

smooth H,

showing

provided Hence

vector cartesian

in E 2 does,

we

have

spaces

:

with

closed.

Other p r o p e r t i e s of that category as well as the c a t e g o r y formed b y the same objects but with only the linear ~ - m o r D h i s m s

are being studied;

in

p a r t i c u l a r d u a l i t y and r e f l e x i v i t y questions. By i n t r o d u c i n g the spaces Ln(E1,E 2) of n - m u l t i l i n e a r

~morphisms

E l ~...~E i ~ E 2 one can of course introduce for a

~P-morDhism ~ ; E i ~ E 2 (n) between convenient smooth vector spaces the maps ~ : E] ~ Ln(EI,E2) w h i c h are also

~-morphisms,

derivatives (n)

and one has the usual relations between the higher

and the higher d i f f e r e n t i a l s dn~.

80

Added in proof. The convenient smooth vector spaces can be identified with the spaces considered by A. Kriegl ("Die richtigen R~ume f~r Analysis im unendlich-dimensionalen", preprint, Vienna 1981, to appear in Monatshefte fur Mathematik), namely the separated locally convex spaces which are

bornological and locally complete.

81

R E F E R E N C E S

[1]

J. B o c h n a k spaces"

2

3

A. F r 6 1 i c h e r

Ac.

9 i0

Notes ii

12

A. Kriegl

XXI/4,

entre

espaces

lisses

engendr@es 1980,

D-

par des

367-375.

et v a r i @ t 6 s

de Fr4che~

p. 125-127. in V e c t o r

of smooth

77,

Spaces w i t h o u t

Norm",

1966.

1979,

p.

functions

d e f i n e d on a B a n a c h

63-67.

and functional

c o n v e x Spaces", Calculus

Springer

analysis",

Mathematics

F.W.

S.H.

Schanuel

Teubner

1981.

in locally

convex

spaces",

Lecture

1974.

glatter

1980.

Mannigfaltigkeiten

and W.R.

Zame

und v e k o r b ~ n d e l " ~

: ~'On C°° F u n c t i o n

Spaces",

1981.

N. V a n Oue and G. Reyes de Whitney",

Recherches

appl.

der F e r n u n i v e r s i t ~ t

1977.

Wien

U. S e i p

diff.

: "Bornologies

: "Eine T h e o r i e

Lawvere,

249-268.

abgeschlossene

Math.

et G4om.

Soe.

Dissertation,

tension

14

Am. Math.

417,

1967, p.

kartesisch

ferm@es

: "Calculus

:"Differential

in Math.

20,

car4siennement

30, S p r i n g e r

26, N o r t h - H o l l a n d

Keller

Preprint 13

in Math.

: "Locally

vector

and of its c o m p o s i t i o n s

Scand.

aus d e m Fachber.

: "A c h a r a c t e r i z a t i o n

H. J a r c h o w

Math.

erzeugte

1981,

and W. B u c h e r

H. H o g b e - N l e n d

H.H.

de Top.

Paris 293,

Proc.

in t o m o l o g i c a l

.

: "Applications

Sci.

Hain

Studies

7-48

Cahiers

A. F r 6 1 i c h e r

R.M.

p.

: "Categories

A. F r 6 1 i c h e r

space", 8

of a function

Seminarberichte

5, 1979,

functions

p. 77-112

of one variable",

Kategorien",

L e c t u r e Notes 7

39, 1971,

: "Dutch M o n o i d e

C.R. 6

"Analytic

: "Differentiability

functions

monofdes", 5

:

A. F r 6 1 i c h e r

Hagen 4

Studia Math.

J. Boman with

and J. Siciak

DMS 80-12,

21,

1981,

des d i s t r i b u t i o n s

8, G4om.

Universit4

: "A c o n v e n i e n t

Algebra

: "Th6orie

Expos@

Settina

diff.

synth,

de M o n t r @ a l

et th@or6~nes d'ex~

fasc.

2, R a p p o r t

de

1980.

for S m o o t h Manifolds".

J. of p u r e and

p. 279-305. S e c t i o n de M a t h @ m a t i q u e s U n i v e r s i t ~ de Gen@ve 2-4, rue du Li6vre CH~I211

GENEVE

24

E n r i c h e d algebras,

spectra and h o m o t o p y limits

John W. Gray O. Introduction.

The purpose of this paper is the same as that of

[5];

to show how certain p r o p e r t i e s of h o m o t o p y limits are consequences of w h a t either are or should be standard facts about categories e n r i c h e d in a closed category. in

The p r o p e r t y to be e x p l a i n e d here is as follows:

[16], T h o m a s o n shows that the d e g r e e w i s e h o m o t o p y limit of a d i a g r a m

of p o i n t e d simplicial spectra is a pointed simplicial spectrum. this the h o m o t o p y limit in the category of such spectra. reasonable to suppose that, category,

in fact,

it is the h o m o t o p y limit in this

but two things have to be proved,

of p o i n t e d simplicial

i) .

The category

Spec K,

spectra is a complete simplicial category,

only such categories have h o m o t o p y jections

He calls

It is e m i n a n t l y

pr n : Spec K, ÷ K,,

e n r i c h e d left adjoints,

limits,

ii) .

for each degree

since

The component pro-

n,

and hence preserve h o m o t o p y

have s i m p l i c i a l ! y limits.

The r e q u i r e d tools are m o s t l y at hand for o r d i n a r y categories in the form of known p r o p e r t i e s of the category for an e n d o f u n c t o r

S

of a category

A.

Dyn S

of algebras

In Section 1 these tools are

s h a r p e n e d and e x t e n d e d to the case of e n r i c h e d categories. Spec A

In Section 2

is d e s c r i b e d for an arbitrary complete V - c a t e g o r y

category)

and a pair of V - a d j o i n t

functors

Z--4Q.

(V

a closed

Finally,

in Section

these results are s p e c i a l i z e d to pointed simplicial spectra.

Note that

the spectra treated here are those for w h i c h phism.

i.

V-categories.

cocomplete,

T h r o u g h o u t this section

symmetric, m o n o i d a l

X n + ~Xn+ 1

V

denotes a complete,

closed category.

category of V - e n r i c h e d categories

is an isomor-

and functors,

V-cat

denotes the

regarded both as a

symmetric, m o n o i d a l closed category itself and as a 2 - c a t e g o r y in w h i c h the 2-cells are V - n a t u r a l transformations; t : F ~> G : A ÷ B the diagrams

between V - f u n c t o r s

A(A,B)

such that for all

commute.

A

and

B,

FA'B > B(FA,FB)

GA,B I B(GA,GB)

i.e., natural t r a n s f o r m a t i o n s

](l'tB) (ti,l)> B(FA,GB)

For basic information,

see

[5],

[8] and references therein.

3,

83

i.i.

Proposition.

Proof:

It is w e l l

to s h o w t h a t If

V-cat

it h a s

A • V-cat

phisms

in

If

• V

that

V-cat

cotensors

then

V.

2 ~ A(f,f')

known

is a c o m p l e t e

2 ~ A

with

2-category.

has

the

limits.

arrow

category

is the V - c a t e g o r y

f : A + B

and

Thus

whose

f'

: A' ÷ B'

d1 - - >

A(B,B')

it is s u f f i c i e n t 2

(cf.

objects

[21]).

are m o r -

are t w o such,

then

is the p u l l b a c k 2 ~ A(f,f')

(f,l)

A(A,A') It is e a s i l y

checked

transformations

t

that

8~ = t

that

8f = f.

1.2.

Proposition.

that

there

: F ~>

where

Q

(i, f,)-> A ( A , B ' )

G

is a n a t u r a l

: A ÷ B

: do ÷ d I

bijection

and V-functors

between

~

in the V - n a t u r a l

V-natural

: A ÷ 2 ~ B

transformation

such such

Let K B

>

B'

Ii be a d i a g r a m F'--4U' tion

~

of V - f u n c t o r s

Then

there

: H U :> U ' K

A

--->

A'

--

H

--

such

that

is a n a t u r a l and

there

are V - a d j u n c t i o n s

bijection

8 # : F ' H =>

KF

between

such

that

F--~U,

V-natural for all

transformaA

and

B

the d i a g r a m s H A(A, UB)

D

:

> A' (HA,HUB)

(I'SB) - - - >

A' (HA,U'KB)

f J

f B_(FA,B)

> B'(KFA,KB) _

- 0-# - i) >

B' (F'HA,KF)

( A' commute Proof: n

(cf. Given

: A =>

the that

[19]) .

UF,

8 : H U ÷ U'K, s : F U => B,

adjunction this

natural

establishes

~'

then

0 # = E'KF

: A' => U ' F ' ,

transformations. a bijection

See

o F'SF

and [4],

as i n d i c a t e d .

s' I, The

o F'HN, : F'U' 6.6

where

÷ _B'

are

for the p r o o f

diagram

D

commutes

84

because bottom

of the are

commute

commutativity

the s i d e s ,

top

of F i g u r e

and bottom

b y the d e f i n i t i o n

1 in w h i c h

of

D.

of V - n a t u r a l i t y .

The The

the

sides,

regions other

top

and

labeled

regions

*

commute

trivially. 1.3.

Definition:

category

S + A

: S A ÷ B.

i)

Let

S

: A ÷ A

is the V - c a t e g o r y

If

~'

: SA'

+ B'

S + A(~,~')

be

whose

then

Pl

+ S

has

The

comma

are m o r p h i s m s E V

is the p u l l b a c k

2 ~ A ( { ,~ ') --

d o

S

A(A,A') A

objects

S + A({,~')

poI Dually

a V-endofunctor.

as o b j e c t s

A(SA,SA')

morphisms

{

: A ÷ SB

and

A + S(~,~')

w

is the p u l l b a c k

of the

diagram A + S(~,~')

.oL

Pl

objects of the

Dyn

S

denotes

are m o r p h i s m s

~

1

d1

2 ~ A(~,~') ii)

> A(B,B')

> A(SB,SB')

the V - c a t e g o r y

: SA + A

and

of S - a l g e b r a s

D y n S(~,W')

--

Dually, objects

coDyn

S

denotes

are m o r p h i s m s

equalizer

of the

~

dlPl

Its

is the e q u a l i z e r

~ A(A,A')

do

the V - c a t e g o r y

: A + SA

and

of S - c o a l g e b r a s coDyn

S(~,~')

two m o r p h i s m s Pl % S(~,~')

A(A,A') d0P 0

1.4.

A.

two morphisms S + A(~,~')

(cf.

e V

in

[6] a n d Remarks:

[i0]) . There

are d i a g r a m s S +A

A_--K--> A_

A+S

A_ ~ >

A_

6 V

in

A. is the

Its

85

-

W v

~

A

A

A

~

v

r~

v

~

r~

A

A

T~I

A

v

! r~ ~J

~D

,r-I ~J

r-~

v

A

f~

.

<

r~ ml

86

which

are

natural are

universal

for

pairs

transformation

of

V-functors

S F 1 ==> F 2

S

coDyn

& --K-> & which

are

universal

transformation 1.5.

for

SF =>

Proposition.

isomorphisms Proof.

The

V-functor s

: X ÷ A

F 1 =>SF2)

.

and

a V-

Similarly

there

diagrams

Dyn

S

A

a single

F

Given

s

V-functor

F

F ~>

over

A

transformation A

determines

~ T s

equations

: A

are

and

: A :>

that

¢ T ÷

: X + A (cf.

~

TS

S

there =

• nU I. that

--

~ = F

--i

are

coDyn

determines

such

to

a V-natural

6.4)

then

Dyn

= T9

S ¢ A _

equivalent

and

[6]

S --J T

x A ~

such

S

A

SF)

a V-adjunction

¢ T

: S + A ÷

adjunction

A_

(resp.,

¢ A = A

natural ~

: ST =>

The

FI,F 2

(resp.

T

Vover

A

a unique Similarly,

8~

= sU 2

since,

' S~.

for

instance @S ~ = (sU 2 =cU 2 =8 so

s

n

Clearly

id.

=

• S~)~ • S(Te

• aSU 1

~

and

A

is

~

=

eU2~

• S~

• ~U I)

= ~U 2

SnU 1 =

e •

restrict

to

• ST@

(~S

• S~U 1

• Sn)U 1 =

give

the

@

second

isomor-

phism. 1.6.

It

is

created

by

U

limits.

a complete

well

Recall

indexed

known

: Dyn that

limit

that

S ÷ A if

{G,F}

Dyn

(cf.

Then

the

moment,

there

is

suppose

G

: I ÷ V A

a diagram

for like

S

then

ordinary

limits

has

[i]) .

e

Dyn

S

is

We

and

show F

here

: I ÷ A

that are

which A

has

a

are indexed

V-functorsthen

satisfies

A(A,{G,F}) For

V-category

V-category.

Proof.

the

If

Proposition.

complete

~

[I,V] ( G , A ( A , F ) )

clarity the

one

that in

S 1.2,

: A ÷

is

a V-functor.

87

AoP

S °p

'l

{-,F}

A(-,F)

and hence

t

a bijection

=> B ( - , S F )

and

: A(-,F) = > B ( - , S F )

tA

: A(A,F)

Then

t

cG

° S

÷ B(SA,SF)

corresponds

between

functors

: S({G,F})

÷

with

S

be t h e

~

Dyn

claim S.

that Then

[I,V]

o {-,F}

transformation

Let whose

BoP

components

tA,i = S A , F i

transformation whose

: A(A,Fi)

c : {-,SF}

components

to an e n d o f u n c t o r to a f u n c t o r

: SF = > F .

=>

÷B(SA,SFi) S o {-,F}

are m o r p h i s m s

S F

: A ÷ A.

: I ÷ A

A V-functor

and

a V-natural

Let

I = {G,~} 6 D y n there

B(-,SF)

in

i = {G,~}

We

]

transformations

=> S

components

codomain

corresponds

transformation

V-natural

have

{G,SF}

>

{-,SF}

to a n a t u r a l

Now we return : I ÷ Dyn

id

between

o S

BoP

{-,SF}

[I,V]

A(-,F)

>

o cG S.

: S{G,F} ÷{G,F} Let

~

: SA + A

be a n o b j e c t

of

is an e q u a l i z e r d i a g r a m

Dyn S(~,I)

--> A ( A , { G , F } )

> A(SA,S{G,F) }

t (l'CG) A ( S A , {G,SF})

J (1,{l,~})

[

A(SA, {G,F}) which

is i s o m o r p h i c

to an e q u a l i z e r d i a g r a m (i, t A)

E-->

[I,V] (G,A(A,F)

(i,

>

[I,V] ( G , A ( S A , S F ) )

(y.i i ~ " ~

(i, (i,~)) [I,V] (G,A(SA,F))

the

crucial

[I,V]

one

step being has

given

by the

commutative

diagram

an equalizerdiagram A(~,~)

--> A(A,F)

tA --->

A(SA,SF)

A(SA,F)

in 1.2.

But

in

88

and

[I,V] (G,-)

preserves Dyn

Hence

I = {G,~}.

1.7.

Proposition.

ates

equalizes

so

S(tg,l)

= E -- [I,V] (G,A(%,~))

S

preserves

If

: A ÷ A

coproducts

S

then

gener-

a free V - m o n a d .

Proof.

See

monad

(triple)

is o b v i o u s to h a v e

[i],

[3],

[9] , [12]

problem.

This

that

them

one

gets

value

on o b j e c t s

whose

V-structure

m

is the

is

S(A)

for d i s c u s s i o n s

simplest We

example.

Let

possible

sketch S

:

~m

A(Sr~A,SmB)

of the case

the d e t a i l s

: A ÷ A

I ] Sn(A) , n=0 by the c o m p o s i t i o n

is g i v e n

m > <SA'~B > ~-

A(A,B)

[16]

a V-monad.

in the m a i n

whose

and

free

in w h i c h

it

in o r d e r

be the V - f u n c t o r

where

S0(A)

> ~-

~ (SmA,

= A,

and

(l,in m)

m

~(

I I SnB)

n

tm I SmA, In I

Sng)

/I

where and

: SmB :>

SB

in the

m'th

s = in I : S ~>

in m

S.

Define

~

whose

components

are

that

formations

~A

° inm,n

and

S =

is an S - a l g e b r a , valent and

induces 1.8. U

:_ I I

I I smsnA m=On=O

= inm+n"

(S,~,~)

then

the

to the r e c u r s i v e

kn+l

= Ii

°Sln

S + A

has

S

>

Let

D = in 0

: Id =>

to be the t r a n s f o r m a t i o n

~,~

algebra

and

: A ÷ A

If

equations

s

I =

preserves given

s

coproducts

[I P

seen

Hence,

: AS ÷ Dyn

by

:

1 1 : SA ÷ A

Im+n = k m o S m l n ) .

adjoint

are V - n a t u r a l

are e a s i l y

I0 = id,

a V-isomorphism

a V-left

I I sPA p=O

is a V_ - m o n a d .

(actually,

If

Then

conditions:

(by c o m p o s i t i o n )

Corollary.

: Dyn

S

the m a p s DA

such

summand.

: S S =>

S. then

trans-

sPA ÷ A

to be e q u i arbitrary, s

: S =>

89 oo

F(A) Proof:

A~ ÷ A

Composing

has

with

s

=

(fA

a I-left gives

:

co

[ I SnA n=l

adjoint

the m a p

given

fA

:

I I SnA) n=0

by

Fs(A)

: S S A ÷ SA

co

fA

>

= ~A

co

[I n=0

SSnA

>

II n=0

SnA co

satisfies

fA o in I n = inn+l"

Rewritting

for

1.9.

Definition.

coDyn

S

1.10.

Let

If

S

The

inclusion

in

= K(~)

the m a p s SA = lim A

qn+l

° Sn~

given

by

=

] ] SnA, n=l

then

has

coDyn

subcategory

sequential

colimits,

of

coDyn

S.

a V-left

adjoint

K

S

is the

- - ~

- - 9

n

n

sequential let

qn

of

A --~> SA. then

whose

value

on

isomorphism.

: l i m SnA ---~> l i m s n + I A ~ S l i m

in the and

the V - f u l l

~:

subcategory

functor

~: A ÷ SA

Let

denote

preserves

is a V - r e f l e c t i v e

an o b j e c t

where

coDyniS

by i s o m o r p h i s m s

Proposition.

Proof:

SSnA

n > i.

determined

coDyniS

co

If n=0

'

fA ° inn = inn

: S S A ÷ SA.

in w h i c h

SnA >

n

colimit

are

: s n A ÷ SA

given

by

be the m a p

sn~ : S n A ÷ s n + I A .

to the

colimit,

so

n = qn"

The maps

sn+IA

that

=

qn+l

induce

the two

sn+IA

=

qn+l

lim s P A

< -

isomorphisms

h n = hqn

I o qn+l

Sqn

lim s P + I A

<

Slim

iff

sPA

Finally,

if

h

: SA ÷ B

is any map,

let

A diagram SA

h > B

L

Sh > SB

SSA commutes

are

>

= Sqn"

: S n A ÷ B.

~

S SnA

>

Hence,

in

~ o h o qn+l

= Sh

I

o I o qn+l

iff

~ o hn+ 1 = Sh ° Sq n = S h n.

90

Step

1.

To s h o w

also

the

square

that

K

is left

adjoint

h0 A - -

4

is an i s o m o r p h i s m For

suppose

In p a r t i c u l a r , follows

that

Conversely, sively

by

then

square

square

hl = 4 -1

(~)

one

i.e.,

that

h0

h n + l = 4 -1

h = n 2.

making o Sh

n

.

square

K

o Sh n o s n ¢ =

makes

is V - l e f t

step.

adjoint

We m u s t

also

is an i s o m o r p h i s m , s(sn~,4)

6 V)

square

E --> A(SA,B) II lim _ A(SnA,B) <------

then

there

hn+ 1

recur-

4 -1 o S(h n o sn-l~)

(~)

commute

to the

(cf.

inclusion

[3],

4.6).

is h a r d e r

because

squares

the V - o b j e c t

of such

squares

(i.e.,

to be the e q u a l i z e r (1,4 -1 ) > A ( s n + I A , B )

is the e q u a l i z e r ( w h e r e

-> A(SSA,SB) II

S > lira <

define

id

(~) S

commute,

it

commutes.

..... > SB

En ÷ A(SnA,B)_ the

h I o ¢ = h0, (0)

..... > B

can be t a k e n A(sn+IA,sB)

For

(0)

14

sn+IA 4

has

square

consider

sn~[

(n)

also

square

hn+ 1 = 4 -1 o Sh n.

= hn-

square

SnA

coDyn

(0)

iff

Then

Then

= 4 -1

: SA ÷ B

Showing

of the r e c u r s i v e

If

commutes

Since

= 4 -I o S(hn_l)

Step

(~)

commutes.

o Sh0.

hn+ 1 ° sn~

Hence

SB

Sh 0 o ~ = 4 o h0; given

consider

I Sh 0 SA - -

If

inclusion,

B

(o)

commutes.

to the

(1'4-1)

...>. A(SSA,B) tl

> _A(SnA,B) id (I,i)

is omitted) >

A(SA,B) iJ

A($n+IA,sB) (l'4-1)>- lJ_mA(sn+IA,B)<-lim(sn~°'l) -> lim A(SnA,B) <

<--

91

The m a p s E

qn

: snA ÷ SA

= lim E n

(since

induce

projection

equalizers commute

maps

with

Pn

inverse

: E

+ En

limits).

and

The maps

<

in this

inverse

illustrated

the e q u a l i z e r fourth

system

in the

rows

of the

show

that

Inspection

shows

that

definition

of

identity

are

first

E n.

because

induced

two rows

by the

i'th

row

there

is also

and the

The

composition

-->

n+l

map.

map

is the

Ei

third

is

and

En+ I. map,

by the

is also

the

diagram (i,~ -I) > A ( S n + I A , B )

~ - - > (i,~ -I)

A(sn+IA,B)

/<

(sn~ , i ) ~

+

identity

(sn+l~, i)

A(sn+IA,B)

diagram The

En

En+ 1 + E n + En+ 1

A(sn+2A,SB)

E

transformations

In this

identity

is a c o m m u t a t i v e

/

2.

an i n d u c e d

E n + En+ 1 ÷ E n

there

natural

of F i g u r e

(sn+l~,l)

A(sn+IA,B)

A(SnA,B) Hence, E

all

= E0;

i.ii.

of the m a p s i.e.,

coDyn. S i

Proof:

Given

: {G,F}

similar in

is V - l e f t

adjoint

to the

inclusion.

coDyn

preserves

G

: I ÷ V

of

shows

pr n

: A

÷ A

limits

then

so

coDyn

S

If

~

~

: I +

in

that

lies

in

coDyn

this

~

coDyniS

: F =>

S.

is the

coDyn

S

where

SF,

consider

A construction indexed

then

~

limit

very

of

G

and

is an i s o m o r p h i s m

{G,~}.

Let

of V - a d j o i n t topological

1.6

and

transformation

= S{G,F}

S.

indexed

isomorphisms,

complete.

{G,SF}

so is

V-spectra.

A

be a c o m p l e t e

endofunctors situation.)

for

A = < A 0 , A I, • ..,>.

n ~ 0. Let

Prn(-) + = P r n _ 1 A) ;

S

to a V - n a t u r a l ÷

the

of

are

V-function

to that

and hence

that

system

If

are

corresponds {G,~}

2.

inverse

Proposition.

and

pair

K

in the

Let

A.

V-category

: A_~ ÷ _ Am n ~ 1

and

and

(The n o t a t i o n

_ A ~ = --~ A n=0 A n = Prn(A)

Write

(-)+ for

of

be

with and,

PRO(-) + = ~

+ = <¢,A0,AI,...>

Z--~ ~

be

a

to r e f l e c t

projection

V-functors

sometimes,

the u n i q u e

i .e. ,

let

is c h o s e n

V-functor

such

(the i n i t i a l

object

92

÷

÷

4r~

I

A

A

A A

,-4

÷

4-

r./l

t"xl +

"F

A

,~ +

I A

A

~

~

÷

~-

I

2

5

m

r.~

2 c-1

,~ +

+

÷

2 ©

v I A

A

A v

I

U?

r~

2

~

+

4-

4-

,~

v I

÷

A

~

A

93

Similarly

(-)

: A~

_

÷

satisfies

A ~

Clearly

(-)+---~ (-)-

Prn(-)-

= Prn+ 1

so

=

is a V - a d j u n c t i o n .

Now

E

and

~

co

endofunctors

of

A ~,

g i v e n by

Z~ = ~ n=0 co

V_-adjoint.

Hence

Z+ =

~

and

-

(-)+ o Z and

also i n d u c e

oo

~

oo

Q

= --~- ~, n=0

which

are

_

= Q

o (-)

are V - a d j o i n t .

Here Z+ 2.1.

Definitions.

subcategory

of

i)

Prespec

Prespec

A

= <@,ZA0,ZAI,...> = <~BI,~B2,...>

A = Dyn Z +.

corresponding

ii)

to

Spee A

--

isomorphism 2.2. of

Dyn Z+ =

Remark. Aco

together with

u n d e r the

1

c o D y n Q-

An o b j e c t

is the V - f u l l

coDyn. Q-

of

of 1.5.

Prespec A

a map

~

is an o b j e c t

: Z+A ÷ A

whose

A = < A 0 , A 1 .... >

components

are m a p s

~ n + l : EAn ÷ An+l" It b e l o n g s to S p e c _A if the t r a n s p o s e # ~ n : An ÷ ~ A n + l are i s o m o r p h i s m s for n ~ 0.

maps

2.3.

Spec A

Theorem.

Prespec

A

If

~

preserves

are c o m p l e t e

sequential

V-categories

colimits

and there

then

are p a i r s

and

of V - a d j o i n t

functors K Spec A <

-

-

F Prespec A <

>

I Proof:

i)

A~

is c l e a r l y

full and f a i t h f u l

functor

Ln >

A~ < --

U complete. such that

The

A

>

__

Prn

left a d j o i n t

(Ln(A)) p = A

if

Ln

is the

p = n

Z-

and

otherwise. ii) adjoint

By 1.6,

it p r e s e r v e s

Prespec

coproducts

A

is c o m p l e t e .

apply,

g i v i n g a left a d j o i n t F such that n UF(A) = I I zPA and the s t r u c t u r e m a p n p=0 n-p c o m p o n e n t s the m a p s n n n+l Z( p=0 I I ZPAn_ p) = p=0 I I zP+IA n-p = I I q=l which

omit

so

~

has a r i g h t

of 1.7 and 1.8

UF(A)=

I I (z+)P(A) . Thus p=o ~UF(A) ÷ UF(A) has as

n+l

÷ I I EqAn+l-q ~qAn+l-q q=0

the f i r s t summand. ii)

limits

Since

so the c o n s t r u c t i o n s

Since

Spec A =

~

has

coDyni~

a V-left

adjoint,

is c o m p l e t e ,

it p r e s e r v e s

by i.ii.

indexed

By h y p o t h e s i s ,

94

preserves

sequential

K

: Prespec

A prespectrum

~:

colimits,

A =Dyn

~+A ÷ A

so 1.10

Z+ ~

applies,

CODyn

corresponds

~- ÷

to

giving

coDyni~-

a left

~ Spec

a coalgebra

~#:A ÷ ~-A

#

components

: An ÷ ~An+l"

A-object

the

The

colimit

reflection

lim(~-)n(A)

into

.

eoDyni~

Thus

has

UK(A) n = l i m

>

The structure

map

UK(A) n = l i m ~ 3 A n + 2.4.

j ~ lim

the

transpose

components

a]+lAn+j+ 1 ~a

i)

The

the

the

isomorphism

isomorphisms

l i r a ~ 3 A n + l + j = a UK(A)n+ 1 >

composition

in h o m o t o p y

FL 0 (A) n ~ 2nA

of

are

>

Remarks.

of i n t e r e s t

is

whose

.

~3An+ j .

j

ZUK(A) ÷ UK(N)

# a-lim(a-)n(A)

as

- - >

n

lim(a-)n(A)

with

-

~n

underlying

adjoint

theory

Q~ = K F L 0

(cf.

[i1]) , is

: A ÷ Spec

given

as

A,

which

is

follows:

so Q~(A) n = lira ~ J ~ n + J A 3

In p a r t i c u l a r ,

Q(A)

= Q~(A) 0 = lim ~ J z J A

is the

stabilizing

functor.

>

ii) indexed H

Let

limits

: I ÷ Spec

Pn = P r n U I

since

A,

all

let

: Spec

three

A ÷ A.

functors

Then

Pn

have

V-left G

H n = pn H.

Then

for any

spectra.

Let

K =

preserves

adjoints.

: I ÷ V,

If

one has

{G,F} n = {G,Fn}.

3.

Pointed

tesian

closed

category with

a map K,

such

that

U

1.6,

x

the

= 1

sets,

and

i.e.,

objects

1 : K ÷ K

denote

sets;

has

a left

follow

coproducts)

Let

(the t e r m i n a l

is c o m p l e t e

not

[A°P,sets] let

denote K, X

the

denote of

K

car-

the together

: 1 ÷ X.

I(X)

K,

does

preserve

of s i m p l i c i a l

simplicial

as a K - c a t e g o r y .

: K, ÷ K

this

category

of p o i n t e d

3.1.

By

simplicial

object

as a K - c a t e g o r y . adjoint

F

given

construction

From

general

.

the

constant

of

K) .

The

underlying

by

f r o m the

the

of

Then

F(X)

= X ~

1.8

since

description

of

K-functor

K, = D y n

i.

functor i,

(although

i(-) D y n S,

does

following: i)

If

Y,,Z,

e K,,

then

K,(Y,,Z,)

e K

not

one has

is the p u l l b a c k

95

- - >

K.(Y,,Z,)

1 = K(l,l)

z[ [(l,z) K_(Y,Z)

- - > (y,l)

ii)

X E K,

then

X ~ Y, 6 K,

1 = K_(X,I) iii)

K,

Z : K(I,Z)

is a l s o

is g i v e n

(l,y~>

tensored

K_(X,Y)

over

X × 1

>

by

being

K, X ~ Y,

the p u s h o u t

1

[ X × Y iv)

One has

the

following

a) K . ( X ~ Y.,Z.) b)

correspond Further,

I,Y,)

as a c l o s e d

K,

X, A

to

If

<X x y , x

X,,Y,

>

Y

> X, A

K,

is a s y m m e t r i c

monoidal

by

1 + K,(X,,Y.)

and ®-product

press

the

UK,(X,,Y,) normal

÷ K(X,Y)

and

closed

functor.

products

(i.e.,

functor.

Hence d)

-al

e)

U

(X ~

closed

equipped UX,

The

(Y ~

preserves preserves

with

and

ii) .

1

adjoint i)

=

by

Y,

with

internal

1 + X,A

hereafter.)

The

Y..

hom (We s u p -

underlying

canonical m a p s

× UY, ÷ U ( X ,

left

i) A

category

given

f r o m the n o t a t i o n comes

a)

x y>

given

: K, ÷ K

1 ÷ K,(X,,Y,)

( f r o m iv)

(i x y)

Then

base points

Let

the p u s h o u t

X

U

e K,,

X, ÷ 1 ÷ 1 ~ Y,

denote

(X x i) "

functor

~ Z.)

@ Y,,Z,) 0

category.

Y.

~ K.(Y.,X

= K(X,Y) .

by adjointness

let

formulas

~ K(X,K.(Y.,Z.))

K,(X,,Y,) n = K,(A[n]

c) K , ( X ~ 3.2.

> X @Y,

× Y,) F =

making

(-) ~

(X × Y) ~

i)

1

to

U

into U

a

preserves

so it is a l s o

a closed

tensors;

i.e.,

(X × Y) m

1 = X @(Y ~

cotensors;

i.e.,

U ( X ~ Y,)

= K(X,Y) .

i)

96

3.3.

Definitions.

i)

Let

S1

be the c o e q u a l i z e r

(in

K)

of the two

maps do A[0] _ _ . ~ dI S,1

and let

If

A

3.4.

tially K . - s m a l l

1 S,

~A = S, ~X E A

: A ÷ A

If

A

objects,

in

then

are defined and are K,-

is called s e q u e n t i a l l ~ K , - s m a l l if

has a s t r o n g l y g e n e r a t i n g

then

S2A

preserves

A,

colimits.

K,.

Hence.

if

{A m}

then

1 1 A(Aa,S , ~ lim B i) -~ K,(S..A(Aa, - - >

lim B i) - - >

= lim K,(SI,A(Aa.Bi))

-~ lim A(A

- - >

-~ A ( A

family of sequen-

sequential

is clearly s e q u e n t i a l l y K , - s m a l l in

is the f a m i l y

K,-category.

sequential colimits.

Proposition.

Proof:

and

An object

perserves

SI

>

is a complete and cocomplete

ZA = S ~ - : A ÷ A adjoint. iii)

q

qd 0 : 1 = A[0] ÷ S 1

have the base point

ii)

A(X,-)

A[I]

,S 1 ~ B i)

>

1 , lim S, ~ B i) - - >

so

S

~ lim B. = lim S, ~ B. _ _ >

3.5.

l

Example:

_ _ >

i "

{A[n] m

i}

tially K, small objects in

is a strongly g e n e r a t i n g family of sequenK,.

However.

~K

(-) = K,(S~.-)

preserves

__W

sequential colimits anyway. 3.6.

Definition.

If

A

is a K. -category.

with the same objects and By category,

[5], 2.2.3, if

then

Furthermore, limits in

U,A

by

A

given then,

U,A(A.B)

is a complete and/or cocomplete K,-

U,A.

there is a close r e l a t i o n s h i p b e t w e e n indexed if

H F

Namely,

the K - f u n c t o r s

spond to K , - f u n c t o r s

is the K - c a t e g o r y

= U(A(A.B)) .

r e g a r d i n g the left adjoint

F : K ÷ U,K,.

U.A

is a complete and/or cocomplete K-category.

[5], 2.4.3,

and in

A

then

H

: I ÷ U,A to

: I ÷ H,A

H # : F,I ÷ A

and

and

U

and

G : I ÷ K

are

as a K - f u n c t o r FG

: I ÷ U,K,

(FG) # : F . I ÷ K,

corre-

and

{G,H}u, ~ = { (FG)#,H#}A. If is defined in

~

is any K-category,

then an indexing functor

[5], 4.5.1, which reduces to

free K - c a t e g o r y on an o r d i n a r y category,

N(I/-)

in case

Z I :I ÷ ~ --is the

such that indexed limits over

97

ZI

are homotopy

limits in

U,A i)

limits.

Hence

for a complete K , - c a t e g o r y

holim H = {ZI,H}u,A(= --

ii)

fiN(I/i) )

--

holim H = { ( Z I ~

-

'

the ¢otensore

sors in 3.7.

U,A

(cf.,

Theorem.

generating

If

are actually

A

functors

Example:

by 2.3,

If

S : k-sp ÷ K

of coten-

Pn

k-sp

denotes

is given as follows:

k-sp(X,Y)

= S(Y x) , Z @ X = IZI × X of pointed k-spaces

if

which has a strongly

objects,

has homotopy

then

Spec A

is

limits that are pre-

: U, Spec A ÷ U,A, n ~ 0. and 3.4.

of k-spaces

1 - I : K ÷ k-sp

structure category

K,-small

2.4, ii),

The category

K-category.

ization and

A

is a complete K , - c a t e g o r y

served by the projection

3.8.

i) ~ H#(i))

i

the same by the construction

U, Spec A

cocomplete

H(i))

[5], 2.2.3).

family of sequentially

Immediate,

~

I)#,H#}A(= I (N(!/i)//

a complete K,-category.

Proof:

homotopy

U.A

--

where

A,

are given by either of the formulas

the singular X, Y • k-sp and

is a complete

denotes

functor, and

Z ~ X = X IZl

is a complete

and

geometric

real-

then this

Z • K,

then

Similarly,

the

and cocomplete K, category.

98

Bibliography [i]

M. Barr, C o e q u a l i z e r s

[2]

A. K. B o u s f i e l d and D. M. Kan, H o m o t o p y limits, Localizations, New York,

[3]

and free triples, Math.

Lecture Notes

in M a t h e m a t i c s

Z. i16(1970),

Completions and

304, Springer-Verlag,

1972.

G. Gierz, K. H. Hofmann, K. Keimel,

J. D. Lawson, M. M i s l o v e and

D. S. Scott, A C o m p e n d i u m of Continuous Lattices, New York, [4]

J. W. Gray, Formal Category Theory:

J. W. Gray,

Adjointness

for 2-categories,

391, Springer-Verlag, New York,

Closed categories,

Pure and A p p l i e d Alg. [6]

Springer-Verlag,

1980.

Lecture Notes in M a t h e m a t i c s [5]

307-322.

lax limits, and homotopy

19(1980),

1974.

limits, J.

]27-158.

J. W. Gray, The e x i s t e n c e and c o n s t r u c t i o n of lax limits, Cahiers de Top. et G~om. Diff.

21(1980),

277-304.

[7]

J. W. Gray, Two results on h o m o t o p y

[8]

G.M.

limits,

(to appear).

Kelly, The Basic Concepts of E n r i c h e d C a t e g o r y Theory,

(to appear). [9]

J. Lambek and B. A. Rattray, Trans.

[10]

Am. Math.

L o c a l i z a t i o n and sheaf reflectors,

210(1975),

E. Manes, A l g e b r a i c Theories, Springer-Verlag,

[ii]

Soc.

J. P. May,

279-293.

Graduate Texts in Mathematics,

1976.

Infinite

loop space theory, Bull. Am. Math.

Soc.

83(1977),

456-494. [12]

D. Scott, The lattice of flow diagrams, A l g o r i t h m i c Languages, 188, Springer-Verlag,

[13]

D. S. Scott,

E. Engeler, New York

Continuous

(1971),

lattices,

[14]

1972,

311-366.

in Toposes,

and Logic, Lecture Notes in M a t h e m a t i c s York,

S y m p o s i u m on Semantics of

ed., Lecture Notes in M a t h e m a t i c s

Algebraic Geometry

274, Springer-Verlag,

New

97-136.

M. B. Smyth and G. D. Plotkin, The c a t e g o r y - t h e o r e t i c solution of recursive domain equations, Edinburgh,

1978.

D. A. I. R e s e a r c h Report No.

60,

99

[15]

R. W. Thomason, gories, Math.

[16]

H o m o t o p y colimits in the category of small cate-

Proc. Camb. Phil.

R. W. Thomason,

Soc.

85(1979),

91-109.

A l g e b r a i c K - t h e o r y and etale cohomology,

Preprint

1980. [17]

H. Wolff, Free monads and the o r t h o g o n a l s u b c a t e g o r y problem, Pure and A p p l i e d Alg.

[18]

M. Tierney,

13(1978),

[19]

233-242.

C a t e g o r i c a l Constructions

A seminar given at the ETH. Mathematics

ZUrich,

87, Springer-Verlag,

G. M. Kelly, A d j u n c t i o n

in Stable H o m o t o p y Theory.

in 1967. Lecture Notes in

New York,

1969.

for enriched categories,

Reports of the

Midwest Category Seminar III, Lecture Notes in M a t h e m a t i c s Springer-Verlag, [20]

1969,

106,

166-177.

L. G. Lewis, The stable category and g e n e r a l i z e d Thom spectra, Dissertation,

[21]

New York,

U n i v e r s i t y of Chicago,

R. Street, F i b r a t i o n s G6om. Diff.

21

(1980),

1978.

in b i c a t e g o r i e s , 111-160.

J.

Cahiers de Top. et

GENERAL

CONSTRUCTION

IN T O P O L O G I C A L

OF M O N O I D A L

, UNIFORM

Georg

CLOSED

STRUCTURES

AND NEARNESS

SPACES

Greve

Abstract/Introduction: In the f o l l o w i n g p a p e r we c o n s i d e r t o p o l o g i c a l s t r u c t u r e s on f u n c t i o n s p a c e s and c a r t e s i a n p r o d u c t s b e i n g c o n n e c t e d by an e x p o n e n t i a l law of the f o r m C(XeY,Z) ~ C ( X , C ( Y , Z ) ) . T o p o l o g i c a l c a t e g o r i e s p r o v i d e d w i t h s u c h a " m o n o i d a l closed" s t r u c t u r e are s u i t a b l e b a s e c a t e g o r i e s for t o p o l o g i c a l a l g e b r a , a l g e b r a i c t o p o l o g y , a u t o m a t a - or d u a l i t y t h e o r y , in p a r t i c u l a r if ~ is s y m m e t r i c or the u s u a l d i r e c t p r o d u c t . We s t a r t f r o m a p u r e l y c a t e g o r i c a l p o i n t of v i e w p r o v i n g an e x t e n s i o n t h e o r e m w h i c h later turns out to be v e r y c o n v e n i e n t for the c o n s t r u c t i o n of m o n o i d a l c l o s e d s t r u c t u r e s in c o n c r e t e c a t e g o r i e s , n a m e l y in t o p o l o g i c a l spaces, u n i f o r m spaces, m e r o t o p i c s p a c e s and n e a r n e s s spaces. E n l a r g i n g a t h e o r e m of B o o t h and T i l l o t s o n [2] it is s h o w n t h a t t h e r e are a r b i t r a r y m a n y (non s y m m e t r i c ) m o n o i d a l c l o s e d s t r u c t u r e s in t h e s e c a t e g o r i e s , h e n c e t h e r e is a g r e a t d i f f e r e n c e t o the s y m m e t r i c case, w h e r e c l o s e d s t r u c t u r e s s e e m to be u n i q u e (cp. C i n c u r a [3], I s b e l l [8]). A f u r t h e r a p p l i c a tion of the e x t e n s i o n t h e o r e m is a c r i t e r i o n for m o n o i d a l - resp. c a r t e s i a n c l o s e d n e s s of M a c N e i l l e c o m p l e t i o n s . Of c o u r s e a s y m m e t r i c m o n o i d a l c l o s e d s t r u c t u r e is u n i q u e l y d e t e r m i n e d by its v a l u e s on a f i n a l l y and i n i t i a l l y d e n s e s u b c a t e g o r y , but also the c o n v e r s e statem e n t is true, i.e. m o n o i d a l c l o s e d s t r u c t u r e s can be o b t a i n e d by e x t e n d i n g a s u i t a b l e s t r u c t u r e f r o m a s u b c a t e g o r y to its M a c N e i l l e completion.

O. P R E L I M I N A R I E S Throughout

this

paper

we h a v e T - i n i t i a l i.e.

for

every

(fi

: A +

= Tgi'

said

to c a r r y

that and

there for

gi

an i s o m o r p h i s m way maps

Ai,

are m o r p h i s m s

in C.

indexed)

that

every

lifted

structure

sources

16OBC

(cp.

so

[13],[7]),

map

y

: TB ÷

TA with

1 : B +

A.

A is

respect

with

a one

to all x.. We a s s u m e i e l e m e n t u n d e r l y i n g set

to be a m n e s t i c

Topological

are T - f i n a l

functor,

is a s o u r c e

to a m o r p h i s m

with

T is s u p p o s e d

iEObC).

there

a topological

TAi)i£ I there

such

object

reasons

implies

are f a i t h f u l ,

(class

can be

initial

is a u n i q u e

technical

of

Set d e n o t e s

(x i : X ÷

Tf i = xi,

: B ÷ the

: C +

liftings

source

Ai)i6i,

TfiY

T

functors

liftings

I is a r e p r e s e n t i n g

of

defined

sinks,

object

( Ti£ObSet

all

for

in this constant

for T, h e n c e

we

101

can

identify

category

elements

a£TA with morphisms

[
finally

(f : E ÷ A ) E 6 O b [ , f 6 C ( E , A ) is " i n i t i a l l y

dense".

is the M a c N e i l l e product T(s®a)

in

= ToxTm,

i.e.

~

® is c a l l e d

H

tor

in C,

(in a n a t u r a l sA B

a and

TaH

lifted

resp. used

here

notion

Note

s~ B

= b.

O.1

Remark

For

a tensorproduct

(cp.

for

PA

C(A,B)

PB

A®I

following

C

C with

a right

~ C(I®A,B)

isomorphisms)

C the

then

underlying

if ® has

[3] T h m

homfunc-

if A ® B ~ B ® A h o l d s are

: AeB + B with

cp.

notion

~ A ~ IoA

to be an i n n e r

there

([4],

sub-

A Tensor-

: CxC +

canonical

symmetric b6TB

[I]). ®

structure

of a m o n o i d a l

(up to n a t u r a l

also

have

dense,

lifted

T ( a sAB) (b) = T ( s ~ B) (a) =

: A ® B + A,

and K e l l y

(cp.

functor

chosen

a6TA,

The definition

coincides

of E i l e n b e r g

w e get

® is c a l l e d

that

The dual

isomorphisms

always

: A + A®B with

projections

TPB(a,b)

so H can

A6ObC.

of T/[

closed

(full)

sink

initially

(automatically)

a monoidal

= C(a,a).

way).

: B ÷ A®B

and

[, resp.

C. O b v i o u s l y

~ TH(A,B), i.e.

every

are n a t u r a l

(AoB)®C w h i c h

: c°Pxc ~

for

: I ÷ A. A

the

and a s s o c i a t i v e

there

and A®(B®C)

C(I,H(A,B))

of

unitary

bijections. adjoint

is f i n a l

a

iff

If [
completion

C is a

dense,

TPA(a,b)

closed with

sections (a,b)

the

= a

structure classical

2.1).

[15]) ®

: CxC ÷

statements

are

equivalent: (i) ® is a m o n o i d a l (ii) F o r (iii)

0.2

structure.

o®A preserves

colimits.

all A £ O b C

o®A preserves

final

epi-sinks.

Remark

Take nal dA

For

closed

all A 6 O b C

® to be 6A

a tensorproduct

: TA + TAxTA,

in C such

6A(a ) =

(a,a),

that

can be

for

all A 6 O b C

lifted

the d i a g o -

to a m o r p h i s m

: A + AeA.

Then

® is the

Proof:

The

usual

universal

product

in C.

property

of ® is

shown b y the

A''

f/

/

f®g

fA®~,

\

following

diagram:

102

I. The

extension

theorem

In the

following

section

sed we

structure

from

introduce

1.1 ®p

the

Definition: : AxB +

Take

THp(B,E)

= C(B,E) closed

(ii)

÷

structure

E
C to be

in a n a t u r a l

structure

on

a monoidal

C. For

clo-

simplicity

way.

with

subcategories

functors Then

respect

with (®p,Hp)

to A,

and

T(A®pB)

= TAxTB

is c a l l e d

B and

and

a partial

E if the

following

hold:

16B and t h e r e for

B
: B°PxE

monoidal

(i)

h o w to c o n s t r u c t

given

following

C,Hp

conditions

we describe

a partially

are n a t u r a l

all A £ O b A ,

B® BcB, P

A®

isomorphisms

A®

P

I ~ A,

1o B ~ B P

B6ObB.

BoA and

there

is a n a t u r a l

isomorphism

P

A®p(B®pB') ~ (A®pB)OpB'. (iii)

There

(®p,Hp) phism

Note in

is a n a t u r a l

is c a l l e d

symmetric

again

that

(i),

C(A®pB,E)

if A = B and t h e r e

the u n d e r l y i n g

(ii)

and

(iii)

YABE(f) (a) (b) = f(a,b) following

maps

of

the

are c a n o n i c a l .

(the f u n c t o r

statement

1.2 E x t e n s i o n Take to

YABE:

~ C(A,Hp(B,E)).

is a n a t u r a l

isomor-

A ® B ~ B® A. P P

1.1

The

isomorphism

is the m a i n

transformations For

T is o m i t t e d ) result

to be a p a r t i a l

A,B,E (cp. 1.1, B
(a) If A is f i n a l l y the d i a g o n a l

closed

E initially

symmetric,

: TB ~ T B x T B

source

then

(C(f,g) C(M,N)

we g e t

Furthermore

(®p,Hp)

structure

For M,N6ObC

Providing

all

a£A,

b£B.

section:

on

: C(M,N) the

an i n n e r

consider

with

respect

can be

and

lifted

can be e x t e n d e d

if for

all

B6ObB

to a m o r p h i s m

closed

dense

structure

and

® exten-

(®p,Hp)

to a s y m m e t r i c

is

monoidal

C.

consider

with

structure

dense

: B + Be B t h e n t h e r e is a m o n o i d a l P d i n g ® , i.e. ® / A x B = ® . P P (b) If A = B, A f i n a l l y d e n s e , E i n i t i a l l y

(1.2.1)

for

this

E
dense,

6B

monoidal

dB

Proof:

of

appearing

we h a v e

theorem:

(®p,Hp)

closed

example

the

~ THp(B,E))b6ObB,E6ObE,f6C(B,M),g6C(N,E initial

homfunctor

the

source

sink

structure H

with

: c°Pxc ~

consisting

respect

C with

)•

to this

H/B°PxE

of all m o r p h i s m s

= Hp.

103

(1.2.2)

TlxTk

and all

sections

Providing

TMxTN

get

a functor

Let

us f i r s t

A® B

®

: T(A®pB) o

m with

l®k

~PcA~B

A£ObA,BEObB,

final C with

o(n)

P

N

on the

roof

I

with .

= ®

P We h a v e (c A "

/

projection hence

PN w i t h

the d i a g r a m

PN ~ s l N =

N,

A® B

l®k

Furthermore M1 > Me1 < mS

~ 1I f p

~

PN

Ip A®

A~ ®

I~A

T s~c

I denotes

N,

sink w e

diagram the

cano-

~s IN is the

section) . hence

we

is a b i j e c t i v e

consider

the

m

have

: B ÷

mS M1

a lifted

retraction,

following

(c B

I

-

to this

morphism,

is final,

i.e.

an i s o m o r p h i s m .

k:B ÷

the f o l l o w i n g

lifted of

M,

respect

: A ÷

nical

>N

sink

I:A ÷

(m,n).

structure

® is u n i t a r y : 0s IN >I®N ~ N

k

=

e/AxE

that

I~B---B The

TMxTN,

÷TMxTN,

the

: Cx C ÷

show

+

: TN

diagram: I canonical,

lifted

section).

< M "~C~

P The

same

argument

as b e f o r e

A is f i n a l l y

dense,

final,

there

Now

hence

let us p r o v e

natural

hence

yields

the

that

® is left

PM"

Moreover

adjoint

H(f,g)

: C(MeN,L)

to H,

i.e.

we h a v e

to find

a

C (M,H(N,L)) . T a k e MN t : MeN + L and d e f i n e t' : TM ÷ C(N,L) by t' (m) := t s , where MN m s : N + MeN is a l i f t e d section. C o n s i d e r the f o l l o w i n g d i a g r a m : m H(N,L)

~M,N,L

projection

on the b o t t o m of the d i a g r a m is M1 s e c t i o n s~ , so we h a v e M®I -- M.

is a l i f t e d

isomorphism

a lifted

sink

+

>Hp(B,E)

(cp.

1.1)

The

source

tr

E60b [

M T

f))

is i n i t i a l ,

r

r A

phism

Tt

= t'Tr. r hence there Now given u' (m,n) A~B

Furthermore

the

is a m o r p h i s m

a morphism

u

:= T ( T u ( m ) ) ( n ) . lek

£

sink

of

: M ÷ H(N,L) This

> M®N <

yields m

tr

all m o r p h i s m s

: M + H(N,L~

with

and d e f i n e the

N

every

is a m o r -

: A + H(N,L) r

with

: A + M is final,

T£ = t'. u'

following

s

so for

: A + M there

: T(M®N)

+ TL by

commutative

diagram:

104

The m u

sink

s MN

of all

: N ÷ M®N

l®k

is final,

: M®N + E with

v is i n i t i a l , ~MNL

tative

Tu

:= ~

to s h o w

B£ObB,

and all

for e a c h v

: L ÷ E there

The

of all v

source

to a m o r p h i s m

u

bijection

® is a s s o c i a t i v e :

Consider

sections is a m o r p h i s m

: L ÷ E, E 6 O b £ ,

: M ® N ~ L.

Defining

because

£ = t.

the

of

following

It

commu-

(1.2.3) : sSIN®L .... N®L %±®kj®,~> (MeN ®L < m

(A® B) ®k ' P > (A®B)®L

(A®pB) ® B '

A6ObA,

~ is a n a t u r a l

that

diagram

hence

= 'Iv u'.

v can be l i f t e d

so u'

by ~MNL(t)

remains

: A ® p B ÷ M®N,

P

/~s J

CA%B)L

(a,b) s M (N®L)

~

T1~'a) M (N®L) ~ s

A O p (BOpB')

i® (k®k')

so L h a s

a right

i.e.

the

sink

BEB,

and

all

of

all

e

sMN®L

m

(a,~)s CA®B) L of

adjoint,

consisting

(AOpB) e k '

the r o o f

preserves

all

is final.

M® (N®L)

(l®k)®L

final

epi-sinks

: (A®pB)®L

Furthermore

because

÷

(cp. O.1),

(M®N)®L,

A6A,

of A® BoA t h e p

sink

: (A®pB)®pB' ÷ (A®pB)®L, B'6B and a l l

: L ~

(A®pB)®L

the d i a g r a m

: (M®N)®L

>

hence of

is final,

is final.

: (TMxTN) xTL ÷ TMx (TNxTL)

aMN L

NL

÷ M®(N®L).

so the w h o l e

The

canonical

therefore

Now

can be

consider

the

sink

appearing

on

associativity

lifted

to a m o r p h i s m

following

diagram:

s M (N®L) A® B

iok

> Mo(N®L)

<

m

N®L

A®pd B A ® p IB®pB)

a~L

(l®k.)®k= (A~pB) e p B

•

d B denotes

the

components

of k, w h i c h

the -I ~L

sink

lifted

on the roof

: TMx (TN~TL) : Me (N®L)

a~LaMN

L =

÷

~

® and

and k N

are m o r p h i s m s

of

the d i a g r a m

(TMxTN) xTL

(M®N)~L.

(M®N)oL,

(b) We d e f i n e

diagonal

(M®N) ®L

i.e.

Of

in Set course

:= pN k resp. in C b e c a u s e

is final, can be we have

kL

:= pL k are

® is u n i t a r y .

so the

lifted aMNLa~L

the Again

associativity

to a m o r p h i s m = Me (N®L)

and

® is a s s o c i a t i v e .

H as in

(1.2.1)

resp.

(1.2.2).

Then

the p r o o f

of

105

(a) shows that ® is u n i t a r y and left adjoint to H. Also the m o r p h i s m aMN L is still availble appearing

(cp.

(1.2.3)). The finality of the sinks

in the following d i a g r a m now proves that the canonical

symmetry TMxTN ~ T N x T M in Set can be lifted to an i s o m o r p h i s m M sym N : M®N ÷ N®M: s A®pB

k®l

B®pA

l®k

> MeN L

m

N~

N

M

It remains to show that the canonical map -I : TMx (TNxTL) + (TMxTN) xTL can be lifted. This is done by the f o l l o w i n g morphism: N

M

M® (N®L) M®symL) M® (L®N)

L

symL®N> (L®N) ®M aLNM ~ L® (N®M)

symN®M> (N®M) ®L N symM®L (M®N)®L.

~

Note that the extended s t r u c t u r e ® above is something "coarsest" m o n o i d a l closed s t r u c t u r e e x t e n d i n g

like the

(®p,Hp), i.e. given an-

other m o n o i d a l closed s t r u c t u r e ®' on C with ~'/AxB natural t r a n s f o r m a t i o n

85~

= ® there is a P : M®N + M®'N such that the u n d e r l y i n g

m o r p h i s m of 8MN is the identity. We get the following

corollary, w h i c h of course

admits further

generalization:

1.3 C o r o l l a r y :

Take BcC to be a class of c a r t e s i a n objects,

every B6B

has a right adjoint.

~B

structure ® on C with MoB = M~B for every M6ObC,

Proof: H

i.e. for

T h e n there is a m o n o i d a l closed B6ObB.

The p r o d u c t f u n c t o r z : CxB + C has a right adjoint

: B°PxC ÷ C and applying the usual trick we can assume

P TH

(B,M) = C(B,M) in a n a t u r a l way (we i d e n t i f y B w i t h the full subP c a t e g o r y g e n e r a t e d by B). For B,B'6B, M£ObC we get THp(B,Hp(B',M)) C(B~B',M)

= C(B,Hp(B',M))

~ C (B~B',M) , thus p r o v i d i n g

w i t h the final structure with respect to the i s o m o r p h i s m

above we get an object Hp(BWB',M) Hp(B,Hp(B',M)).

Of course H

w h i c h is i s o m o r p h i c to

(B~B',D)

is a functor and we get for

P

M,N£ObC:

C(M~B~B',N)

C(M,Hp(B~B',N)).

~

C(M~B,Hp(B',N)) ~ C(M,Hp(B,Hp(B',N))) ~

That means:

Finite p r o d u c t s of c a r t e s i a n objects

are

106

cartesian.

N o w take

~e to be the closure

then ~ : CxB e ~ C has (~,Hp)

is a partial

assumptions

closed

tesian

closedness.

pact open

topology.

a monoidal

Hence

closed

as a g e n e r a l i z a t i o n (cp.

[2], T h e o r e m

regularity

Take

objects

approximation

every

category

which

of compact

theorem

a monoifor car-

Top of

compact

Hausdorff

is g i v e n by the Hausdorff

So corollary

it is shown

yields

com-

spaces

1.3 can be u n d e r s t o o d

of B o o t h

that

and T i l l o t s o n

the c o n d i t i o n

of

is superfluous.

closed

closed

completion

structure

structure

with

of A
if there

respect

as a p a r t i a l

to A

to A,A,A).

closed

iff there

isomorphism

A,A',A''6ObA,

where

We have

class

Moreover

and a n a t u r a l

Procf:

C to De the

that for every

adjoint

structure.

monoidal

respect

(b) C is c a r t e s i a n

taking kno%~

C to be the M a c N e i l l e

monoidal

(i.e. with

cartesian is a good

of a c o r r e s p o n d i n g

2.6).

a symmetric

symmetric

of all

of course

a right

in that t h e o r e m

1.4 Corollary: (a) C has

closed

it is w e l l s~B has

finite products,

H e : Be°Pxc + C and o b v i o u s l y P s t r u c t u r e f u l l f i l l i n g the

theorem.

Furthermore

spaces,

the functor

yields

which

of B under

adjoint

the subclass

structure

topological space

monoidal

of the e x t e n s i o n

In p a r t i c u l a r dal

a right

to prove

is a functor

C(A~A',A'')

H : A°PxA + C P ~ C(A,Hp(A',A'')) for all

~ is the p r o d u c t

in C.

(b) : (1.2.1)

(1.2.2)

and

with

®

= n yield P

a pair

of adjoint

functors

(®,H).

to show that for each M6ObC phism

in C. C o n s i d e r f

A dA~

f,f

~d M

that because

symmetric

the M a c N e i l l e cal functor gories w h i c h

Io4

closed, In[13]

a MacNeille

have

is a mor-

diagram:

it is useless

get again

monoidal

of the sink

functor.

to apply

1.2

(b). E s p e c i a l l y its s t r u c t u r e

it is shown,

completion,

a symmetric

to a s e m i t o p o l o g i c a l

is s u f f i c e s

yields

the assumption.

completion.

has

0.2

: TM + T M x T M

f : A ~ M, A6ObA,

of 0.2

we w o u l d

of remark 6M

The f i n a l i t y

>M®M

completions, already

the f o l l o w i n g

>M

A~A

Note

Because

the d i a g o n a l

monoidal

(a) to M a c N e i l l e if A in 1.4 is

can be e x t e n d e d

that every

semitopologi-

so 1.4 can be applied closed

structure

to

with

to caterespect

107

Finally 1.4(b)

it s h o u l d ~sing

statement sidered

be m e n t i o n e d

quite

different

as a g e n e r a l i z a t i o n

MacNeille

"cartesian

(cp.

In the

following

gories

of t o p o l o g i c a l

spaces

(Mero)

iff

identity.

closed

state

the

Lemma:

(= r e a l tion

or

Take

here

of s e c t i o n

I to the

con-

so c a l l e d

(Near),

spaces

Note

that

a class

(or " s t r o n g l y

spaces

they

are

to get

mentioned

closed

[9]) or an

explicitely

different

above.

structures

cp.

constant

considered

are u s e d

A of com-

rigid"

is e i t h e r

cate-

merotopic

in

monoi-

In o r d e r

Top

to

we h a v e

Hausdorff

1
Then

space

there

and x i £ X , Y i 6 ~

is a c o n t i n u o u s

func-

= Yi"

g

n ~ {x i} is a d i s c r e t e and c l o s e d s u b s p a c e of X, h e n c e i=1 : M ÷ LR, g(xi) = Yi is c o n t i n u o u s , h e n c e t h e r e is a c o n t i n u o u s m a p

f

: X + tR w i t h

Proof:

M

:=

2.2 Lemma: resp. with

Then

Proof:

For

A subbase

Now

of

(2.2.1) Because

set of open

Cco(X)

of

the

suppose

are o p e n

the

finite

= {f

the

= Yi b e c a u s e

X to be

compact

gence.

base

f(xi)

Take

Cpw(X) the

U(K,O)

to

lemmata:

a compact

f (xi)

results

categories

simple

x i ~ xj,

f : X ÷ IR w i t h

via

t w o of t h e m

on m o n o i d a l

X to be

, who

categories

nearness

rigid

of c o m p a c t

in the

theorem

this

Uni6.

(Unif).

spaces

[14],

following

numbers),

the

is c a l l e d

structures

our m a i n

consider

2.1

[9]

of A. F r 6 1 i c h e r

object

and

(T0p),

map between

classes

in

apply

spaces

spaces

Rigid

for e x a m p l e dal

shall

continuous

found

He g o t

[5]).

and u n i f o r m

Hausdorff every

we

independently

(not p u b l i s h e d ) .

of one

Top, Near, Mero

to

R. B ~ r g e r

of a t h e o r e m

completions

monoids"

2. A p p l i c a t i o n s

pact

that

methods

a compact all

that

sets

resp.

topology

of

of

X is not

Cco(X)

topology

consists KcX

and d e n o t e from

by

Cco(X)

X to IR e q u i p p e d

of p o i n t w i s e

finite,

take

We a s s u m e

conver-

0

:= ]0,1[

~ U({xi},Oi). i=I of U ( { x } , O ) D U ( { x } , O ' ) = U({x},ODO')

= Cpw(X).

and O c ~ open.

of U ( { x } , O ) ,

ISiSn-1,

Cco(X)

of sets

is c o m p a c t

consists

~ ~ O.c~R and x.£X, i n-1 i

U(X,O)

functions

the

is o b v i o u s .

where

Cpw(X)

space

X is finite.

X the e q u a t i o n

: X ~ IR 1 f(K)cO}

Hausdorff

continuous

topology

= Cpw(X ) iff

topology

X is T 4.

and

xcX,

A sub-

Oc~R open.

assume

that

there

with

=

for x6X,

O,O'c~R open,

we

108

can assume x i # xj for l
and given Yi6Oi,

Now there is an element Xn6X , Xn#Xi,

1
there is a continuous f u n c t i o n

f : X + IR w i t h f(xi) = Yi' 1
thus we have a c o n t r a d i c t i o n to

= Cpw(X).

2.3 Le~ma:

Take

McObTop to be a rigid class of compact Hausdorff spaces

and denote by Me t h e

class

of

all

finite

products

of

spaces

of

M. T h e n

all continuous m a p p i n g s between the spaces of M e are either constant or n canonical, i.e. e v e r y m o r p h i s m f : T J ] M i + M, M.6M1 i s c o n s t a n t or a projection.

Proof: We prove the assumption proof is obvious.

for n = 2, then for a r b i t r a r y n the

So consider spaces X,Y,Z6M,

Y # Z. For a continuous ^

m a p p i n g f : X~Y + Z d e f i n e f : X + Cco(Y,Z) (Cco(Y,Z)

by f(x) (y)

:= f(x,y).

is the set of all continuous maps from Y to Z with the com^

pact open topology).

The exponential

law yields the c o n t i n u i t y of f.

M o r e o v e r we have Cco(Y,Z ) ~ Z, hence f is constant or an isomorphism, so we obtain that f is either constant or a projection.

Now assume

Y = Z. If X # Z we get the statement above by i n t e r c h a n g i n g X and Y. It remains to consider the case X = Y = Z, w h i c h is already m e n t i o n e d by H e r r l i c h [6], p. 134.

2.4 Theorem:

Take M to be a rigid class of non finite compact Hausdorff

spaces. Then there are as m a n y d i f f e r e n t m o n o i d a l closed

structures

in

Top as there are n o n - e m p t y s u b c l a s s e s of M. Proof:

Take M to be a rigid class of non finite compact Hausdorff

ces, ~ #

1.3 we get m o n o i d a l closed structures ®F and ®N with right adjoint resp.

spa-

NcM, ~ # PcM, N ~ P, say N6N, N~P. Then according to corollary

H N. HP(N,tR) is the set

HP

Top(N,IR) provided with the initial struc-

ture with r e s p e c t to the source of all maps HP(N,fR) ..... HP(f, IR) ~Cco(P,iR), p£pe,

f : p ÷ N, w h e r e pe is the class of

all finite products of spaces of P. Of course all f : P+ N must be constant,

so for KcP compact and O c R open we get HP(f,IR)-I(u(K,O)=U(f(K),O)

hence HP(N,~)

carries the topology of the pointwise

convergence, w h i c h

10g

by lemma

Next form

2.2 is d i f f e r e n t

let us c o n s i d e r

spaces

defined

to be the

44). The

uniform

on Unif

F : Unif ÷ Top dorff

spaces

there

ture

as there

of K. T h e n

then

struc t e d

with

tions

: K ÷ X.

is initial compact tial

closed

structure

for u n i f o r m

where

in

spaces

the

Cu(K,Y)

space

space

of

closed

For compact

Haus-

so it is easy

of non finite monoidal

which

monoidal

is e x t e n d a b l e

Take

structure source FC

u

X the function

FHK(x,~)

of M yield

of u n i f o r m

func-

convergence,

(K,FY), i.e. for a co FHK(xgR) carries the ini-

FHK(f'IR) > Cco(K,IR) , K6K e,

coincides

different

continuous

= C

space

in the proof

adjoint

FHK(x,y)FHK(f'Y)>FCu(K,Y)

(K,Y)

to all FHK(x,IR)

i.e.

struc-

by 1.2(a)

space HK(x,Y) is conHK (f'Y) > Cu(K,Y) to

of all HK(x,Y)

we have

in

products

closed

H K to be right

X,Y the f u n c t i o n

the

Hausdorff

structures

of all finite

a partial

Unif.

on

compact

closed

of M.

yields

that the

constructed

subclasses

K

with

Moreover

f : K ~ X continuous,

empty

[8],

functor

[8], p.46),

by K e the class

source

topology with respect

function

Isbell

is

induced

structure

a monoidal

is the set of all u n i f o r m

It follows

Top.

Hausdorff

class

to C, K e, C,

from K to Y p r o v i d e d

K£Ke,f

(cp.

sources.

(cp.

subclasses

®/(UnifxK e)

respect

by r e q u i r i n g

be initial,

initial

are as m a n y d i f f e r e n t

are n o n - e m p t y

to a m o n o i d a l to ®K,

Z

forgetful

For uniwhich

uniformity

to the

and yields

The

= K~K',

M to be a rigid

Take ~ # KcM and d e n o t e

(®p,Hp)

spaces

to p r e s e r v e

we have K®K'

Take

of spaces

initial

X®Y,

following

spaces.

Proof:

the

is left adjoint

[8], p. 46).

is known

2.5 Theorem: Then

with

spaces.

product

from XxY to spaces

product

on f u n c t i o n (cp.

K,K'

to set up the

Unif

functions

semi-uniform

of u n i f o r m

"semi-uniform"

set XxY p r o v i d e d

convergence

structure

the

= Cco(N,R).

Unif

the c a t e g o r y

X and Y we have

by all s e m i - u n i f o r m p.

from HN(N,R)

with

of 2.4,

monoidal

the

hence closed

corresponding distinct

non

structures

in U n i f . Now

let us have

categ o r y

Near

peat

the r e l e v a n t

pair

(X,~)

fying

the

a look at the c a t e g o r y

of nearness facts

spaces

on these

(cp.

Mero of m e r o t o p i c and at the

for example

categories:

[7]).

A merotopic

such that X is a set and ~ is a set of covers following

conditions:

We shortly space

re-

is a

of X satis-

110

(i) If A£~ AcB, (ii)

#~

(iii) The f

and

A6~,

: (X,~)

, i.e.

for each AEA

there

~ are

(Y,~)

called

uniform

is a set m a p

covers,

a uniform

f : X + Y satisfying

f is f i n a l

iff

the

f-1(A)6~.

For

a set I and m e r o t o p i c

iff

is an B6B

with

A^B = { A D B I A 6 A , B £ B } 6 ~ .

implies

of

+

A
{X}6~

B£~

members

A6H.

and

then

following

condition

holds:

spaces

continuous

f-I(A)£H A cover

(Xi,~ i)

map

for

all

A of Y is in

(X i d i s j o i n t )

X. := ( U Xi, I I ~i ) , w h e r e ~i i is the set of all c o v e r s of iEI 1 i£I i-~ X. w h i c h are r e f i n e d by a c o v e r ~ Ai, Ai6u i t o g e t h e r w i t h the i£I 1 i6I c a n o n i c a l i n j e c t i o n s is the c o p r o d u c t resp. the sum of the s p a c e s (Xi,Hi).

The

Unif

cal. (X,H)

is u n i f o r m

a cover with

forgetful

functor

is a b i r e f l e c t i v e

V£H

A merotopic condition

iff e v e r y

such

St(V,V)=

that

holds:

int A = { x l B V 6 H ,

(cp.

is c a l l e d

every

have

2.6

there

of A lying

is

in H

A

iff

the

following

:= {int A I A 6 A } 6 u are

that

by o r d i n a l

a full

the

with

bireflective

reflector

induction

in the

way:

Remark:

He+1

int

space

spaces

to check,

R : Mero ~ N e a r can be c o n s t r u c t e d following

topologispace

[7]).

Nearness

it is e a s y

and

a merotopic

a star-refinement,i.e,

a nearness

A£U we

St(x,V)cA}.

of Mero,

subcategory

admits

~ ~}

(X,U) for

A6U

is a m n e s t i c

Mero,

of

{St(V,V) {V V} is a r e f i n e m e n t

U{W6VIWNV

space

: Mero + Set

T

subcategory

For

(X,u)

:= { A £ ~ e l i n t H

:= n { ~ B l S < e } 0 canonical

ObMero

A£H e}

Then

morphism

and

R(X,~)

NX

: (X,~)

define for

~o

:= ~" For

a limit

:=

an o r d i n a l

ordinal

e set

(X, _ ~ ~ ) is a n e a r n e s s

+ R(X,~)

has

the

e take

space

universal

and the

property

of

a reflection. For

every

covers

symmetric

of X g e n e r a t e s

Ro-spaces

are

(2.6.1)

adjoint

be

like

is d i f f i c u l t

with

the

to the

to o b t a i n

following

But

(Ro-Space

structure.

functor

inclusion monoidal

a semi-uniform

in Unif.

X

J

product

in a b s e n c e

an e x p o n e n t i a l

of V

Near,

all o p e n

to p r o v e

i.e.

we

have

and

law.

in Near

a structure

the

that an right

Near.

structures

of

set of

: Near ~ ToPo b e i n g

: TOPo ~

closed

the

It is easy

subcategory

topology"

of g e t t i n g

to d e f i n e

convergence it

way

space

a nearness

a bicoreflective

"underlying

An o b v i o u s would

topological

and Mero

of u n i f o r m

star-refinement

So we h a v e

to g e t

axiom along

111

2.7 Proposition:

For

(X,~),(Y,~)6ObMer0

product of X and Y by X®Y

covers refined by some cover A®(B A) @ is a monoidal Proof:

closed

Obviously

every

structure

(Y,v)£ObMero

According

! .) ~ X i = ( b; X ., =i_ ~ i6I i£I i i6I l Then(~®~ is generated by iEI ± U Aie(HA. ) = {AiXBA JAi6Ai, iEI z l AiEui,BA 6v , hence ( ~ W i ) ® ~ l iEI Now take e : (X,u) + (X',u')

be a cover of X'xY with A®(B A) of 06W'®9,

:= {AXBATA6A,BA6BA} , AE~,

(e®y)-I(D)

Mero,

in

to 0.1

o®y preserves

Take

where W®~ is the set of all

coproducts

to show, that

to show,

that for

and final eplmorphisms:

to be the coproduct

of

(Xi,~i)£ObMero.

all covers !, B A 6B A ,i6I} = ~, (Ai®(B A ) with i i i£I i = ~ (wi®v) and o®Y preserve~coproducts. i6I to be a final e p i m o r p h i s m and assume D to

(exY)-l(~)6U®v.

Then there is a refinement

in U®9 and e(A)®(B A) is a refinement

of 0, i.e.

so e®Y is final.

we have to prove 2.8 Lemma:

(X,~),(Y,v)6ObMero.

Assume

(b) X , Y 6 O b N e a r

(a) Take

spaces

Then the following holds:

= int~A®(int~B A) for A6~,

implies

Y6ObNear

to 2.7 for nearness

the following

(a) int mg(A®(BA))

Proof:

it remains

it suffices

In order to state a result corresponding

(c) For

BA6~.

on Met0.

® is a tensorproduct

® has a right adjoint.

we define the semi-uniform

:= (XxY,~®~),

BA£~.

XeY6ObNear.

we get R(X®Y)

= RX®Y.

(x,y)6int ®v(AXBA),

then there are covers U£~,

V 6u,

for each UEU such that St((x,y) ,U®(Vu))CAxB A. That means i'

~{UxVuJ (x,y)EUXVu}CAXBA, xEint~A,

hence St(x,U)x

~ St(y,Vu)CAxB A and we get U,xEU yEint~B A. Other way round given (x,y)6int Axint B A there are

covers UEU , V6~ with St(x,U)cA,

St(y,V)cB A, so

St((x,y),UeV)cSt(x,U)xSt(y,V)cAXBA,

i.e.

(x,y)6int~®9(AXBA).

(U®V= {UxVIUEU,V6V}). (b) is an immediate (c) We show

consequence

of

(a).

(~®v)~ = ~a®~ for every ordinal a

(cp. 2.6).

For ~ = O

there is nothing

to prove. For an arbitrary ordinal e we get Because of A® (BA) E (~®~) ~+ I iff A®(BA)£~ ®~ and intu ® v ( A ® ( B A ) ) 6 ~ ® 9 .

(a) this is e q u i v a l e n t

to A ® ( B A ) 6 ~ ® ~

and i n t A ® ( i n t ~ B A ) E ~ e

®9, hence

112

int

A£~,i.e.

ous.

Of

get ~(~®9)~

The

A6~

course, =

~(~

following

uniform

+ I and

for

®9)

(~e)®~

X£ ObNear

o®X p r e s e r v e s

epi-sink

the

same

product

our

we

assumption.

closed

is a m o n o i d a l

in Near.

structure

of

closed

:Yi®X + Y®X)

by 2.8

R preserves

that

for

there

Take

that

(f. : Y. ÷ Y') 1 1 = Y, h e n c e

RY'

by

2.8

each

(f. : Y + Y) 1 1 is a m e r o t o p i c s p a c e Y'

as Y such

and we g e t by O.1

(fi®X

and o b v i o u s l y

to s h o w

epi-sinks:

Then set

category

It s u f f i c e s

final

underlying

in Mero

epi-sink

proves

the m o n o i d a l

topological

in Near.

® is a t e n s o r p r o d u c t

a final

is o b v i hence

spaces:

semi-uniform

Near is an a m n e s t i c

Proof:

yields

direction

= ~ ®~ h o l d s ,

(~®~)~

and this

for n e a r n e s s

The

The other

~

on Near.

structure

with

=

ordinal

proposition

convergence

2.9 P r o p o s i t i o n :

A@(BA)£~e+I®9.

a limit

to be

is a f i n a l

=

(Rf.®X : Y ®X ~ RY'®X) = (R(fi®X) : Y . ® X + R(Y'®X) 1 1 1 f i n a l e p i - s i n k s , h e n c e the sink of all f . ® X 1

is final.

2.10

Theorem:

Then

there

as t h e r e

Proof:

ces,

i.e.

X~oK

Therefore

clude

VC

being get

for

which

Y,Z the

functor

argument responding

covers

® denotes

to

source V

of

proof

function

2.5

space

shows of 2.4

of K o b v i o u s l y

adjoint

for

A6(B A) of R o - s p a -

product to ® we

(2.6.1))

in

can

con-

because

we h a v e

to the

Then

proof

K ~ K = KeK, of 2.5 w e

closed

s t r u c t u r e (®p,Hp) K s t r u c t u r e ® . For n e a r n e s s

HK(y,z)

HK(f,Z)

that

category

semi-uniform

(cp.

closed

space

all HK(y,z)

spaces.

in Near

X an R o - S p a c e .

• Furthermore

: Near ~ TOPo p r e s e r v e s

in the

the

monoidal

function

compact

structures

in the

so a n a l o g u o u s l y

to a m o n o i d a l

adjoint

X~oK

to be r i g h t

Cco(X, V)

@ # KcM a p a r t i a l

the

space,

for Y£ObNear

in Near,

finite

closed

B A, A£A,

product

Cn(D,Q)

(K,VY)

co adjoint

product

each

quiring The

assuming

is e x t e n d a b l e

spaces

where

of n o n

of M.

Hausdorff

the d i r e c t

= C

left

the

of

= X®K.

(K,Y)

n is

a compact

class

monoidal

subclasses

A of X and o p e n

cover

Year.

J(D~oK)

a rigid

different

empty

K to be

cover

is an o p e n

M to be

as m a n y

are non

Take

an o p e n

Take

are

~ C

initial VHK(X,~)

(X c o m p a c t ) ,

is c o n s t r u c t e d n

(K,Z)

to be

sources, coincides thus

by re-

initial.

hence with

different

the the

cor-

non-empty

113

subclasses

2.11

of M y i e l d d i f f e r e n t

Theorem:

Then there as there

Proof: for

Take

M to be a rigid

are a s many d i f f e r e n t

are n o n - e m p t y

a class

monoidal

Finally

of T r n k o v a

a proper

2.12

class

of rigid

Corollary:

noidal

closed

If there

to the c o n g l o m e r a t e

only a small 2.10 resp. gence

spaces

closed

[14],

that the

and p r o b a b l y

It should

pletions

of classes

algebras

which

cal

c~tegories

of compact

mentioned

the e x i s t e n c e

the mlmber

of

in this section

corresponding

for p r o x i m i t y any more

the

complete

over

subcategory

give

to 2.4,2.5,

of g e n e r a l

comve2-

spaces. symmetric

to M a c N e i l l e

or in a special

closed

of mo-

is e q u i p o t e n t

1.4 can be applied

are c a r t e s i a n the full

~ ~ P, NcM

class.

the c a t e g o r y

spaces

a monoidal

®P and ®N in ~4ero.

to consider

that

Hence

the c o r r e s p o n d i n g

cardinal,

can be o b t a i n e d

be m e n t i o n e d

can c o n s t r u c t

of a proper

for

epi-sinks.

Near and Mero

A result

holds

spaces.

in Mero

the f o l l o w i n g

Unif,

categories

of examples.

obviously

final

constitutes

is no m e a s u r a b l e

In this p a p e r we d o n ' t want case.

which

yields

in Top,

compact

structures

so for d i f f e r e n t

structures

of all s u b c l a s s e s

choice

2.11,

closed

R® K yields

of 2.10).

spaces

structures

Let us remark,

of non finite

spaces w e

such that

(cp. proof

we get d i f f e r e n t

a result

structures.

of M.

Hausdorff

in Mero

K

in Near

structure

class

monoidal

subclasses

K of compact

structure

closed

R : Mero + Near p r e s e r v e s

The r e f l e c t o r

closed

monoidal

com-

form to Heyting

lattices,

hence

topologi-

{~i}<Set.

REFERENCES

[I]

Adamek,

[ 2]

Booth,

Herrlich,

Comment.

Strecker:

Math.

Tillotson: categories 35-53,

[ 3]

Cincura:

[ 4]

Eilenberg,

Univ.

Monoidal

Least

and

Carolin. closed,

of t o p o l o g i c a l

largest

initial

2:0,1, 43-58, cartesian

spaces,

1979.

closed

Pacific

competions,

and c o n v e n i e n t

J. Math.,

88,

1980.

Tensorproducts

Comment.

Math.

Kelly:

Algebra,

in the c a t e g o r y

Univ.

Closed

La J o l l a

Carolin.

categories, 1965,

20,

of t o p o l o g i c a l 431-446,

Proc.

Springer

Conf.

Verlag,

spaces,

1979. on C a t e g o r i c a l

Berlin,

1969.

114

[ 53

Fr61icher:

Durch Monoide erzeugte kartesisch abgeschlossene

Kategorien, Seminarberichte aus dem Fachbereich Mathematik der Fernuniversit~t, Nr. 5, 1979.

[ 6]

Herrlich: Topologische Reflexionen und Coreflexionen, Lecture Notes in Math.

78, Springer, Berlin,

1968.

[ 7]

Herrlich: Topological structures, Math. Centre Tracts 52,

[ 8]

Isbell: Uniform spaces, Mathematical surveys No. 12, American

[ 9]

Kannan, Rajagopalan: Constructions and applications of rigid

[10]

Linton: Autonomuous equational categories, J. Math. Mech.

[11]

MacLane: Categories for the working mathematician, Springer,

Amsterdam,

59-122, 1974.

Mathematical Society, Providence, XI, 1964.

spaces, Advances in Math.

637-642,

29, 89-130,

1978. 15,

1966.

Berlin, 1977.

[12]

Pavelka: Tensorproducts in the category of convergence spaces,

[13]

Tholen: Semitopological functors I, J. Pure and AppI. Algebra 15,

[14]

Trnkova: Non-constant continuous maps of metric or compact

Comment. Math. Univ. Carolin.

13, 4, 693-709,

1972.

53-73, 1979.

Hausdorff spaces, Comment. Math. Univ. Carolin. 283-295,

[15]

13,2,

1972.

Wischnewsky: On monoidal closed topological categories I, Proc. Conf. on Categorical Topology, Lecture Notes in Math.

540, Springer, Berlin, 676-686, 1976.

Georg Greve Fachbereich Mathematik Fernuniversit~t Hagen Postfach 940 5800 Hagen I Fed. Rep. of Germany a.l.

THE FUNDAMENTAL GROUPOID AND THE HOMOTOPY CROSSED COMPLEX OF AN ORBIT SPACE P.J. HIGGINS and J. TAYLOR

I. Introduction Let G be a group acting on a topological space X and let X/G be the orbit space. Armstrong and Rhodes

[1,2,10Jhave

shown how to determine ~I(X/G, *) in special

cases from information on HI(X , *) and the action of G.

We show here that the use

of fundamental groupoids in place of fundamental groups provides a more functorial setting for these results and thus simplifies both their statements and their proofs. This in turn opens the way to analogous results in higher dimensions, the fundamental groupoids being replaced by the homotopy crossed complexes of suitable filtrations of the spaces, as in the higher dimensional Seifert-Van Kampen theorem proved in E6,7]. We first list the cases in which the fundamental group of X/G is known to have an easy algebraic description, and the reader will notice that in each case the action of G is assumed to be discontinuous in some sense.

It is this assumption

that isolates the algebraic aspects of the problem from the topological ones, and we will not consider more general types of action except for the purpose of setting up general machinery and formulating some natural questions. We write q:X ÷ X/G for the quotient map, and x.g for the image of x E X under the action of g E G (with x.gh = (x.g),h).

The orbit g(x) = x,G will also be

denoted by x. Case I.

Let X be connected and let the action of G be properly discontinuous

(that

is, every x E X has a neighbourhood Nx such that, for g ~ l in G, N x ~ N ~ g = ~). Then G acts freely, the quotient map q:X ÷ X/G is a regular covering projection, and HI(X/G , x o) is an extension of the group ~l(X,Xo) by the group G.

(See, for example,

Spanier Eli, p.87] ). Case 2.

(Armstrong [I~).

be its polyhedron.

Let K be a connected simplicial complex and let X = ]K]

Let G act on X by simplicial maps.

If X is simply connected

then HI(X/G , Xo) ~ G/H, where H is the normal subgroup of G generated by those elements which have fixed points. Case 3.

(Armstrong [2]).

Let X be a path-connected locally compact metric space.

Assume that G acts so that (i) the stabiliser of each point of X is finite, and (ii) each point x ~ X has a neighbourhood Nx such that Nx N Nx-g = ~ for all g not in the stabilizer of x. Case 4.

If X is simply connected, then ~l(X/G,Xo) ~ G/H as in Case 2.

(Rhodes EI0~).

For any group G acting on a space X, define the fundamental

group of the G-space X as follows.

A path of type g in X at the base-point x is a o -I pair (~,g), where g @ G and ~ is a path in X from x to x -g For fixed g, let o

o

[~,g] denote the set of paths of type g homotopic to ~ relative to end-points.

These

homotopy classes form a group o = o(X,Xo,G ) with respect to the composition defined by

116

[~,g] + [B,h] = [e + B'g-l,gh]. Let o" be the normal subgroup of o generated by all elements L~ _ ~.g-I g] where ~ is a path from x

of the form

to x I and g flixes x I.

If X = IKN is a

o

connected polyhedron with G acting simplicially assumption

that X is simply connected)

These examples relation

the

then ~I(X/G,~ o) = o/o'.

suggest that the fundamental

to group actions,

introduction

as in Case 2 (but without

group behaves rather erratically

in

but we will show that order can be restored by the

of the fundamental

groupoid.

The indications

Izhat groupoids may be

helpful are the following. (i)

The fundamental

group is a functor on spaces with base-point,

whereas

we certainly want to consider group actions G ÷ Top, where Top is the category of spaces without base-point.

To abandon the base-point means

instead of loops and therefore to use fundamental (ii)

Rhodes'

construction

really using groupoids (iii)

seems unnatural

to consider paths

groupoids. in the context of groups; he is

in disguise.

A simple example shows that, even in the case when G fixes the base-point

the fundamental

group does not behave well on passing

suitable fundamental 2, its non-trivial

groupoid does.

Let X be a circle and let G be cyclic of order

element g acting by reflection in a diame1:er.

are Xo, x I then the fundamental

group ~l(X,Xo)

a group of order 2, whereas

the fundamental

groupoid of X with respect

If the fixed points

is infinite cyclic, with generator ~,

say, and G acts on this group with ~.g = - ~.

the fundamental

to the orbit space, whereas a

The result of killing

this action is

group ~I(X/G,~ o) is trivial.

However,

to the pair of points x0, x I (see §2 below

for definitions)

is a free groupoid with two vertices Xo, x I and two generators ~, B

from x

The group G acts on this groupold with ~-g = B, B'g = ~ and the result

to x I. O

of killing this action is a free groupoid with two vertices from x

to ~I.

Since X/G is a closed interval

O

groupoid

~o' ~i and one generator

joining x

and Xl, its fundamental o

relative

to x

and Xl, is also free on one generator and we see that the O

algebra models

the geometry more closely.

obtained by retricting attention 2.

Fundamental

The fundamental

group of X/G is now

to a particular vertex after

killing

groupoids

We recall that a groupoid is a small category

in which all morphisms

We refer to vertices and arrows rather than objects and morphisms, notation.

the action of G.

The fundamental

groupoid ~I(X,X o) of a space X relative

(to be thought of as a set of base-points) from x to y is a homotopy class, relative

has X

and we use additive to a subspace X °

as its set of vertices;

o to end-points,

have inverses

an arrow

of paths from x to y.

Addition is induced by the usual addition of paths. Given a G-space X, let X fundamental

o groupoid ~I(X,Xo).

be a subspace stable under G.

Then G acts on the

Also, G acts trivially on the groupoid ~I(X/G,Xo/Go),

and the quotient map q:X ÷ X/G induces a morphism of G-groupoids

117

q" : ~I(X,X o) ÷ ~I(X/G,Xo/G). Now, if T is any G-groupoid, groupoid

we can "kill the action of G" to obtain a new

F H G on which G acts trivially.

There is a canonical morphism of G-

groupoids T : F + Fff G and the construction property

: every G-morphism P : F ÷ A, where A is a G-groupoid with trivial action

of G, factorises co-completeness generators

through T.

The existence

of such a groupoid is ensured by the

of the category of groupoids.

and relations,

x E F, g ~ G.

However,

be warned that, although is not necessarily

then

If F is presented as a groupoid by

r ~ G is obtained by adding the relations x = x-g for

the reader unfamiliar with the algebra of groupoids the canonical morphism

r : F ÷ F~G

[5]).

should

is surjective,

a quotient groupoid of F by a normal subgroupoid.

is in fact a fibration of groupoids of F H G

is characterised by the universal

FffG

(The morphism

We shall give an explicit

description

later.

Using this purely algebraic

construction we now have, in all cases, an induced

morphism of groupoids q, : ~I(X,Xo)~ G ~ ~I(X/G,Xo/G)2.1 THEOREM.

Let X be a CW-complex and let X be its O-skeleton. Let the group G o act on X by cellular maps in such a way that if an element g of G stabilises a cell then it fixes that cell pointwise

(in other words,

element of G is a subcomplex of X). q, : ~ I ( X , X o ) H G

the fixed-point

set of every

Then

÷ ~I(X/G,Xo/G)

is an isomorphism of groupoids. Proof.

The fundamental

group of a CW-complex

if the complex has more than one 0-cell, of a maximal statement

tree in the l-skeleton.

is more straightforward.

can be computed as its edge-path group;

the resulting presentation

For the fundamental

involves the choice

groupoid the corresponding

First choose an orientation

for each l-cell and

2-cell, and a base-point in X for each 2-cell. Then the groupoid nI(X,X o) has a o presentation with one vertex for each 0-ceiL one generator for each oriented 1-cell (with end-points

as in the complex)

by the attaching map on its boundary

and one relation for each oriented 2-cell, (starting at the base-point).

deduced from the standard result for the fundamental using R.Brown's

and base-points

theorem

without changing

[4].

in the theorem, we can choose orientations

which are compatible with the group action.

the characteristic

[12]).

group, or it can be easily proved

groupoid version of the Seifert-VanKampen

If G acts on X in the manner described

given

This fact can be

We can also re-choose

maps of the cells so that they are compatible with the action of G,

the topology of X (see Bredon

With this renormalisation,

[3], Ch. II; the details are given in

two things are apparent.

Firstly,

the orbit

space X/G is a CW-complex with one n-cell for each G-orbit of n-cells of X, and with characteristic

maps induced by the normalised

characteristic

acts, not only on ~I(X,Xo) , but on its standard presentation

maps of X. described

Secondly, above;

it

G

118

permutes the vertices,

the generators and the relations.

The result of killing this

action is therefore a groupoid with one vertex for each G-orblt of 0-cells, one generator for each G-orbit of 1-cells and one defining relation for each G-orbit of 2-cells, given by the attaching map for any cell in the orbit. standard presentation of ~I(X/G,Xo/G).

But this is just the

[]

Any simplicial action of G on a polyhedron X = IKI satisfies the hypotheses of Theorem 2.1 if we take as its cell structure a suitable subdivision of the simplicial complex K (the first derived complex of K will suffice).

To recover the theorems of

Armstrong and Rhodes in Cases 2 and 4 above it is therefore enough to describe the groupoid F H G ,

where F = ~I(X,Xo) , in such a way that its vertex group at a particular

point can be recognised as the group ~I(X/G, ~o ) in the form described by them.

To

do this we use semi-direct products. If a group G acts on a groupoid F, the semi-direct product F ~ G is a groupoid with the same vertices as F, whose arrows from x to y are pairs (~,g), where g @ G -I and ~ is an arrow in F from x to y.g Addition of arrows is defined by (~,g)+ (~,h) = (~ + y'g

-I

, gh).

[A more symmetric form of this definition is obtained by taking as the arrows all triples

(~,g,B) where ~'g = B.

target is the target of B.

The source of this arrow is the source of ~ and its

Addition is defined by

(~,g,~) + (y,h,6) = (~ + y-g

-i

, gh, B'h + ~).

The asymmetric definition omits the superfluous last entry in each triple.]. 2.2 PROPOSITION.

If a group G acts on a groupoid F, then

FHG

= (F ~ G)/N,

where N is the normal subgroupoid of F ~ G generated by all arrows

(Ox, g) , 0 x being

the zero element at a vertex x of F. Proof.

Write E = F ~ G and let ~ : F ÷ E be the canonical injection ~ ~---+ (~,l).

The group G acts on E by the rule (~,g).h = (~.h, h-lgh), and ~ is a morphism of Ggroupoids.

The normal subgroupoid N of E generated by all arrows

a G-subgroupoid,

since G permutes these arrows.

(Ox,g) is clearly

Hence E/N inherits an action of G,

and the quotient map p : E ÷ E/N is also a morphism of G-groupoids.

We therefore

have a canonical morphism of G-groupoids T = p O ~ : F ÷ E/N and we show that this has the universal property necessary to identify E/N as F H G. (This gives an independent proof that F H G exists). First,

the action of G on E/N is trivial because,

in E,

(~,g).h = (~.h, h-lgh)

= (0x, h -l) + (~.g) + (Oy,h) for appropriate vertices x,y of F, and the first and last terms of this sum are in N.

119

If now 0 : F + A is any morphism

of G-groupoids, e ---+

F

where

G acts trivially

on A,

&

Z

;L 2/N 0

then the map 0 ~ : E ~ A defined 0"(~,g)

is a morphism of groupoids is unique

by

= e(~)

of G-groupoids.

The kernel

subject

surjective.

to this last equation,

But,

for an appropriate

we only need to show that T : F ÷ Z/N is

(~,1)

+ (Ox,g)

vertex x, so

p(~,g)

= p(~,l)

= r(~).

[]

For any vertex x ° of F, the vertex from Xo to Xo, that is, all pairs

(0x,g), where

group E(x o) of E at x

(~,g) where

group N(x o) of N is easily

Xo in E of loops

N, so there is a morphism

O, o T = ~. To see that e,

in E,

(~,g) =

The vertex

of e ~ contains

0, : E/N ÷ A such that e, o p = e ", whence

consists of all arrows o -i ~ is an arrow of F from Xo to Xo'g

seen to be generated

g fixes

the vertex x.

u = (~,h) + (Ox,g) + ( - ~ ' h ,

by all conjugates

Such conjugates

lying at

are of the form

h-l),

-i where h is an arbitrary element of G, and ~ is an arrow in F from x to x h -i -i o -I Writing x I = x-h , gl = h g h , we see thatgl fixes x I and u = (~ - ~'gl , gl)It is now clear that, precisely

that ~l(X/G,Xo) theorem

in the case F = ~l(X,Xo),

Rhodes' groups o(X,XoG) ~ o/o',

but in the more general

in Case 2 is an immediate

The results

in Cases

namely,

large G-stable groupoid

q, : ~ I ( X ) H G

computational

simplified),

in terms of groupoids

and the conclusion

is the same in all

is an isomorphism

for sufficiently

÷ ~I(X/G) is an isomorphism.

(*) is equivalent, + gI(X/G,Xo/G)

by a purely

algebraic

is an isomorphism

argument,

to the assertion

for all G-stable

for each g E G, at least one point of each path-component

set of g (a condition

possible.

Armstrong's

X . In particular, writing ~I(X) for the full fundamental o we have, in Cases I-4 and under the conditions of Theorem 2.1,

that q, : ~I(X,Xo)ffG

point

of CW-complexes.

1 and 3 also admit a reformulation

that q, : ~ I ( X , X o ) H G ÷ ~ I ( X / G , X o / G )

The assertion

which contain,

context

recover his theorem

subspaces

~I(X,X),

(*)

E(x o) and N(x o) are

We therefore

corollary.

(though the proofs are not thereby cases,

the groups

and o'(X,Xo,G).

purposes

which

is satisfied

by the X

subspaces

X°

of the fixed-

of Theorem 2.1). For o one would use this form of the assertion with X as small as o

120

The interesting question remains: conditions condition follows)

is the assertion

that X be simply connected if X satisfies

space exists. 3.

if a group G acts on a space X, under what

(*) valid?

For example,

as indicated

in Case 3 can be omitted

local conditions

ensuring

in [2], the

(and the conclusion

that a simply-connected

It is not clear whether these conditions

(*)

covering

are really necessary.

Crossed complexes The groupoid version of theSei[ert-VanKampen

fundamental groupoid functor preserves

suitable

result and is of the same general type.

theorem

Theorem 2.1 uses this

It was proved in L7] that the homotopy

crossed complex of a filtered space also preserves generalisation

E4] states that the

colimits.

suitable colimits,

giving a

of the Seifert-Van Kampen theorem to higher dimensions.

that crossed complexes

can be used similarly

We now show

to prove a higher-dimensional

analogue

of Theorem 2.1. If X is a filtered space X complex

C X I C ... C X C ... C X, the homotopy crossed o n of ~ is an algebraic structure with the following constituents. In

~

dimension

| lies the fundamental

groupoid

~i~ = ~1(Xl,Xo)-

In dimension n > 2 lies

the family of relative homotopy groups ~ n X~ = {~n(Xn'Xn-I 'x); x ~ X o }which we view as a (disconnected) : ~n X~ +

groupoid with X ° as its set of vertices.

~n-IX

(n ~ 2) and an action of ~I~ on

There are boundary morphisms

~n~ (n > 2).

The algebraic

laws

which hold in this example and are used as the defining laws of a crossed complex are written out fully in [83 in this volume.

In sursnary, they say that ~2~ is a crossed

module over ~I~, that ~n2 X acts trivially on

z X (n ~ 3), that 6 preserves

the action

n ~

of ~i~ and that ~6 = 0. If a group G acts on X by filtered maps,

that is, each X

is G-stable,

then G also

n

acts on ~ . category,

Since crossed complexes are equationally

defined,

they form a co-complete

so we can kill the action of G, as in Section 2, to obtain a crossed complex

(~X)~ G with the obvious universal property.

Again,

induces a morphism of crossed complexes q, : ( ~ ) ~ G 3.1 THEOREM.

the quotient map q : X ÷ X/G + ~(~/G).

Let G be a group acting on a CW-complex X and satisfying

of Theorem 2.1.

the conditions

Let X be the filtered space determined by the skeletons X (n ~ 0) of n

X.

Then q, : ( ~ ) ~

G ÷ ~(X/G)

is an isomorphism of crossed complexes. Proof.

As in the proof of Theorem 2.1, we may choose base-points,

characteristic

maps,

orientations

for all cells of X, compatible with the action of G.

and

Then X/G

becomes a CW-complex whose cells are the orbits of cells of X with base-points orientations

and characteristic

maps induced by those of X.

Now the cell structure of X gives rise to a standard presentation complex ~Xp which can be described as follows.

of the crossed

~i~ is the free groupoid with one

vertex for each 0-cell and one generator for each (oriented)

1-cell.

~2~ is the free

121 crossed module over ~I~ with one generator corresponding

to its base-point,

the boundaries fundamental generator

of the 2-cells.

for each 2-cell,

located at the vertex

with ~ : ~2~ ~ ~I~ given by the attaching maps of We write H for the groupoid ~iX/6~2~

(which is the

groupoid ~I(X,Xo)).

for each n-cell,

For n > 3, ~ X is the free H-module with one nlocated at the vertex corresponding to its base-point,

with

6 : ~ X + ~ X given by the attaching maps of the boundaries of the n-cells, using n~ n-i ~ the homotopy addition lemma. This description of ~X follows from classical results, or it can be deduced from Corollary 5.2 of [7]. compatible with all the constituents presentation n-cells.

(~/G).

It is clear that the action of G is

of this presentation

of the same type with one generator

This latter presentation

so that ( ~ ) H G

has a

in dimension n for each orbit of

is equivalent

to the standard presentation

of

[]

In order to apply this result one needs to analyse

the crossed complex

(~)H

G.

If C : ...

> C

~ > C

n

~ ...~

n-I

C2

~

> C1

is any crossed complex and G acts on C, we form the crossed complex D = C ~ G as follows.

First, D I = C I ~ G as defined in Section 2.

For n > 2, D n = Cn, with

boundary map D 2 + D I given by x~---+ (6x, l) and the action of D I on D n given by x(~,g) = x~.g = (x.g) ~'g . The following extension of Proposition

2.2 is easily verified

(the details are in

DE]). 3.2 PROPOSITION. complex D = C ~ G

If the group G acts on the crossed complex C, then the crossed is a quotient complex of C ~ G given by

(i) D I = (C I ~ G ) / ~ w h e r e all elements

(Ox,g)

N 1 is the normal subgroupoid of C 1 ~ G generated by

(x E Co, g E G) as in Section 2;

(ii) D 2 = C2/N2, where N 2 is the normal Cl-subgroupoid

of C 2 generated by all

elements cU-c (c E C2, u E NI) ; (iii) for m > 3, D m = Cm/Nm, where Nm is the Cl-submodule elements

of Cm generated by all

cU-c (c E Cm, u E NI) ;

(iv) the boundary maps and the action of D I are induced from those of C. In particular, Applying

the canonical map qn : Cn + Dn is a surjection this proposition

to the situation

for

n ~ 2. [ ]

of Theorem 3.1, and translating

the

result into the language of groups we obtain the following recipe for the second relative homotopy group of the orbit complex. 3.3 COROLLARY.

If G acts on the CW-complex X as in Theorems

2.1 and 3.1, then

~2(X2/G, XI/G , ~) ~ ~2(X2, El, x)/K, where K is the normal ~l(Xl,X)

- subgroup of ~2(X2,Xl,X)

c-g-c and ca-c, where c ~ ~2(X2,Xl,X), a ~ Ker[~l(Xl,X ) ÷ ~l(X1/G,x)]. In principle,

~2(X2/G,x)

generated by all elements

g is an element of G fixing x, and

[]

can now

be

found as the kernel of the boundary map

122

: ~2(X2/G,XI/G,x) ÷ ~l(Xl/G,x), but this may be difficult because the relative group is usually rather large and the Reidemeister-Schreier method is needed to obtain a presentation of the absolute group. We observe that Corollary 3.3 and similar results in higher dimensions will be deducible whenever the assertion (**)

q, : ( ~ ) N

G ÷ ~(~/G) is an isomorphism

is true, and it is natural to ask under what more general conditions this is so. For example, it would be interesting to know whether conditions similar to Armstrong's in Case 3 together, perhaps, with a homotopy fullness condition on X, would be sufficient for (**) to hold. Acknowledgement.

The second author is grateful to the Science and Engineering

Research Council for a Research Studentship covering the period when this work was done. References I. M.A.Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965) 639-646. 2. M.A.Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968) 299-301. 3. G.E.Bredon, Equivariant Cohomology Theories, Springer Lecture (1967).

Notes in Math. 34

4. R.Brown, Groupoids and van Kampen's theorem, Proc.London Math. Soc. (3) ]7 (1967) 385-401. 5. R.Brown,

Fibrations

of groupoids, J.Algebra 15 (1970) ]03-132.

6. R.Brown and P.J.Higgins, On the algebra of cubes, J.Pure Appl. Algebra 21 (1981) 233-260. 7. R.Brown and P.J.Higgins, Colimit theorems for relative homotopy groups, J.Pure Appl. Algebra 22 (1981) 11-4]. 8. R.Brown and P.J.Higgins, Crossed complexes and non-abelian extensions, this volume. 9. P.J.Higgins, Notes on Categories and Groupoids, Van Nostrand Mathematical Studies 32 (1971). I0. F.Rhodes, On the fundamental group of a transformation group, Proc.London Math. Soc. (3) 16 (1966) 635-650. I]. E.H.Spanier, Algebraic Topology, McGraw-Hill Series in Higher Mathematics

(1966).

12. J.Taylor, Group actions on m-groupoids and crossed complexes and the homotopy groups of orbit spaces, Ph.D. thesis, Univ. of Durham (1982).

Department of Mathematics, University of Durham, Science Laboratories, South Road, Durham DHI 3LE, U.K.

Minimal

Topological

Completion

KBanI ÷ KVecx)

of

Rudolf-E.Hoffmann

The concept

of a "norm"

real numbers)

or

"topological"

notion

obvious

K=C

forgetful

from the c a t e g o r y K-linear

maps

of K - v e c t o r since

(i.e.

(in a n o n - t e c h n i c a l

KNorm +KVec Norm) of

KNorm

(or

balanced Thus

spaces

and K - l i n e a r

there

category

it seems

X

maps

fails

However,

the

a

the

is no t o p o l o g i c a l

(proposition

natural

to ask

of the faithful completion

functor

E:A ÷ B with V ~ T . E functor

functor

, then there exists

1 -join

functor,

([S] 11.3,

Norm + X into any

functor

of the

"minimal

Norm ÷ V e c .

functor

of a faithful

functor

T : B ÷ C and a full and

such that,

T':B'÷ C

and

(or Vec )

1o).

completion"

of a t o p o l o g i c a l

the

for a d e s c r i p t i o n

topological

consists

spaces

to be a t o p o l o g i c a l

in Norm,

the product

topological

E':A +B'

sense).

normed K - v e c t o r

The m i n i m a l

topological

(i.e.

is c e r t a i n l y

one into the c a t e g o r y ~Vec

of norm at most

Indeed,

space V over K = R

numbers)

functor

it does not p r e s e r v e

p.196).

on a v e c t o r

the complex

whenever

and some a full and

VST'.E'

V:A ÷ C faithful

for some

full and faithful faithful

functor

functor F:B ÷ B'

such that T'-F~T, Up to e q u i v a l e n c e s functors), it always raining

the given

topological

though

relies

The key in

and natural completion

transition

universe

isomorphisms is u n i q u e l y

to a higher

U as an element)

need not be U-legitimate. [He],

F.E~E'.

(of categories)

the m i n i m a l exists,

and

upon

ideas

of P h . A n t o i n e

[Ho5]

and

[A]

given

(cf.also

[He] is based

U+

(con-

may be necessary,

The construction,

idea of

universe

(of

determined;

in [Ho5]

since and

[W] ).

upon an o b s e r v a t i o n

[HoI~ :

A functor

T:A ÷ ! is a t o p o l o g i c a l

a com p l e t e

lattice

functor

(made into a c a t e g o r y

iff A is e q u i v a l e n t

in the usual way,

to

cf.

[ML]

1.2,p.11) - where ! denotes It is readily must

extend

of faithful

the c a t e g o r y clear,

the M a c N e i l l e functors

then,

consisting

completion

V:A ÷ C

of a unique morphism.

that the above m e n t i o n e d of a poset

P

(note that the unique

construction

[MN] to the functor

level

A ÷ ; is

~) The main result

of this paper

is b r i e f l y

announced

in

~o7]P.214.

124

faithful

iff A is e q u i v a l e n t

partially

ordered

Indeed, completion

many

carry

following

Suppose ful

functor i)

has

T'E E:A

E:A

of the r e s u l t s over

result

its r o o t

÷ B is T - d e n s e ,

over

E:A

(E,T)

of V:A

the

characterizing

topological

i.e.

of

functor

-

for e v e r y

[BB] ) :

÷ C, a full ÷ C such

B£ObB

and

faith-

that

the c o - c o n e

B)

U-small)

÷ B) iA6ObA,

The

(cf.also

T:B

÷ B in M o r S I A 6 O b A } ,

the M a c N e i l l e

completion.

[B]

V:A

functor

(not n e c e s s a r i l y

set

f6Mor~}

I)

÷ B is T - c o - d e n s e

is e q u i v a l e n t

This

result,

in the

The

basic

identifying [HOl]

Other

U-small

(i.e.

E ° P : A °p ÷ B °p

to the m i n i m a l

is T ° P - d e n s e ) ,

topological

completion

+ C.

needed

in

necessarily

in a t h e o r e m

a V for a f a i t h f u l

{ (A,f:E(A)

ii)

[B~

+ B and a t o p o l o g i c a l

is T - i d e n t i f y i n g

then

in

to the m i n i m a l

({f:E(A] indexed

to a - not

set).

obtained

terminology

cone/lift",

(see e.g.

authors

independently

used

(following

Bourbaki's

"...-co-identifying"

instead

of

"...-identifying".

such

that

[Ho5] and

"quasi-norm"

("topological

[Ho2] §o for an e a s i l y

of

A

here

"V-(co-)discrete

instead

I.

in

~e] , w i l l

be

following.

on a v e c t o r

and

object")

accessible

terminology) "...-final"

space

functor", is that

V over

K=R

"V-(co-)

introduced

reference).

prefer

"...-initial"

(or " . . . - c o - i n i t i a l " )

or ~<=C is a m a p p i n g

LI? ii:v ÷ [o,~] i)

ll~xll<_ l~l'llxlf 2)

ii)

Jlx+yll < Itxll+ilyLl

for all

x,y6V,16

~< - w h e r e

o.~=o, a . ~ : b + ~ = ~ + b = ~

for a,b6[o,~]

with

a+o

I) In o r d e r to o b t a i n an a p p r o p r i a t e d e f i n i t i o n of a T - d e n s e f u n c t o r E:A ÷ B w i t h r e g a r d to a n o n - f a i t h f u l f u n c t o r T:B + C , o n e has to u s e - a n i n d e x c a t e g o r y ~ w h o s e c l a s s of o b j e c t s is{ (A,f:E(A) ÷ B) IA6ObA, f6Mor~} and w h o s e m o r p h i s m s (Al,f I) ÷ (A2,f 2) are i n d u c e d by t h o s e A - m o r p h i s m s g : A I ÷ A 2 w i t h f 2 E ( g ) = f I. For C=I, this is the c l a s s i c a l d e f i n i t i o n , due to J . R . I s b e l l [I] (for full e m b e d d i n g s E:A ÷ B, "left a d e q u a t e s u b c a t e g o r i e s " ; cf. also [U] w h e r e the term--"dense" is used).

125 The

category

expansive

m a p s 3)

of

"quasi-normed"

is d e n o t e d

K-vector

spaces

and non-

~QNorm or, m o r e simply,

by

by

QNorm. It c o n t a i n s

the c a t e g o r i e s

KNorm of n o r m e d tively, maps

vector

as

full

of n o r m

usually

spaces

over

subcategories

at m o s t

one

~Ban I

and

K and

Banach

- where

spaces

over

the m o r p h i s m s

3) . In o r d e r

to s i m p l i f y

K,

are

respecthe K - l i n e a r

notation

we

shall

write

Norm and Ban I instead

~Norm and EBanl, r e s p e c t i v e l y .

of

The

obvious

forgetful

functor

from

~N~minto

the c a t e g o r y

Vec of v e c t o r s p a c e s o v e r ~ and K - l i n e a r m a p s is d e n o t e d by U:QNorm ÷ Vec. 2.

Lemma : For ~<-linear E-linear

spaces

maps,

and

A and A i ( i 6 I ) , quasi-norm~[l.lli

((A, IJll),{fi is a U - c o - i d e n t i f y i n g

a family

fjA ÷ Ai(i6I)

on A i for e v e r y

of

i6I,

=(A,tl [I) + (Ai,[[.lli)}iCm)

cone

QN~m iff ll.ll

in

is g i v e n

by

x=sup{ llfi(x)Iri [ici} for e v e r y

(The p r o o f

x6A.

- which

uses

standard

techniques

- m a y be

left

to the

reader.)

It m a y quasi-norm

be n o t e d

that,

for

on the ~<-vector

I=~,

space

A,

we o b t a i n defined

the

"U-co-discrete"

by

~Zx~J = o for e v e r y

N O W one

2)

3)

x6A.

easily

deduces

Equivalently, or

"maps

the

following

lllxII=llI.Ilxll.

of n o r m

at m o s t

I",

IIf (x)[J 2 for e v e r y

x6V.

result

i.e.

Sllx 111

([Ma], [Ho3]).

126 3.

Proposition: The

4.

functor

a)

÷

Vec

is a t o p o l o g i c a l

functor.

Remark:

By the d u a l i t y clear

U:QNorm

theorem

for t o p o l o g i c a l

that all U - i d e n t i f y i n g For v e c t o r

quasi-norm

spaces

lifts

functors

([A], JR]) , it is

exist.

A and B, a linear map g:A ÷ B, and a

II.IIA on A,

g: (A,IE.IIA) ~ (B, It.lIB) is a U - i d e n t i f y i n g

morphism

iff

ll. IIB is d e f i n e d

IIyIIB:inf { IIu IIAIu6A, for e v e r y b)

The

y6B - w h e r e "U-discrete"

g (u) =y}

inf~=~. quasi-norm

II-ll on a v e c t o r

5.

are r o u t i n e

A quasi-norm

space A is g i v e n by

I~II=o.

IIxll=~ iff x+o and (The proofs

by

and may be left to the reader.)

ll.II on a v e c t o r

space v is said to be s e p a r a t e d

iff ilxll = o implies for e v e r y

x = o

x6V.

For a q u a s i - n o r m e d is a linear

subspace.

linear

The

space V over E, Vo:={x6VIIlxll=o}

linear

quotient

space

.=V/ Vsep" receives

the U - i d e n t i f y i n g

Vo

quasi-norm

ll.llsep

~[x]IIsep= IIx llEvidently,

Vse p is s e p a r a t e d

projection

map

with

(v, ll.il) ÷ is a U - c o - i d e n t i f y i n g

6.

regard

to

II.IIsep. M o r e o v e r ,

the

(Vse p, II.IIsap )

morphism.

Lemma : Let

(V, II.II) be a q u a s i - n o r m e d

countable projection

family

of norms

space over E. There

ll.lln(n6N)

maps p: (V,ll.ll) +(Vsep,l]. IIn)

form a U - c o - i d e n t i f y i n g

cone

indexed

is a

on Vse p such that the

over N.

127 Proof : In v i e w

of

5., we m a y

We c h o o s e

a basis

of V such

that

assume

w.l.o.g,

that

(V,I].II) is s e p a r a t e d .

{viii 6 I}tJ {Wkik 6 K} {vili

6I}

is a b a s i s

of the

linear

subspace

v' = {v ~ vl llvll
l[~jw k tln:=n'zlljl 3 we d e f i n e

a norm

]I.]in on the

{Wklk 6 K }

for e v e r y

3

3

linear

subspace

W generated

by

n 6IN.

Obviously,

llv+wlin := IIvll + llwlln where

v 6V'

for e v e r y

and w £ W ,

x 6V,

there

defines

a norm

is a u n i q u e

on V for e v e r y

representation

n 6[q, since,

x=v+w

with

v 6V'

and w 6 W. It is r e a d i l y

II

.lln

(n 6 N)

Since space,

clear

that

]l.li is the

every

thus

normed

space

established

(V, II.II) of

that

is U - c o - d e n s e that

this

7.

Lemma :

relative

embedding

(V, ]].II) be

exists

family

of n o r m s

to

÷

÷

of a B a n a c h

induced

norm,

Vec

It r e m a i n s

.

to s h o w

U-dense.

a quasi-normed

a U-identifying

the

QNorm

U:QNorm

is a l s o

is a s u b s p a c e

(v, II.li) , w i t h

the e m b e d d i n g

Ban I

Let

of the

.

the C a u c h y - c o m p l e t i o n

we h a v e

supremum

linear

space

o v e r ~.

Then

co-cone

({gi: (K, i. I) ÷ (v, li.il)}i~i' (v, ll.fl)) where

[. I d e n o t e s

the o r d i n a r y

norm

on ~.

Proof: Let I={x 6 V I IIxIl~o,~} u { (y,n) 6 vx[~ly 6v, liyll=o} For

x 6V

with

ilx]l~o,~,

we d e f i n e

gx:E 1

gx(1) for e v e r y

I 6K.

For y £ V

with

IIY[I = o

= ~

and n 6[q, let

.x

÷ V

by

there

128

g(y,n) (I) = for e v e r y

(n.l)y

I 6 K.

It is e a s i l y

seen that e v e r y g i ( i 6 I) is K - l i n e a r

expansive.

In o r d e r

to s h o w that

identifying

co-co n e ,

let

({gi}i6i, (V, I!.!I))

f

and non-

is a U-

(w, IIILw)

K ~

Tu (v, II.II)

commute

for some q u a s i - n o r m e d

linear

space

(W, ll.!IW) , some l i n e a r

m a p u : V ÷ W and a f a m i l y of l i n e a r n o n - e x p a n s i v e For x 6 V

with

IIxII+o,~,

u(x)

maps

fi:[ + W.

we h a v e

= Ugx(iIxii)

= fx (HxH),

hence

llu(x)IIw = IIfx(IIxEI)I1w
f X

is n o n - e x p a n s i v e . u(x)

for e v e r y n 6 ~ ,

If

IIxII:o,

then

= U g ( x , n ) (1) = f(x,n)(1)

hence I

[lu(x) iiW = for e v e r y n 6 N ,

Hf(x,n ) (1)IIW< - n

hence

IIu(x)llw In all this completes

says that u: (V, ll. II) ÷

8.

o

above

of the m i n i m a l

together with

topological

completion

6. and 7. n o w y i e l d s

Theorem: The m i n i m a l

topological

completion

~Ban I

÷

of

~Vec

is

~QNorm +~Vec where

This

the proof.

The c h a r a c t e r i z a t i o n tioned

:

(W,II.IIW ) is n o n - e x p a n s i v e .

~BanI

is

canonically

embedded

into

~QNorm.

men

129

9.

Remark: i) It may be noted that a quasi-normed

"separated"

iff every U - c o - i d e n t i f y i n g

vector space

(V, HI.If) is

cone

((v, II.rl) '{fi: (v, il.il) ~ (xi, It.t1i ) }i~i ) in QNormwith domain equivalently, domain

(v,iI.il) separates

the points of

iff every U - c o - i d e n t i f y i n g

(v, il.li)

morphism

(v, ll.ii) or,

in QNorm

with

is one-to-one 4).

ii) A quasi-normed

vector space

(V, li.ii) enjoys the property

that

lixil< ~ for every x 6V U-identifying

iff

(v, il.ll) is "co-separated",

i.e.

iff every

co-cone

({gi: (xi' H .ili) ÷ (v, il.H) }i~' (v, it.11 ) in QNorm is

(i.e. induces)

an epimorphic

equivalently,

iff every U-identifying (v, II.ID is onto 4) iii) A quasi-normed

vector

that every Cauchy sequence is separated

space

co-cone

in Vec

m o r p h i s m with co-domain

(V,~.II) is complete

converges

, or,

in the sense

to a unique point iff

and every U - c o - i d e n t i f y i n g

(v, II.II)

e p i m o r p h i s m with domain

(v, li.ll) in the category

sepQNorm of separated quasi-normed

vector

spaces over K and non-expansive

linear maps is an isomorphism. It results

from

(i)-(iii)

and theorem 8 that Banl ÷ Vec

though it fails to be right adjoint, from its M a c N e i l l e

completion,

is externally

an exceptional

,

reconstructible

phenomenon

for which

no strict parallel is known in any other situation. (The present example plays a great role in the theory of "separated" objects with regard to a topological

functor,

as well as in the theory of "complete"

developed

objects developed

in [Ho3 ]5) in [Ho4]).

It may be also worth pointing out that the separated quasinormed vector topological

spaces and non-expansive

hull of Banl over

Vec

functors and E,M = {epimorphisms} tively)

and that the complete

(in which

Ban I is coreflective)

linear maps

5) A brief indication

(E,M)-

(with regard to the obvious and

{monomorphisms}

(+ separated)

in Vec, respec-

quasi-normed

form the semi-topological

4) This latter concept of a "(co-)separated" functor U:X + Ens

form the

is essentially

due to

of the results

in

vector spaces hull of

object with regard to a [Br] (cf.also [Ho3]).

[Ho3] may be found in

[Ho4] §2.

130

Ban I

Vec

over

(iv) tion

The

does

not

Ban I ÷

cannot

that

(= a m o r p h i s m

abelian

8, s i n c e

U-small

be r e m e d i e d

fibers

by any

an e p i m o r p h i s m

It is easy

completion

Vec

to see that

Vec

has

a

construction.

change

of t h e s e

unchanged.

iff e v e r y and

comple

and

the d o m a i n s

C is b a l a n c e d

is b o t h

the

reasonable

, leaving

topological

by the v e r y

Norm ÷

of

Vec

a category

category

of the m i n i m a l

in t h e o r e m

indeed

co-domain

which

an i s o m o r p h i s m . every

"size"

of t o p o l o g i c i t y

or their

Recall

us

domain,

failure

Vec

functors

of the

concern

U-legitimate

The

- cf. [Ho6]P.73.

question

bimorphism

a monomorphism)

is b a l a n c e d

is

(indeed

is b a l a n c e d ) .

Proposition:

1o.

There

is no t o p o l o g i c a l

functor

KNorm

into

a balanced

÷ X

category

T

orKBan

I ÷

X.

Proof: Suppose,

on the c o n t r a r y ,

is g i v e n

- where

A is e i t h e r

By

1.3 and

2.1,

[Ho 2]

(V, ~.II) o v e r

that

K is T - d i s c r e t e

a topological

functor

T:A + X

or B a n I .

Norm

a normed

(V, II.ll) is an i s o m o r p h i s m .

such

vector

space

iff e v e r y

Clearly,

(resp.,

bimorphism

the n a t u r a l

a Banach

with

space)

co-domain

map

(v, 2. li.lr) + (v, 11. If) (mapping

every

an i s o m o r p h i s m T-discrete

point

identically)

iff V~{o},

object

as one

if and o n l y

is a b i m o r h p h i s m easily

if

Va{o}

checks.

in A.

Thus

This

is

(V, ll.ll) is a

(the v a l i d i t y

of

"if"

is s t r a i g h t f o r w a r d ) . Since T-discrete

This T:A

there

contradicts ÷

is an i s o m o r p h i s m

objects

and

the

fact

I is a t o p o l o g i c a l

lattice

([Hol]) .

between

the

full

subcategory

of

X, we o b t a i n

that

T is a t o p o l o g i c a l

functor

iff

functor,

A is e q u i v a l e n t

since

to a c o m p l e t e

131

References A

Antoine,Ph.: Etude @l&mentaire des categories d'ensembles structur@s. Bull. Soc.Math. Belgique 18, 142-164 and 387-414(1966). Banaschewski,B.: HHllensysteme und Erweiterung von QuasiOrdnungen. Z.Math. Logik Grundlagen Math.2, 117-13o (1956).

BB

Banaschewski,B. and G.Bruns: Categorical characterization of the MacNeille completion. Archiv d.Math.18, 369-377(1967).

Br

Br~mmer,G.C.L.: A categorial study of initiality in uniform topology. Ph.D.thesis, University of Cape Town,1971.

He

Herrlich,H.: Initial completions. Math. Z.15o, 1oi-11o(1976) (also in: "Kategorienseminar" (Hagen,1975), pp.3-25(1976)).

Ho I

Hoffmann,R.-E.: Finaltopologie.

Die kategorielle Auffassung der Initial- und Dissertation, Universit~t Bochum 1972.

Ho 2

--: Topological I-7(1975).

functors and factorizations.

Ho 3

--" (E,M)-universally topological Universit~t DUsseldorf 1974.

HO 4

--: Topological functors admitting generalized Cauchy-completions. In: Categorical Topology. Proceedings of a Conference (Mannheim, 1975), pp.286-344. Lecture Notes in Math.54o. Berlin-Heidelberg-New York: Springer 1976.

Ho 5

--: Topological completion of faithful functors (1975; unpublished). Summary (=§o) in: "Kategorienseminar" (Hagen, 1975), pp.26-37(1976).

Ho 6

--:

Ho 7

--- Note on universal topological completion. gie G&om. Diff@rentielle 2oo, 199-216(1979).

Note on semi-topological

functors.

functors.

Isbell,J.R.: Adequate subcategories. 541-552(196o).

Archiv d.Math.26, Habilitationsschrift,

Math. Z.16o,

69-74(1978).

Cahiers Topolo-

Illinois J.Math.4,

Ma

Manes,E.G.: A pullback theorem for triples in a lattice fibering with applications to algebra and analysis. Algebra Universalis 2, 7-17(1972).

ML

MacLane,S.: Categories for the working mathematician. Heidelberg-New York: Springer 1971.

MN

MacNeille,H.: Partially ordered sets. Trans.Amer.Math. Soc.422, 416-46o(1937).

R

Roberts,J.E.: A characterization of initial functors. 8,181-193 (1968) o

S

Semadeni,Z.: Banach Spaces of Continuous Functions. PWN-Polish Scientific Publishers 1971.

Berlin-

J.Algebra

Warszawa:

132

U

Ulmer,F.: Properties of dense and relative adjoint functors J.Algehra 8, 77-95(1968).

W

Wyler,O.: Are there topoi in topology? in: Categorical Topology. Proceedings of a Conference (Mannheim, 1975), pp.699-719. Lecture Notes in Math.54o. Berlin-HeidelbergNew York: Springer 1976.

Rudolf-E.Hoffmann Fachbereich Mathematik Universit~t Bremen D-28 Bremen Federal Republic of Germany

ON

THE

FREENESS

OF W H I T E H E A D - D I A G R A M S

Michael

I.

H~ppner

Introduction Diagrams

R-Mod

a ring

with

of f u n c t o r Mitchell

a partially

over

severa~

some

[7 ] and in

projective

diagram

analizing

is r e l a t e d

the

following of p o s e t s

the

proof

i.e.

set

there and

values

additive

has

been

sum

apparent

as W h i t e h e a d ' s

as m o d u l e s

category).

theory

diagrams.

a direct

in a c a t e g o r y

be c o n s i d e r e d

of m o d u l e

it b e c a m e

is k n o w n

I with

also

a small

spirit

is free,

to what

R may

(i.e.

~ the

properties

ordered

ring

objects

categories

homological

when

over

of m o d u l e s

was

some

As was

The

that

by

interest

in

shown

[5]

of r e p r e s e n t a b l e s , freeness

property

study

initiated

in

over

any

and

of a d i a g r a m

in a b e l i a n

group

theory. It is w e l l - k n o w n [4],

but

in g e n e r a l

the

usual

ZFC-set

that

the

the

situation

diagrams

are

exactly

there

are

obtained

some

the

ZFC.

abelian

group

Theorem

W. L e t

without

divisible

studied

e~ements,

in

then

over

abelian

track

where

~

one.

and ~

loss

(I,

Therefore,

and

respectively. without

with

of d i a g r a m s

semi-simple

(Proposition

at the

case

2.3.)

Whitehead-diagrams

derive

also

[3]).

Whitehead-

and

easily

following

in For

therefore to be

analogue

of

the

D is

of

countable

rank

free.

ordered

set

of i n t e g e r s

~ have

already

[6 ].

To keep

dimension

hereditary

be a W h i t e h e a d - d i a g r a m

D: I ÷ F - M o d

over

in

we

the

is free

theory.

.~. W h i t e h e a d - d i a g r a m s

categories

In

rank

is u n d e c i d a b l e

of S h e l a h [ 10 ] (see

diagrams

of n o n - f r e e

However,

Whitehead-diagrams been

flat

of c o u n t a b l e

of a W h i t e h e a d - g r o u p

by a r e s u l t

is as f o l l o w s :

examples

within

freeness

theory

diagrams

any W h i t e h e a d - g r o u p

the

poset

denote So I has

the

hereditary group

R-Mod)

theory

that

by a r e s u l t

of

neither

ordered

sets

we will

we

have

to r e s t r i c t

are h e r e d i t a r y ,

I shall

to be a t r e e

of g e n e r a l i t y

posets

Brune

[2]

contain

of n a t u r a l

with

the

assume

of g l o b a l

ring

R has

to be

( ~ + I ) °p nor ~ x ~ numbers

wel~-ordered

further

to

i.e.

R=F

and

interval's a fie~d.

~I<2~, and

134

For every constant

subset

diagram

special

F-Mod

diagrams

diagrams

As

EiX with J=(i}

I i~j}.

is left

to e v a l u a t i o n

adjoint

the

0 elsewhere.

S.X with J={jBI i

÷ (I, F-Mod)

X we define

X on J and value

the q u a s i - s i m p l e

cases we note

representable

S.:

J of I and any v e c t o r s p a c e

AjX with value

and the

Because at i, the d i a g r a m s

i

S.X are projective and every

projective

diagram

P is of the form

1

P= @ SiQ i for some ieI The f o l l o w i n g

spaces

Qi'

definition

i.e.

P is free [5].

is in o b v i o u s

analogy

with abelian

group

theory. 2.1.

Definition.

( W - d i agram)

A diagram

if Ext

By h e r e d i t y Of course, diagram,

the class

any p r o j e c t i v e

because

2.2.

of W - d i a g r a m s diagram

SiF ÷ (SiF)

the o b v i o u s We will

Lemma. L e t

duality

is closed

see in a m o m e n t

by taking

subdiagrams.

but so is every

with

respect

from

the spliU~ing of the

, where

with

a Whitehead-diagram

is called ieI.

is a W - d i a g r a m

every

canor~cal e m b e d d i n g and ieI.

for every

S.F is i n j e c t i v e i The last fact f o l l o w s easily

sequences.

denotes

D: I ÷ F-Mod

(D, SiF)=O

:(I,

F-Mod)

÷ (I °p, Mod-F)

D i = H O m F ( D i , F) for every that there

flat

to p u r e - e x a c t

diagram

D

are no other W - d i a g r a m s .

D: I ÷ F-Mod b e a W - d i a g r a m ,

then

D is

a diagram

of

monomorphisms.

Proof.

Suppose

subdiagram

there

is a n o n - t r i v i a l

U of D g e n e r a t e d

0 ÷ V ÷ SiF

÷

U

2.3. the

fo~owing

1) D i s 2) Ext

are

4) D i s

(D, EiF)=

0 for

D: I ÷ F-Mod

every

K of I b o u n d e d

0 for every

keK,

a

be a d i a g r a m

of

monomor~hisms.

ieI

a monomorphism

for

every

ieIand

J= { j e I

f~at

(D, S I X ) = 0 f o r

Proof.

I)=> 3)

subset

equivalent

÷ D is J i

every

ieI

and s p a c e

X.

2) by heredity.

Set ~ = { j e I

the

o

5) Ext

2)=>

(U, SkF)=

consider

sequence

V= @ SkF for some keK Ext

then

a W-diagram

3) l~m D jeJ

(V, SkF)=

Let

Proposition.

(Di÷Dj),

by x and an exact

÷ O. B e c a u s e

by i, we have F= Hem contradiction,

xeKer

i j~i}

and c o n s i d e r

0 ÷ EiF ÷ A~F ÷ Ajf ÷ 0

and

the exact

sequences

j
Then

135

+ Hom (D, Note that

&TF) ~ Hom (D,

f may be i d e n t i f i e d

with

l&m D + D w h i c h t h e r e f o r e jeJ J i 3)=>

components

induction

on the n u m b e r

bounded 4)=>

+ Ext dual

(D,

of

EiF)

the

sum taken

over

the

of J. Now use the f l a t n e s s of g e n e r a t o r s

2.4.

Corollary.

then

0 is

(right

morphism

filtered

criterion

of a l e f t - o p e n

criterion

)

in [6 ] and

subset

for p u r e - i n j e c t i v i t y

has a lot of i n t e r e s t i n g

Let

D: I + F-Mod

K of I

above,

o

consequences.

~e a f i n i t e l y

presented

W-diagram,

free.

For any W - d i a g r a m (I, F-Mod)

to be free

has to be perfect.

this is e q u i v a l e n t (I °p, Mod-F)

(projective)

By a w e l l - k n o w n

to the c o n d i t i o n

become

the c a t e g o r y

that all

theorem

By P r o p o s i t i o n is a W - d i a g r a m

iff for every so being

is an a n a l o g o u s

result

studying

freeness

category

(I, F-Mod)

The f o l l o w i n g

F:

becomes

iff in

I is

diagrams

Note

I with

that

artinean.

[6] a d i a g r a m

of D to J= {jeI

is a Loca~ property.

posets

by M i t c h e l l

Since

D

I j~i} there

[9],

we may

a maximal element o when

by this r e s t r i c t i o n

the

noetherian.

without

divisible

are all n o n - f r e e

~op

criterion

the r e s t r i c t i o n

for p r o j e c t i v e

of W - d i a g r a m s .

Counterexamples.

b) D(2)=Z~

ieI

a W-diagram

to h e r e d i t a r y

3. W h i t e h e a d - d i a g r a m s

8]

(S i F)ne ~ n

and the f l a t n e s s

is a W - d i a g r a m ,

ourselves

[I,

chains

stationary,

2.3.

restrict

of d i a g r a m s

of Bass

descending

2.5. C o r o l l a r y . Anu W - d l a g r a m i n ( I , F-Mod) i s f r e e

3.1.

+

canonical

by i.

5) by the s p l i t t i n g The p r o p o s i t i o n

in

the

must be m o n o m o r p h i c .

4) l~m Dj= @ ( U D.) with jeJ Z jeZ J

connected

&jF)

elements

W-diagrams.

a) D (I)= H S F: ne]N n

~op

÷ F-Mod.

+ F-Mod.

~N°P c) Let I=

]Nx7 be o r d e r e d

the s u b d i a g r a m elements Since examples

by

(n,i)<

(m,j) iff m
D (3) of D=

H S(n,2)F: I ÷ F-Mod ne]N (1,1 .... )eD(n,1 ) and leD(n,2 ) • any n o n - a r t i n e a n

may

be t r a n s f e r e d

poset

I must

to any such

contain

generated

~op

Then

consider

by the

the first

set by K a n - e x t e n s i o n s .

two

136

The any

second

abelian

in a free discrete diagram

example

direct

summand

valuation D (2),

divisible

is not

ring

see

a counterexample

G is separable,

W-group

of G. the

On

the

quotient

elements

to o b t a i n

free

of d i a g r a m s

be seen

3.2.

but

non-trivial a)

should

Definition.

there

b) for

3.3.

cannot

Let

element

in

the

be b r a n c h e d

D: I + F - M o d

countable and

many

ne~ there

of G is containe,

for

( which

While

third

same

as

any

we have

the

that

complete

is a n a l o g o u s

seems

light.

as w i l d

to the

to e x c l u d e

second

example

to be a s p e c i a l i t y

Note

(1,1,

that

an

element

)eD(3~" "'" ~ o,I)"

(of m o n o m o r p h i s m s ) .

A

dZv~sZb~e, if e i t h e r chain

are

J in I such

pairwise

x, eD. such K Jk

Let D: I + F - M o d

Proposition.

Remember

element

hand,

be a d i a g r a m

xeD. is c a l l e d l descending

non-trivial

really.

So in g e n e r a l

the

is a c o u n t a b l e

jl,..,Jn
field

modules.

counterpart,

diagram

every

other

[6]) is a W - m o d u l e .

has a m o d u l e - t h e o r e t i c

of a free

i.e.

that

that

xe R D. or jeJ J

non-comparable x=x1+...+x

be a W-dZagram,

n

elements

.

then the f o ~ o w Z n g

are equivalent I)

D does not

2)

For e v e r y

have ×eD i

d~visib~e there

are pairw~se

~ Dk such and X l e D g•l ~ k<Jl 3) D i s Proof. 2)=>

e~ements.

that

non-comparable

3)=>

Because

I)=>

2)

are

direct

obvious.

^

decompositions

D is a W - d i a g r a m

there

Djl

. Djl~

=

the

n

n

morphism ^

D ÷

that

n

^

f:

n

D=

D +

XleDjl

@ S. D. 1=I J1 31

^

@ Eo D. along 1=I J1 Jl

"

@ S. O. ÷ ~ E. O. . M o r e o v e r 1=I J1 J1 1=I J1 J1

the p r o j e c t i v e

^

~ S. O. ~ Ker 1=I J1 31

f

and

^

n

xe

^

obvious

Z D k such k<j I

is a t,omomorphism n

lifts

cover

jl,..,Jn~i

separable.

3) C h o o s e

which

e~ements

x=x1+. " .+x n .

@

S.

i=I

D.

J1

In

.

J1

the

following

sense

the

remaining

first

example

is of

too g r e a t

cardinality. 3.4.

Definition.

the rank of D.

Let

D:

I ÷ F-Mod

be a d i a g r a m .

Then

dim F D O is c a l l e d

137

Now Proof

we are

prepared

of T h e o r e m

by i n d u c t i o n

W.

there

Let are

for

the

(x

I n e ~ } be a g e n e r a t i n g set for D . Then n I ~ o free direct s u m m a n d s p~nj of D such that

D= P ( 1 ) e . . . P ( n ) e D ( n )

and

is a pure

of D the

(D/P)

o

subdiagram

= 0 we must

In a b e l i a n

Pontryag~n's well

for

have

group

.,xnep(1) . . .e quotient

points

..ep(n) diagram

Because is flat.

P= @ p(n) ne~ Since

D= P.

o

theory

Theorem

crZterZon for f r e e n e s s

downward

branching

xI .

filtered

posets

W is u s u a l l y [4].

For

or for

proved

diagrams

posets

with

by e s t a b l i s h i n g

this

seems

finitely

to work

many

only.

REFERENCES [I] H.

Bass:

Finitistic Trans.

Brune:

On p r o j e c t i v e

[2]

H.

[3]

P.C.

Eklof:

Monthly [4]

L.

Fuchs: 1970,

[5] M.

[6]

dimension

rings.

83

(1976)

Infinite

and

Math.

Whitehead's

Soc.

a generalization 95

(1960)

representations problem

of s e m i - p r i m a r y

466-488

of o r d e r e d

is u n d e c i d a b l e .

sets.

Amer.

to appear

Math.

755-788

Abelian

Groups

I,

II.

Academic

Press,

New

York

1973

H~ppner

and

ordered

sets

-,-

Amer.

H. Lenzing: are

: Diagrams

theory.

over

Abelian

Springer [7] B. M i t c h e l l :

free.

Leet.

J.

pure

ordered

Group Notes

Rings

Projective

with

appl.

sets:

Theory Math.

Alg.

a simple

(Proc.

874

several

diagrams

over

20

(1981)

model

for

Oberwolfach

(1981)

objects.

partially 7-12 abelian

Conf.

group

1981),

417-430 Advances

Math.

8 (1972)

1-161 [8]

: Some Bull.

[9]

Math.

: A remark (1981)

[10]

applications

Amer.

S.

of m o d u l e 84

Shelah:

on p r o j e c t i v e s

Infinite

abelian

Israel

groups,

Mathematik-Informatik

Universit~t-Gesamthochschule-Paderborn D-4790 Germany

to f u n c t o r

categories.

867-885

in f u n c t o r

J. Math.

HUppner

Fachbereich

theory

(1978)

categories.

J.

Alg.

69

and

some

24-31

constructions.

Michael

Soc.

Paderborn

Whitehead's 18

(1974)

problem,

243-256

APPLICATIONS

OF

CATEGORY

THEORY

M. Hu.~ek,

Applications usually

two

application one

must

here

one

the

the

structures,which

use

special

such

the

with

of

some

new

and

ordered

stigation

Mannheim

in

theory

in

the

conglomerate

natural

set

Unif

from on

two

or and

category

all

for

emon

topological

the

end

lower I will

subcategories

Unif

of of

inve-

Top. Topology

held

in

monotransformations, properties

some

arranged

Set. By ~

over

monotransformations

n:F----+1K

be

the "greatest

Categorical epi

spaces)

may

possibilities or

show

or b i r e f l e c t i o n s of

closed

and to

topological

coreflection).At

a topological all

Top

(or

definitions

monomorphisms)

in

ordered

(e.g.that

"lattice"of main

straightforward

(also

suggest

coming

a

the

of

and

Conference

K be

structures. I want

any

have

convenient

be

coreflections

a

its

the

spaces

be

structures

and

not

cartesian

Unif

into

repeat

are

not

special

of

properties

need

intrQduced

me

nX

uniform

STRUCTURES

need

spaces

all

concerning

Top. Let

and

components the

ago

I have

Unif the

additional

years

for

of

of

all

some

background

coreflections:

some

category

[Hua].Let

uniform

Unif

functors

see

note

in

conglomerate

results

to

properties

and

coreflections

add

Six

procedure

reflections

category

bound

theory

UNIFORM

Praha

a categorical

in

into

spaces)

category

often

concerning bedded

of

components:

TO

we

de-

(i.e.all

the

F,together

with

order nI<~2

Clearly,k{ bound

and

if

nl=D2o£

for

is a complete

lattice

(i.e.,every

a greatest

glomerate complete

and

The

situation:

dual

lower

bound

in

C of all coreflections the

least

upper

The conglomerate in

the

DI<~2

if

~2=eoU1

the

£.

usual

of

DeC

meet-stable

subfamilies

in

in

K=Unif

~( a n d or

in

has

a

sense)

in M

C and

least and

upper

its

(i.e.,C

8ubconis

also

~4 c o i n c i d e ) .

E of all epitransformations

order for

some

e;

its subconglomerate R of all epirefleetions stable in E . W e s h a l l s h o w n o w a m e t h o d h o w is n o t

~cM

is join-stable

bounds

n:lK---+F is complete

some

R

is

K=Top.

not

ia complete and meetto

find

Join-stable

in

examples E even

that for

C

nice

139

At from

tegory by

first

KI

G'

is

may

of

mean

of

such

mean

X

that

the

into

of

full

constant

which

concrete

G:A

functor

Since

S

is

a clear

strongly way

the

subcategory

of

A generated

G of

be

defined

can

ctively

is

generated

that

G'

is

the

is

the

G'

is

fact

by

G'

The

that

coarsest

the

any

functor

G2

and

maps

follows

that

for

T

for

is

the

than

Y

we (i.e.

a concrete

every

proves IA,then

from

the with

functors

the

order,G'

gene-

G t)

from

full

into

A.The

the

by

shall

than

domain

of

1A follows

the

same

satisfy

the

that

for

same

directly that

G2X

any

G' is extending

if we

domain,and

G2

~(G2X,G'T)) • If

to

a coreflection

construction

that

G t (realize

G2oG2=G2.Similarly,if is

G 1 indu-

Gt.Suppose

prove

f6A(X,T) b e l o n g s to A(G2G2X,G'T)

GI

the

extension

functor

generated IA.We

is b i g g e r the

by

G'

to a

or smaller

or a coreflection.

between

than

map G' from

have

G'.

two

O'
the

(i.e.,Gi
relation

i=1,2). The

constructed

G t (provided fact

extremal

in

f belongs

G',H t in o u r P r o p o s i t i o n

corresponding

G2 from

that

map

than

directly

~

and bigger

also of

proJectively bigger

fEA(X,T) ,hence G2oG2
It

class

can be extended

preserves

domain

such

smaller

G'

(denoted

fact

the

and

by

and

T
object

and

the order

functor

as

idempotent,then

idempotent

the

map

categories

finer

G~

we

I IX I is a m o r p h i s m

to be a bireflection rigid

idempotent

a bireflection.

from

is

map

~A. If G' is idempotent

than 1A, then G can be found

now

G extends

by

in concrete X

K

category

rigid class. Then any concrete

into o b A preserving

G'

generated

the

of

Y~obK1.Dually

a concrete

that

and

subca-

generated

object

functor

categories,F:A

a part of F-i(S)

in

that

o n l y . A concrete

means

Let A,E be topological

functor, ScobE be a strongly

rates

is

G'Y,where

of

order

coincide

functors

a full

Y) .

PROPOSITION.

Proof.

into

maps an

special from

functors. The

subcategory

sets. B y X<[

GX

objects

and

of

a functor

G:K---+K projectively

fEK(X,Y)

the

IXI , IYl

is

X~obK,then

generated class

relation sets

construction O'

functor

if

all

underlying

underlying

from

by

identities

preserving

the

the

inductively rigid

the

c a s e . If

follows:

generated

a class

always

our

K,then

as

a 8trongly

consists

briefly

for

K into

define

By

map

repeat

defined

proJectively one

we

[HUll,adapted

mentioned

reflection The cation

of

G'

is

above

extending following the

functor

G2

idempetent that

in

that

Gt.Dually construction

Proposition

is

and

into

for

the

biggest

bigger case

than there

bireflection /A) ;it is

also

follows the

extending from

the

smallest

bi-

coreflections.

prepares

the

~[ni~ a n d

Top.

situation

for

our

appli-

140

CONSTRUCTION. functor,P that as

Let

be

F(sup

A,~

be

a strongly

T)=F(inf

topological rigid

T)=P

and

categories,F:A

object

+E

a concrete

IPI>I,TcF-I(P) be s u c h SeT,define G~:Tu{supT} ~A

in

~ with

infT~T.For

follows

GS (T)=c'(su°~S ~ T)=S G~(T)=T Then

the

inductive

{GsIS~T} ction

extension

G~

is

a coreflection,the

T,and

G=inf{Gs[S~T}

is

~)=inf{Gs(sU

p T) I S E T } = i n f

T

is o r d e r - i s o m o r p h i c

to

family

not

a corefle-

because G(sup

T)=inf{Gs(inf

G(inf (in

fact,the

one

can

last

proceed

object

(instead In

of

the

uniform

the

to

to

T

we

such

of

T on

IPl) •

{HsISeT} of b i r e f l e c t i o n s sup{HsIS~T} is n o t a b i r e f l e c t i -

that

sup

the

URif

object

a family

assume

identify

subcategory

T) I S E T } ~ i n f

A-discrete

obtain

T~T we m u s t

inf

following

bireflective

is

similarly

in A o r d e r - i s o m o r p h i c on

G S of

T>S, T~S.

if if

~T)

.

Prox w i t h

category

composed

of

all

totally

the

full

bounded

spaces.

Let (Z,_<) be an ordered set without the least (or biggest) element. If K=Top or Prox or Unif,then there is a family {Gzlz~Z} of coreflections (or of bireflections) in K such that {z ~Gz } i8 an order-isomorphism and i n f { G z Iz~Z} in ~,~ is not a coreflection (or, sup{Gziz~Z} in E is not a bireflection, resp.). Proof. Suppose f i r s t t h a t K = U n i { a n d F:I~nif ~lep is t h e s t a n d a r d

THEOREM.

functor. By rigid At

the

space

first

P

we

preceding in

Top

shall

such

add

least

IZI

spaces

of

set

with

be

Zo the

infinite

closed

(by

[T],there

are

u S be

S(z)=[0,~]

for

indeed,if

z
a~S(z)-S(x)

the

in

BNZ=~,BnA=~,then A is

is

far

not

from

finer

B

K=Prox or K=Unif.

embedded

into

smallest

element

rigid

of

topological

space

rigid

topological

.Now,we

shall

identify

cardinality)

P

family

on P

from

and

to

(since

8P b y

corresponding

with

is

US(z)

to

in

ADB=~

completes

US(z) in

S

finer

to

a

than a

to

(Z,~) .

US(x).If

BcC w i t h

B~0

ANBNS(s)~)

but

6P,AnBnS(x)=~),hence

US(z)

the

(since

the

8sP.Then,if

order-isomorphic is

proof

for

P

at

ScZ let 8SP

shrinking

A~a,AAS(z)=~;take B

cardinality

~C-C=cSC-C.For

in

{ U S ( z ) Iz~Z}

S(z)cS(x) is A c C

US(x).This

(F-I(P),<).

be

strongly

proximity

in U S ( x )

strongly

metrizable

A is p r o x i m a l

than

a

C of

obtained

there

find

subspace

of

Z,then

,then

to

the

strongly

subset

z~Z,the

can

Z as

discrete

compactification

suffices

discrete

large

a closed

it

(Z,-<) 0 to

a metrizable

an

arbitrarily

point,and

that

a point

Zu(O)=Zo.Then w e t a k e containing

construction

the

case

when

141

If

K=T0p,we

Top .... y~P-C

with

trace

of

shall

)Top.Take the

use

for

the

again

the

previous

neighborhood

BP-neighborhood

logy

on

IPI

coinciding

Then

[IyNN{UslsES } F(IPI,ts)=P,t S is

functor

system

system with is

U

y

at x

that

the

F

the

sequential

situation

of

P

and

tS(x)

contains

COROLLARY

no

member

of

denote

ScZ

let

on

IPI-(y)

(Z,s)

that

and

ts(z)

There are infinite well-ordered

I.

x the

y.

is

finer

neighborhood is

topo-

that

order-isomorphic

tS(z)

some

U

be

such

at is

by

tS

and

system

again

Ua,SO

the

x~Z

to { ( I P I , ~ Iz~Z}:if z~x in Z , t h e n S(z)cS(x) ts(z) , a n d if t h e r e is s o m e a~S(s)-S(x),then in

a point

for

neighborhood

Hausdorff

modification

P,C,Z,choose

P.For

,and on

with

not

finer

arbitrarily

than of

than

y

tS(x).

long families

of coreflections (or bireflections) on Uni~,Prox or Top the meet in ~ (or join in E) of which is not a coreflection (or a bireflection, resp). By

the

same

F,G

reflections

way such

GoF) . S u c h

functors

GX)

the

T

to

is

be

one that

may

space

a countable

obtains

for

sup(F,G)

be

is

described

proJectively

coarsest

instance not

an

generated

T1-space);then

F~E,~

an

now

that

arbitrary

Fe{.Then

ordinal

F°=1 K, F~+i=FoF ~ , Clearly,Fe
8 for

we

may

the

number,in

define the

e<~.Dually

we

define

by

we

bi-

FoG~ FX (or where

X

of

reals

base

at

xEX,one

the

indiscrete

induction

following

way:

for

~

limit

F ~ for

put

C(X,T)

(or

set

two

(hence

Top

C(X,R)

by for

F~=sup{Fal~E~}

and

of

a bireflection

directly,e.g.in

([x-e,x+~]~{rationals}) u{x} ,e>O,as a n e i g h b o r h o o d GX=X,FX=E t h e s t a n d a r d s p a c e of r e a l s , a n d GR is Suppose

example

F~{

the

with has

space. functors

.

F~>F 8

(then

for

~<8) .

For any ordinal ~ there is a concrete functor G in Unif, P~ox or Top such that l~G~@~...~O~=@~+~and similarly for decreasing

COROLLARY

2.

families. Proof.

Take

functor gical

from space

isomorphic to

~.It

functor

for the P to

proof

there the

is

of a

our

~:K

proJectively

Using

strongly which

now

rigid

"never

rable coreflections case:

put

Pro×

Tap

or

Theorem. For

well-ordered to

+K

or

subset

suffices

a functor

first

Unif

K either

set

~+I

of for

generated

by

proper

some

{T I ~ + l } c F - 1

GtT~= T~+[

stops",or

and

all

for

strongly

(P)

+Top

F:!f

rigid

which

ordinals

is

less

~E~,G'T~=T~,and

the

topolo-

orderor to

equal take

in Top,one c a n c o n s t r u c t e.g. proper olas8 of mutually incompa-

classes a

the

G'.

or bireflections.we

shall

prove

here

only

the

142

There are concrete functors G,H in Unif,Prox or Top such that G a ~ G S , H ~ H 8 whenever ~<8. The

proof

rigid in

is

similar

class

[K]

{PKI~

such

ty

K.It

is

a

that

follows

strictly

Then

the

to

P from

the

the

H.

Some [Hu2].I The is

not

the

F

the

above

the

above for

then

there the

map

e.g.for

has

J~

no

fact

and

~X

in

closed?

the

there

=X required

progets

investigated

such

when,for -

it

in

that

of

supF

a functor

suffices

properties

the

H,

to p u t functors

in

Y the

if

is

like

connected

in

not

either

to

mention

two

first

Categorical

one

epireflective answer

is

no

- the

answers

the

V o l u m e ) .In

fact,we

that point

last

assertion

in

the

E are

one

follows

in of

details

proved have

the

the

posed last

[Inif w h i c h

are

by

year:Is is

contained

same

proved

concer-

assertion

in

cartethe for

Top

more:

There is no nontrivial evireflective subcategory of an epireflective 8ubcategory of Unif or Top which is cartesian closed. THEOREM.

COROLLARY.

E,

empty.

question

Conference

who

in

J2=2a,

(realize the

"categorical"results

subcategory

and

reals

FX=X or FX is i n d i s c r e t e .

new

H.Brandenburg

of

.If we

]J1^J2,21[,]J2,J1V--~2[

with

]JIAJ2,J1[,]Ja,J1vJ2[

Ottawa

this

intervals

indiscrete)

FX>X t h e n

space

To-space)

X

exists

E,J2=sup{F~EIFX=X}

in

together

space

x~U t h e r e

(e.g.the

modification

line

spaces. The

with

the

U and

two-point

indiscrete}

long

JIY

a topological

set

fx=a,f(X-U)={b}

the

the

The

paper

(published

OwX

bireflections

gaps. Take

bireflective

intervals

a nontrivial

joint

cardinali-

notes.

a bireflection

many

with or

the

and

that

would

uniform

F.Schwarz

of

generation,one

examples

and

a strongly

constructed

F-I(P<)

in

G t has

two

a,b~X,any o p e n

bireflections

indiscrete

the

Now,I

sian

contain

points

f:X

means

X=R

Similarly,the

there

two

T1-space

are bar

but

F of

various

not

by

inductive

following

motivation

JI=inf{FEEIF>ITop,FX

where

ning

also is

that define

M,C,E,R w e r e

of

the

families

the

lattices any

coarsest

define

from

for

the

take

spaces

sup{FERIHF=H}) .

type

that

and

must

subspace

Theorem

generated

here

of

discrete

our

properties add

examples

[Hu2]

a continuous or

to

sup{FcRIHF=H}

(see

The such

lattice

like

one

paracompact

{XK,~l~E~,and

families

a bireflection,give

functor

H=sup of

other

would

of

family

G proJectively

decreasing

one,only of

a closed

proof

increasing

functor

preceding cardinal}

contains

perty. Using functor

the

infinite

There is no cartesian closed nontrivial epireflective

subcategory of uniform or topological Hausdorff spaces.

143

The uniform with

other

result

concerns

spaces

having

a base

respect

concrete space take

to

some

all

upper

order).Unlike

with

monotone

respect

maps

bounds),the

to

spaces

to

order

<

greatest is

there

(X,<) (for

lower

quite

(i.e.those

composed

in T 0 p , w h e r e

[fni{

in

covers

assigns

the

preserving

situation

uniform

uniform

Ord-------+Top t h a t

functor

orderable

orderable of

of is

the

convex

exactly

sets one

topological

morphisms bounds

different

O~d

in

and

least

(see

[Hus]

we

for

details):

There i8 a proper class of concrete functors Oad +UnZ{ that assign to (X,<) a uniform space orderable with respect to <.

THEOREM.

We

add

that

between

the

formity

of

to

(X,<)

finest

the

the

the

the

connected

which

orderable

proximity

with can

if

there

from

the

uniform

say

that

is

(X,<)

topology

on

(X,<) , a n d

the

like

biggest

and

to

K is p r o J e c t i v e l y for

mention

two

the

in

fine

functor

uni-

assigning (i.e.,the

categorical

problems

spaces.

category

objects

generated

situated

completion

in

by

K over

K and

Set

is

bounded

of

a cardinal

<

such

that

C;

{Xili~I}cobK,CcC,f~K(~{XilieI},C) gEK(K{Xili~J},C) s u c h t h a t IJI~K,f=goprj ( i . e . , f (ii)

are

topological

(X,<) .

topological

C of

the

Dedekind

of

a topological

a class

Theorem

to

of

I would

last

assigns

compactification)

end

We

(i)

functors

functor

orderable

At

type

all

each

there

is

depends

JcI,

a

on

at

most

K

coordinates). We

say

It

is

then

that

known of

metrizable

is

of

unbounded

if

K is

on

f(g)=a

K is

of

PrOX

and

spaces

(perhaps

the

the

are or

type

of

the

simplest

cardinal,C

proJectively

choose

points

function" if

f

of

the

type

(C is

interval,resp.)

proof

ICl>1,we

two

K.

countable unit

infinite

with

"characteristic means

{~i{

uniform

type

a given

a C£C

category

[Hu 4] t h a t

class

take

the

is

the

the

,but

Top

following:

generates

Top,we

a,b in C a n d t a k e t h e O={g~ (K+)K+Ig(gO)=gO},that

set

EeG,f(g)=b

if g ~ G ; t h e n

f: (K+) <+

~C

does

are

features

categories

of

bounded

not

depend

K coordinates).

QUESTION Can

these

rical the

I.

properties

category Another

with

What

is

by

be

of or

not

of

In

convergent

more

categories

bounded

investigation

structures.

~+1,the

of

characterized

topological is

possible

sequential

generated

special

categories

type?

categorically?What

catego-

depend

that

on

the

fact

type? in

topological

TSp,sequential sequence,and

categories

spaces the

are

deals

inductively

connected

two-point

144

To-space In

which

on

the

Uni~,sequential

uniform also

spaces

every

n a t e s . In

other

spaces

which

again

can

use

~EC((~+I)<,~+I)

depends

QUESTION

the

be

used

for

topological

projectively

inductively

proJectively

Can

also on

above

a general

the

generate

many

perhaps

description

on

of

the

by

the

whole

Top.

metrizable

whole

countably

factorization

countably and

generate

generated

~EU(HZ~,M) ,M m e t r i z a b l e , d e p e n d s

T0p,one

2.

hand

are

Uni{,and

many

property:

coordi-

every

coordinates.

some

additional

sequential

properties

structures

in

any

category?

REFERENCES

[BH]

H.Brandenburg r i e s of U n i f

[Hul]

M.Hu~ek, Construction of s p e c i a l f u n c t o r s and Comment.Math. Univ. Carolinae 8 (1967) 5 5 5 - 5 6 6 .

[Hu23

M.Hu~ek, Lattices of r e f l e c t i o n s and coreflections in c o n t i nuous structures, Categorical Topology (Proc. Int. C o n f . M a n n h e i m 1975), Springer L e c t u r e N o t e s in M a t h . 5 4 0 (1976) 4 0 4 - 4 2 4 .

[~u3]

M.Hu{ek, Categories of o r d e r a b l e Int. C a t e g . C o n f . O t t a w a 19g0.

[Hu b ]

M. H u s e k , F a c t o r i z a t i o n s of m a p p i n g s ( p r o d u c t s of p r o x i mally fine spaces), Seminar Uniform Spaces 1973/74, Academia Prague (1975), 173-190.

[K]

V.Koubek, Each concrete paracompact topological 15 (1974) 6 5 5 - 6 6 4 .

IT]

V.Trnkov~, Hausdorff

and M . H u ~ e k , and Top, this

N o t e on Volume.

cartesian

spaces,

to

closed its

appear

subcatego-

applications,

in

Proc.

category has a representation b y T2 spaces, Comment.Math. Univ. Carolinae

Non-constant continuous mappings of m e t r i c or c o m p a c t s p a c e s , Comment.Math. Univ. Carolinae 13 (1972) 2 8 3 - 2 9 5 .

MathematiCal Institute Sokolovsk~ 83 186 00 Praha Czechoslovakia

of

Charles

University

A Cate$orical

Framework

for Interpolation

Theory

Sten Kaijser and Joan Wick Pelletier*

0.

Introduction Interpolation

theory, a branch of functional analysis which has applications

to partial differential has at its core

numerical

a very categorical

and various duality Briefly,

equations,

theorem

analysis,

and approximation

(the Aronszajn-Gagliardo

theorems, making it irresistible material

the traditional

formulation

considers

by taking pairs of bounded linear maps

linear sum maps

A 0 + A I , B0 + B I .

A 0 N A I ~ A ~ A 0 + A I , where polation space with respect category restricts and Gagliardo (Ao,AI)

to

F

patible couples to the category of Banach spaces interpolation and

F

and

T:

T: A ~ A .

functors

spaces with respect H

to

are, respectively,

(X0,XI))

and

H

A ,

only, is called an inter(A0,AI) .

(A0,AI)

in the

The results of Aronszajn

(see [I]) show that given an interpolation

there exist interpolation

which yield bounded

Banach space

if each map

(A0,AI)

(the category of compatible

~ denotes a monomorphism

to a bounded linear map

for category theorists.

(Ti: A i ~ Bi)

An intermediate

(A0,AI)

theorem '~-G")

pairs of Banach spaces

contained in a third vector space, made into a category couples)

theory,

space

A

with respect to

on the category of com-

(i.e. F(X0,XI) such that

,

H(Xo,XI)

F(A0,AI)

are

= H(Ao,AI)

minimal and maximal among interpolation

= A

functors

having this property. We generalize

the above setting in a natural way by considering

doolittle diagrams of Banach spaces. again a doolittle diagram, not necessarily

Since the "dual" of a doolittle diagram is

this eliminates

the classical difficulty

being a "dual couple" of the Banach couple

shall show that the

A-G result in this context yields

theory of dual functors, *Both authors acknowledge

introduced

(A0,AI)

of .

(A~,A~) Moreover, we

functors related by the

by Mityagin and Svarc [6].

partial support

Research Council of Canada.

the so-called

from the Natural

Science and Engineering

146

I.

A new setting for interpolation theory Let

~

denote the category of Banach spaces over

field) and norm-decreasing linear maps. diagrams in

6

By

~

I

(= real or complex

we denote the category of doolittle

, i.e. objects are pushout-pullback diagrams A0 ~

X~ =

A1

X0

~

~

XI

E0

E1

and morphisms

T: X ~ Y

E0OTooA0 = ElOTlOgl

.

are pairs

( ~ : X i ~ Yi)i=0, I

(We neglect to index the maps

satisfying gi,Ei

by

X

or

~ , hoping

that the context will clarify the meaning.) I.I.

Example.

Let

(X0,XI)

be a pair of Banach spaces each continuously

embedded into some Hausdorff topological vector space section

X0 n X I C F

and let

Z

be the sum A

~

~ .

Let

X0 + XI ~ F .

A

be the inter-

Then

X0

XI ~ is an element of

F •

E

Thus, we see that our setting for interpolation theory con-

tains the usual setting. Given an element respectively,

A

and

X E B E

,

X

and

X

being functors

the embedding functor which sends

X

will often be denoted by

~ ~ B •

In fact, letting

AX,ZX , 6: B ~ ~

be

to the identity doolittle diagram, we have

the following easily verified proposition. 1.2.

Proposition.

6

It is well known that L: B °p × B ~ ~

has left adjoint

E

and right adjoint

A.

~ is a closed monoidal category with Hom functor

(the unit ball of

L(X,Y)

is

B(X,Y)).

Happily, we see that

inherits this feature. 1.3.

Preposition.

is a B-category and, moreover,

~

is a closed monoidal

category. Proof.

We define the ~-valued Hom functor

L: ~op X ~ ~ B

to be the pullback

147

L(X,Y)

of the diagram L (Xo,Y 0 )

L(XI,Y I) The~-valued

lifting

L(XA,Y E) is then obtained by completing the diagram below by

L(X,Y)

forming its pushout: L(X,Y)

~

L(Xo,Y O)

L(XI,Y I) Similarly,

X ® Y E ~

is defined to be the pushout of

XA ® YA

~

XO ® YO

XI ® Y1 where ® denotes the projective tensor product in

~

and

X ® Y E ~

is defined by

taking the pullback of the following diagram to obtain a doolittle diagram: XO ~ YO

X1 @ Y1 " We shall use the notation out that the

T

=

~I

~

X ® Y

X ~ Y = ~(X,Y) = ~ Y

and

X ® Y = ~(X,Y)

of 1.2 is in accord with this notation since X~ = ~

= ~

, pointing , where

.

The adjointness relation

~(X ~ Y , Z) ~ ~(X,L(Y,Z)) is a pushout-pullback

exercise left to the interested reader. ~(X ® Y,Z) ~ ~(X,L(Y,6Z))

Finally,

the adjointness

follows from the above result by using the composition of

adjoints:

L (Y,-)

8

148

It now follows according functors

for functors

~ ~ ~ .

sense that the assignment decreasing.

by

natural way.

,

Let us for a moment A = (Ao,AI) E ~

Conversely,

DF

L(X,Y)

of

F

to

L(FX,FY)

to be strong in the

is continuous

is defined as follows:

transformations

for

and norm-

X E ~

,

made into a Banach space in a

We shall see that the dual functor turns out to be useful in describ-

The Aronszain-Ga$1iardo

Banach space

from

the set of natural

ing the minimal and maximal

Let

Functors will always be assumed

F

The dual functor

DF~ = NAT(F,E~)

2.

to Linton [5] that there is available a theory of dual

functors

consider

and let

A = FA every

A-G functors.

the most general formulation of the

~ = IAI = ~ .

which is an

L(~)-module

A functor

L(A)-module,

A

F: ~ ~ ~

i.e. if

gives us a functor

.

theorem.

picks out a

T: A ~ A ,

~ , ~

A-G

then

T: A ~ A.

Now the restriction

functor U:

~

has both left and right adjoints, inclusion

~ = ~ , La~

calculate

these funetors directly.

2.1. and

Ra~F~

Theorem.

, Ra~

the left and right Ken extension along the

Let

Proof.

F 6 8~

= LL~(L(X,~),A),

module tensor product over

~ as they are usually denoted.

, X £ ~

where LA

The first statement

®LA

and the

, A = F~ . and

LL~

LA-module

Then

In fact we may

La%FX

= L(~,X) ~L~A

denote, respectively, linear maps

(see [4]).

follows from the two isomorphisms

of which is proved in [4], the second of which follows

the Banach

below, the first

from the Yoneda lemma avail-

able in this setting [5]: NAT ~(L (X, -) ®I~A,G) ~ LI~(A,NAT (L (A,-), G)) ~ LL~(A ,G~) = LLX(FA,GA ) =

NAT~N(F , UG) .

To prove the second statement we note as above that NAT~(Ug,F)

= LI~(GA,FA ) = LL~(G~,A ) ,

and we proceed

: NAT ~(G,LL~(L(-,A),A))

to define an isomorphism

~ LI~(GA,A)

149

and its inverse where

y .

We define

~(t)(a) = t~(a)(l~)

t: G ~ LI~(L(-,~),A) a E GA , S ~ LL~(GA,A)

to the reader the tedious verification that

~

and y(S)~(x)(g) = (S,Gg(x))

, x ~ GX ,

and

~

g E L(~,A) .

,

We leave

are inverses.

[]

We remark that the Kan extensions give us a situation similar to the A-G theorem since functor

(I) e a ~ F ~

G: ~ ~ ~

= A = Ra~F~

such that

(we omit this proof) and (2) given any other

G~ = A ,

there are natural transformations

(not nec-

essarily monos) ea~F

~ G ~ Ra~F

corresponding to the identity of Ra~F

LL~(A,A)

via adjointness.

However,

Lan~F

and

are net exactly what we are seeking for our generalization of the A-G theorem,

as we shall see below. We now specify that a generalized interpolation space will be an property

l~-module. FoB = I~ .

A

A generalized interpolation functor We note that for all

X E ~

with respect to F

must satisfy the

there are maps

X~ ~ FX

deriving from the above property, such that the following diagram commutes:

XA

-

X0

Fg

Given a generalized interpolation space FAX

A

with respect to

~ , we define

to be the coimage of the map

Z~

d: L(A,X) ®L~A ~

defined by

d(S~a) = S ~ a )

makes sense since

,

S: A . X

where

~: A -~ E A

gives rise to

is the above map.

S: ZA-~ E~.)

HA~

following pullback: HA~

....... ~

I

LL~(L (~,A),A)

~

q,

~l~

~ e

LL~(L (~,~)

LL~(L (X,A) ,EA--)

,0)

,

(This definition

is defined to be the

150

where

e(x)(S) = S(x) .

It can be verified

that

FAO5

= I~

and

HAO6 = I~ .

We now give our version of the A-G theorem. 2.2.

Theorem.

then the functors

If FA

functors satisfying

eralized

is a generalized

and

FA

interpolation

functor

G

such that

L(A,X) ®L~A

~

GX.

f(S~a) = G(S)(a) FA~

to

LI~(L(X,A ),~)

space with respect to

and

GA = A ,

h

interpolation

HA

are, respectively,

We note that for any genwe have maps

LL~(L(~,~),A)

,

h

, h(x)(T) = G(T)(x)

G~ , and

A,

functors.

Only the second assertion needs discussion.

interpolation

map from

Furthermore,

such generalized

f

where

interpolation

H A , defined above, are generalized

FAA = A = HAA .

the minimal and maximal Proof.

A

.

Moreover,

together with

f

~: GX ~ ~ X

yielding a map, by definition of

HAX, from

clearly lifts to a equalize G~

to

e

and

HAX .

Hence, we

have FAX ~ G~ ~ H 7

3.

Duality results Duality results in interpolation

the elements of intermediate following

~

, e.g.

space

result,

A , e.g.

A E ~

,

~

Proposition. A(A)' = ~(i')

Proof. pullbacks.

is dense in A~

A

is dense in

,

meaning in

A .

is dense in ~ and

A0

and

conditions

A I , or on the

is true in the tradition-

A .

is closed under formation of dual diagrams, Z(i)'

The second assertion

on

One reason for this is that the

true in our context without hypotheses,

al context only when 3.1.

AA

theory usually require density

= ~(~')

i.e. if

.

follows from the fact that duals of pushouts are

Although duals of pullbacks need not be pushouts in general,

examination

of the doolittle diagram shows that duality does preserve them, hence, proving the first assertion. Our main result of this section is to show that the maximal and minimal functors are connected

by the notion of dual functors.

A-G

151

3.2.

Theorem.

such that such that

AA

AX

Proof.

Let

A

be a generalized

is dense in

A

and

interpolation

A ~" E A .

for all

X E

We must establish an isomorphism

We first observe that there are maps ~: NAT(L(~,-) ~ L ~ , A , ~ ) that

~ ZX

L(A,A) ® ~

,

and

~(T)(g)(a)

L(A,I) ® ~

(LI~,((Y~)' , ~)o]]) (T) (g) (a) = g(r~(a)) (thinking of

a

.

, A')

= g(r~(a)) and ~(r) = T~(1)

= I .

, ,

Moreover,

and

However,

both as an element of

((~a)oT~)(1)

.

~: NAT(L(A,-) ~ L A A,E~) ~ LI~,((Y~A)'

given by

= A

~ HA,(X)

(eo~) (T) (g) (a) = e(T~(1)) (g) (a) = g(r~(1) @ a)

have

DFA~ = HA,X

is dense in

: NAT(FA, ~ )

recalling

Then

space with respect to

A~

by the naturality of

and as a map

= T~(1) ®a = ( T ~ L ( A , a ) = T~(a)

T , we

I ~ A)

~A)(1)

:

TT L (A,I) ~A

,L

L (A, a) ~A

eo~ = LL~,((Y~A)' , ~)a~] ,

q0: NAT(L(~,-) ®L~A

, Z~) ~ HA,(X)

~a

~- ~ A

L @,A) ~

Hence,

l

T~

and, thus, there is a map •

Let

~ = ~0 o Dd: NAT(FA,E~) -. HA,(~)

.

A map 4: is given for

HA, (~) -~ NAT (L (~,-) ®LA--A, E~)

(T,x) E HA ,(~) ,

S E L(~,~)

~(T,x)~(S~a) We note that if

As an important dual functors on

~

= x i ® S(a) i ,

d(S~a) = 0 , then

~: HA,(X ) -~ NAT(FA,E~)

.

~

and

~

, a E AA , by i = 0,I

~ (T,x)~(S~a) = 0 , so we get

may be readily checked to be inverses.

[]

corollary of 3.2, we obtain by means of a duality theorem for a result similar to Theorem I of Janson ~3].

(and, of course, the generality)

differ somewhat, however,

The hypotheses

but we shall not explore

152

here the exact differences. putable

F: ~

~

F~ = lim IF~I~ finite dimensional subspace of

X!

and

to be com-

(~ la [2]) if

A finite dimensional element AT

Briefly, we define a functor

Z~)

~

of

~

are finite dimensional.

is one in which

Y0

and

Y1

(and, hence,

Our result is similar to Theorem 1.9 of

Herz-Pelletier [2]. 3.3. Theorem. such that

~

If

is dense in

3.4. Corollary. Proof.

F: ~-~ ~

If

is computable, then

DF(~') = (F~)'

for all

~ . FA

is computable, then

This follows from 3.2 since

(FAX)' = HA,(~' )

o

DFA~' = HA,~' -

We remark in closing that 3.4 is a slightly stronger result than Theorem 1 of [3].

Hence, our setting promises to be advantageous to the study of interpolation

theory. References [I]

J. Bergh and J. LBfstrom, Interpolation Spaces, Grundlehren der mathematischen

Wissenschaften 223, Springer-Verlag Berlin, Heidelberg, New York (1976). [2]

C. Herz and J. Wick Pelletier, Dual Functors and Integral Operators in the

Category of Banach spaces, J. Pure Appl. Alg. 8 (1!)76), 5-22. [3]

S. Janson, Minimal and Maximal Methods of Interpolation, Report No. 6, Institut

Mittag-Laffler

(1980).

[4]

S. Kaijser, On Banaeh Modules I, Math. proc. Camb. Phil. Soe., to appear.

[4

F.E.J. Litton, Autonomous Categories and Duality of Functors, J. Alg. 2 (1965),

315-349. [6]

B.S. Mityagin and A.S. ~varc, Functors in categories of Banaeh Spaces, Russ.

Math. Surveys 19 No. 2 (1964), 65-127.

Uppsala University

York University

Uppsala, Sweden

Toronto, Ontario, Canada

Toposes are monadic over categories. by J. Lambek O) McGill University

and University of Oxford.

In [L2, LSI, LS2] a study has been made of the category Top whose objects are toposes with canonical subobjects

and whose morphisms

are (strict)

logical functors which preserve these canonical subobjects on the nose. auxiliary notion,

the concept of a dogma, was introduced

the algebraic aspects of toposes.

functors in [L2]).

Actually,

Top

closed) dogmas are all monadic

Burroni

closed monoidal

(tripleable)

in a wider sense, as we

over

categories and (Cartesian Grph,

the category of graphs,

They are all examples of the "graphical algebras" of Albert

[B] in a very specific way:

operations

is already equational

(orthodox

Cat of categories.

Cartesian closed categories,

Cat. 2)

Dogmas are the objects of a

are also called logical functors in [LS2]

shall see, over the category

or

[L2] to capture

In the present context it will be convenient

to assume that dogmas are Cartesian closed, l) category Dog whose morphisms

in

An

and equations.

they are graphs

(or categories)

with

The notion of graphical algebra as envisaged by

Burroni is much more general:

it includes

toposes, provided these are

"equipped with a choice of finite limits and colimits and even 'parts' - a representative

choice of monomorphisms".

to give an independent verification is monadic

(tripleable)

monadic over

Cat,

over

Grph.

It would therefore be of interest

that the category

Top

discussed above

Instead, we shall prove that it is

in the sense that the forgetful functor

U: Top ÷ Cat

154

has a left adjoint functor

F

K: A --+A T

with adjunctions

N

is an isomorphism,

and

where

e

such that the comparison

T = (UF,n,UeF).

In what follows, we shall suppress explicit mention According

to Beck's Theorem

Given two arrows

in

A ~

Top

having a split coequalizer

EH_+ C

in Cat is a coequalizer

Dog is monadic over Cat, we may assume that Dog.

It remains

A ~

C

is a topos with canonical subobjects,

(If)

H

is a coequalizer

(Ia)

On the way to proving

of

(F,G)

and, incidentally,

A

have canonical equalizers

category

.~._~. and

essentially

Top.

Since

is a coequalizer

a pair

in

in

Top.

(I), we shall first show that

equalizers B

B -~ C

in

in Cat,

to show:

(I)

and

U.

[M], we must then verify the following statement:

F,G: A =z~ B

each (split) coequalizer

of the functor

that

H

preserves [L2].

them.

C

has canonical

We know of course that

To be precise,

let

E

be the

QA: AE --+A

the functor which sends any object of

f,f':

AI

A0 ~

of arrows in

A, onto its canonical

equalizing object: QA(f,f')

= Ker(BAl < f,f' >)

.

Now consider the following diagram in Cat: FE AE

~

GE

QA

A

----'~. G

HE

I h I

CE I I I

I QB t I

I

BE

B

QC C

H

AE

155

Since

F

and

left com~Qute. equalizer of

G

preserve canonical equalizers [L2], the two squares on the

Since

H

is an absolute coequalizer of

(F,G),

(FE,G E) , hence there exists a unique functor

such that the square on the right commutes. sends any object of

QA

h,h':

KA: A - - + A E.

The same is true for

QB" H

in

may call "canonical".

nC

and

Moreover

is

nC

and

QA KA eC

and as

e C = HEeB SE.

eC ' so that H

QA

Cat:

It is then routine to derive the equations which render with adjunctions

C I , onto

hA: IA ~

We define

is the given splitting of

n C = H~BS ,

KC

QC

This fact amounts to a

pair of equations involving the natural transformations

S

CO ~

may be expressed by saying that

right adjoint to the constant functor

follows, where

QC: cE -'~ C

Qc(h,h').

The defining property of

~A: K A Q A --+ IAE "

is a co-

It remains to show that

C E , namely a pair of arrows

its equalizing object

HE

C

QC

right adjoint to

has equalizers also, which we

preserves canonical equalizers in the

sense that HQ B = QC HE , H~ B = ncH , HE~ B = ecHE . The calculations, which are omitted here, also make use of the arrow

T: B--+ A

of the given split eoequalizer diagram in Cat. Incidentally, the above argument establishes that categories with canonical equalizers are monadic over Cat.

Similar arguments apply to categories

with canonical coequalizers, limits, colimits and any combination of these, even if additional equations are prescribed.

156

(Ib)

Next w e shall p r o v e that

C

is a "quasi-topos" in the following sense,

p r o b a b l y equivalent to the n o t i o n of P e n o n kernels such that char(ker h ). = . h [L2]

[P]:

for all

it is a dogma w i t h c a n o n i c a l h: C --+ ~

in

C.

We recall

that in any dogma the c h a r a c t e r i s t i c m o r p h i s m of a m o n o m o r p h i s m

m: C O --+C

is defined by

rchar m n

For any

h: C --+ ~ Ker h

= . {zeCINz0eC 0 z = m z0 } .

we put

=

Qc(h,TOc).

This is just the e q u a l i z i n g object; w h i c h is related to For any

in the topos

and recall that H(char(ker

N o w the logical functor also kernels.

kerh: Ker C--+~

ec(h,TO C) .

g: B - + ~

B = S(C), g = S(h)

it comes e q u i p p e d w i t h an a r r o w

H

B, we h a v e

HS = i C • S(h)))

char(ker g)

= . g .

Take

Then

. = . h

.

preserves c h a r a c t e r i s t i c m o r p h i s m s and, b y

(la),

Therefore char(ker h)

. = . h ,

as required. (Ic)

To show that

is, u p to isomorphism,

Let w e have

IC: C --+ PC

C = H(B).

C

is a topos, it remains to verify that e v e r y m o n o m o r p h i s m

of the form

ker h.

be the s i n g l e t o n morphism.

N o w in the topos

B

Putting

B = S(C)

as above,

157

IB . ~

hence,

applying

the logical

functor iC . ~

. ker(char

IB) ,

H, we obtain . ker(char

IC).

This shows that there is an equalizer diagram [L2] that the canonical

logical functor

topos generated by the dogma that description holds in

C --+ PC m:~ ~, and we know from

HC: C --+ T(C), where

C, is then full and faithful,

C, in particular,

T(C)

is the

or, equivalently,

internal equality

implies ex-

ternal equality. N o w let

m: CO--+ C

be any mono in

C

and put

h

. = . char m , that

is~ rh7

We shall prove that

m

{zcC[~zo~C 0 mz 0 = z } .

is an equalizer

of

h

and

ing square is a pullback.

C

4

lY

h

~

TO C , that is, the f o l l o w

158

Indeed, suppose

g: C

s

--+ C

is such that

hg

o = . TO

. Let

zt

be an

C" indeterminate arrow

i --+ C ~ , then

h g z" . = . T.

Therefore

I- g zt~ r h l , gt

that is,

by

l - V z ~ C S N Z 0 e C 0 mz 0 = g zp • Suppose for the moment that we can replace

~

in this.

Then, b y description,

Vz, EC , m f z s = g z" . actually have

mf

there exists a unique

. = . g. N

by

N~ ?

It suffices to show that

l-Vx,y~CO(mx = m y ~ x = y). L o o k at the proof of

is shown in a topos.

However,

teriBtic m o r p h i s m s are u n i q u e To show that

m: C O --+ C

c a n o n i c a l if

check the conditions For example, of

C

[L3, L e m m a 12.4], w h e r e this

the proof holds in any dogma in w h i c h equalizers

(la), internal e q u a l i t y implies e x t e r n a l equality

(Id)

in

C

such that

Since i n t e r n a l equality implies external equality, w e

W h y can w e replace

exist

f

C m

(see above) and charac-

(an easy c o n s e q u e n c e of (Ib)). has canonical subobjects, we call a m o n o m o r p h i s m

. = . ker h

(i) to (v) of

for some

h: C - - + ~.

It remains to

[L2,§9].

let us v e r i f y that the p r o d u c t of two c a n o n i c a l subobjects

is a c a n o n i c a l subobject.

Now a canonical subobject of

C

in

has the form Ker h = Qc(h,TO C) = HQB(S(h),S(TOc)) where

k . = . ~S(~) < S(h)'S(TOc) >

"

The result now follows f r o m the fact

that the p r o d u c t of two c a n o n i c a l subobjects of (II) Top.

W e c l a i m that

H: B - - + C

= H(Ker k),

S(C)

in

is a c o e q u a l i z e r of

W e k n o w it is the coequalizer in Dog, so suppose

m o r p h i s m in

T o p such that

H ' F = HtG.

B

is one. (F,G):

A :~ B

He: B --+ C t is a

Then there exists a u n i q u e logical

in

C

159

functor

M:

C--+

C~

such that

MH

= H"

It remains to show that

.

Now

T o p , that is, that it preserves canonical equalizers. canonical equalizers,

that is,

HQ B = QC HE etc.

H

Recalling

M

is in

does preserve

that

HS = 1 C , we

therefore have MQc = MHQBSE = Ht QBSE = Q

H' EsE C'

= Q

~HEs E ¢"

C" Similarly one shows that

Mq C = qC' M

and

~C

E M-,

E

and so

M

pre-

C' serves canonical equalizers

POSTSCRIPT. The theorem stated in the title is one of those results which are easily proved once one is convinced

of their truth.

In particular,

it is not necessary

to base its proof on the notion of a dogma, with which the author happens to be familiar.

A direct proof along some of the lines indicated in [B] is also

p o s s i b l e . 3) In the published version actually asserts

[B] of his talk at Amiens,

that toposes are tripleable

see why the present argument

Burroni

over graphs.

fails to yield this apparently

even if dogmas are eliminated.

It is instructive stronger result,

to

160

To s i m p l i f y m a t t e r s and to vary the approach a little, let us confine attention to the related assertion are m o n a d i c over graphs. f,g: A nzn~ B. unique arrow

Let

U

A

to the functor

denote the "canonical" equalizer of

h: C - ~ A

C --> source

To equip a category adjoint

~(f,g)

Then, for each B(f,g,h):

that categories w i t h canonical e q u a l i z e r s

such that

~(f,g)

such that

F: A - - + A "=:~"

given by

A--~A A , F(a) = a | | a ,

IA

and

a: A ---+ A" ~: FU ÷ id

.

junction equations relating volving

~

and

A+ ' ~ A

N o w it is easy to express

in terms of

a

and

~ .

q

and

g

U

that the category

e q u a l i z e r s is m o n a d i c over Cat.

B(f,g,h)

and the adjunctions U

~: id ÷ U F

and the ad-

may then be w r i t t e n as equations in-

B •

present article,

Grph

+"

The functoriality of

It readily follows from these considerations,

over

~(f,g) B(f,g,h) "=" h .

w i t h canonical e q u a l i z e r s is to provide a right

1A F(A) = A ~

where

fh .= • gh, there is a

Equ

of small categories w i t h canonical

W h a t goes w r o n g in trying to show m o n a d i c i t y

is that the condition involves the fact that

as in the m a i n text of the

fh -= • g h A

p r e s u p p o s e d in the d e f i n i t i o n of

is a category, not just a graph.

Indeed, in a t t e m p t i n g to v e r i f y Beck's condition for m o n a d i c i t y over we assume that

F,G: A ~:::$ B

in

Equ

h a v e a split c o e q u a l i z e r in

w a n t to s h o w that each (split) c o e q u a l i z e r e q u a l i z e r in A ~

B ---+ C

Equ.

A f:-~ B ~

C

in

Grph

Grph

and

is a co-

(Since Cat is m o n a d i c over Grph, w e m a y assume that

is a coequalizer in Cat, but not a split coequalizer.)

Grph,

161

A crucial part of the argument be the arrow splitting

H

in Grph,

Bc(f,g,h) for any

f,g: A

forced upon us if

4 B H

in

such that

fh "= • gh.

Indeed,

canonical equalizers,

we cannot deduce

S: C ---+ B

this definition

provided

is

the right

this from

fh .= • gh,

is an arrow only in Grph, not in Cat. the surprising

discovery

are needed, whose defining conditions y(u,v,p): and

6(u): then defines

source

source

B(f,g,h)

Beck's condition, position,

u ---+

C.

source

Referring

in case

However,

to the split coequalizer

fh = gh. resolve our p r o b l e m with ¥

in Grph and using the equations

still involves

following on

C

a suggestion by and write

H

~C

and

SH = GT, one easily

that 4)

from the given fact that

~c(u,p)

diagram

HF = HG, HS = i, FT = 1

source Yc(U,V,p)

com-

S

~B-G

namely

a(u,v)

like the other operations

A

satisfied,

~(u,u).

of the source of

T

calculates

source

observation won't

in Cat not in Grph.

we treat composition

for up in

are automatically

= y(f,g,h)6(fh)

as the description

an operation

that only two special cases of

a(up,vp) ~

At first it seems that this ingenious

Burroni,

Let

= H(~B(S(f),S(g),S(h)))

Unfortunately

Burroni has made

One

~C"

For this to be the case we must check that

S(f)S(h) • = • S(g)S(h). S

of

then we try to define

is to preserve

hand side makes sense.

since

C

is the construction

= source ac(~c(U,p),

~c(V,p))

162

source 7B(S(u),S(v),S(p))

Thus, borrowing

= source ~B(UB(S(u),S(p)),~B(S(v),S(p)).

some of Burroni's

ideas, we may recapture his result

by our method.

For a quick exposition of the difference between the methods of [B] and note that any functor

AxB + C

may be used to obtain a structure-semantics

adjointness

between concrete categories

particular,

if

U@

@: B ÷ T

is bijective

exploited respectively. Footnote 2.)

over

A

and theories under

B. 5)

In

on objects and one forms the pullback

Ae

÷ CT

A

+C B ,

will satisfy Beek's condition,

In [LI] and [B] the functors

[LI],

although it won't in general have a left adjoint.

CatxCat Cat

> Cat and

Grph×Grph~ lln

~ Sets

are

(The methods of [LI] should be amended in accordance with

163

REFERENCES

[B]

A. Burroni, Alg~bres graphiques, 3~me colloque sur les categories, Cahiers de Topologie et G~om~trie Diff~rentielle 23 (1981), 249-265. 6)

ILl]

J. Lambek, Deductive systems and categories II, Lecture Notes in Mathematics 86(1969), 76-122.

[L2]

J. Lambek, From types to sets, Advances in Math. 36(1980), 113-164.

[LSI]

J. Lambek and P. Scott, Intuitionist type theory and the free topos, J. Pure and Applied Algebra 19(1980), 576-619.

[LS2]

J. Lambek and P. Scott, Algebraic aspects of topos theory, 34me colloque sur les categories, Cahiers de Topologic et G~om4trie Diff~rentielle 22(1981), 129-140.

[Li]

F.E.J. Linton, An outline of functorial semantics, Lecture Notes in Mathematics 80 (1969), 7-52.

S. MacLane, Categories for the working mathematician (Springer, New York, 1971).

164

FOOTNOTES 0)

The author wishes to acknowledge support from the National Science and

Engineering Research Council of Canada and from the Social Sciences and Humanities Research Council of Canada.

He is endebted to Albert Burroni for an exchange of

ideas and to Max Kelly for insisting on the footnotes and for uncovering an error in [LI]. i)

This paper was first presented in Sussex in November, 1980.

We briefly recall the definition of a dogma in [L2].

While dogmas there

were not required to be Cartesian closed (only partially so), we shall insist on this property here.

A (Cartesian closed) dogma is a Cartesian closed category with

a specified object and

VA' ~A:

~

and with specified arrows

PA + ~,

where

PA = ~A.

T, i: i ÷ ~, A, V, ~: ~x~ ÷

These arrows satisfy a number of conditions

to assure that, when interpreting higher order logic, all intuitionistic theorems are equal to the arrow

T.

It was shown in

[L2, p. 124] that these conditions can

be expressed as identities, in fact, as equations between constant arrows, e.g., A = i~, where

HA,B: AxB + A

and

A = <~' ,~ > ~,~ ~,~

~' • A×B ÷ B A,B"

are the usual projections.

There are 17

such equations or families of equations. 2)

For example, a Cartesian clesed category

finite products and a functor such that

expA(-,A')

is right adjoint to -xA'

qA(-,A'): id A ÷ exp(-xA',A') adjunction equations.

expA: AxA °p ÷ A,

and

let

is a category with canonical usually written

eA(-,A'): expA(-,A')xA' + id A

A --+ B

One wishes to show that

F,G: A --+ ~

H ) C H

be a split

in Cart

Cat.

hence absolute coequalizer in

is a coequalizer in

exPc: CxC °p ÷ C

satisfying the usual

the category of Cartesian closed

Cart, in particular,

the structure of a Cartesian closed category and that p.102], we define

expA(A,A') = A A'

by virtue of natural transformations

We shall verify monadicity over

Given a pair of arrows categories,

A

H

that

preserves it.

as the unique functor such that

Cat. C

has

As in [LI,

exPC(HXH °p) = H exp~

165

What was said about natural transformations

in [LI, p.102]

is incorrect,

instead

one should define ¢c(C,C') and similarly

for

~C"

and similarly

for

~,

ations satisfying

in

Dog

VC, H C

From this it easily follows that and that

the adjunction

To show that

= HEB(S(C),S(C'))

Dog

eC(-,C')

and

qC(-,C')

C.

is monadic over

One defines,

= sc(H(B),H(B')),

are natural transform-

equations. Cat, we assume that

and have to show how to define the additional in

HCB(B,B')

F,G: A --+ B

structure

~, T, ±, A,V, =,

for example,

~C = H(~B)'

TC = H(TB)'

^C = H(AB)

and Vc(C) = H(¥B(S(C)) , where we have written One easily verifies equations hold in

3)

According

V(C)

that

H

in place of preserves

¥C

and

S

the additional

is the functor which splits H. structure and that the 17

C.

to the revised version of [B], a topos is a bicartesian

category with equalizers and coequalizers the canonically specified object

associated ~

there is associated

arrow

and arrow

such that for every arrow

Coim(f) + Im(f) T: 1 + ~

a unique arrow

Im(f)

~

~

f: A ÷ B

such that the following square is a

pullback:

B

f: A ÷ B

is invertible and also with a

such that for every arrow

B + ~

closed

1

166

4)

Indeed,

we have

source

~(u,v,p) = source c"

HYB(SU,Sv,Sp)

= H source YB(SU,Sv,Sp)

= H source

~B(~B(SU,Sp),~B(SV,Sp))

= s o u r c e H ~ B ( F T ~ B ( S U , S p) ...... )

= s o u r c e H F ~ A ( T ~ B ( S U , S p) ...... )

= s o u r c e H G ~ A ( T ~ B ( S U , S p) ...... )

= s o u r c e H ~ B ( G T ~ B ( S U , S p ) , ..... )

= s o u r c e H ~ B ( S H ~ B ( S U , S p ) ...... )

= s o u r c e ~ C ( ~ C ( u , p ) ...... )

5)

theme,

implicit

in the w o r k

m e at the M i d w e s t

Category

Seminar

McGill

6)

This

of F r e d L i n t o n ,

in W a t e r l o o

was expounded

and in a g r a d u a t e

by

seminar

at

in 1968.

There

Burroni's

exists

also a m o r e r e c e n t m a n u s c r i p t

t a l k at C a m b r i d g e

in J u l y 1981.

with

the s a m e t i t l e c o n c e r n i n g

ESSENTIALLY MONADIC ADJUNCTIONS b y John M a c D o n a l d and Arthur Stone

In an earlier p a p e r

[9] the authors show h o w the canonical r e g u l a r epic decom-

p o s i t i o n of a m o r p h i s m is p a r a l l e l e d by an analogous d e c o m p o s i t i o n for a d j u n c t i o n s - the canonical m o n a d i c d e c o m p o s i t i o n

(cf.

[2] and

[7]).

The first section of this p a p e r shows that for a faithful right a d j o i n t

U

the essential length of the a s s o c i a t e d a d j u n c t i o n is the same as the r e g u l a r length of the counit, w h e n both are defined. The "higher" Beck theorem, c h a r a c t e r i z i n g e s s e n t i a l l y m o n a d i c a d j u n c t i o n s is p r o v e d in the second section. be w r i t t e n as a

(canonical)

An e s s e n t i a l l y m o n a d i c a d j u n c t i o n is one w h i c h can

composite of m o n a d i c adjunctions.

Such a d j u n c t i o n s

have faithful right adjoints.

i.

Lenqth of a faithful a d j o i n t

This section shows that the essential length to a faithful right a d j o i n t

U

F o r n o t a t i o n a l reasons we recall that with e •

F :~ + ~

fined as in as in

left a d j o i n t to

Given a d j u n c t i o n s [9].

1

of the a d j u n c t i o n

N : X + A

c o n s i s t s of functors

U , t o g e t h e r w i t h a choice of u n i t

N~ : X ÷ Y

and

N8 : v ÷ Z

U s i n g this c o m p o s i t i o n a c a t e g o r y

the composite

associated

Adj

F n

E .

and

U ,

and counit

N = N~AN ~

is de-

of a d j u n c t i o n s is d e t e r m i n e d

[i0], page 102. Let the ordinal

l

be i d e n t i f i e d w i t h the o r d e r e d set

o r ordinals less than

l

and w r i t e

A c h a i n of object length of

N

is equal to the regular length of the counit

~

~

a8

for the unique m o r p h i s m

in the c a t e g o r y

~

is a I chain c o m p o s i t e of m o r p h i s m s

colimit p r e s e r v i n g c h a i n An a d j u n c t i o n

F

of o b j e c t length

N = X ÷ A

chain c o m p o s i t e of m o r p h i s m s

(considered as a category)

is a functor N Z ~ (a < ~;8=~+i) I + 1

with

a ÷ 8

when

F : I ÷ ~ .

if there is a

N Z 8 = F(~8)

and

N = F(0I).

has a canonical m o n a d i c d e c o m p o s i t i o n if it is the NaS(e < g;8=a+l)

in the comma c a t e g o r y

(Adj,A)

p i c t u r e d as in

X0

~ < 8 <

A morphism

N =

NO

A

168

and subject to the following conditions. N~

is a first monadic component

of

First of all, when

N ~ , that is,

I

N~

_

F ~8

= USF ~ .

XJ

as the category of algebras

Secondly when

<

N__ e

is a limit ordinal

the adjunction and

_

This leads to the well known universal for

6 = e+l

= N6AN ~8 , q~ = ~ e6

property and identification

(cf. Beck's results cited in

~K

is the limit in

right adjoints U~6 (~< 6 < <) , with F e< 6 q has an inverse but q does not, for

= U
Cat and

[9] and

of

[i0]).

of the d i a g r a m of qe< = e

The ordinal

~

and, finally,

is called the

monadic length. An adjunction over

~ )

if

N

N : X÷A

is essentiall~ monadic

(and

has a canonical monadic decomposition

parison adjunction

N6

determines

an equivalence

~

is essentially monadic

i.i in which the last com-

of categories.

Ue6 Lemma 1.2:

Let

...X_~

XJ...

with limit If the

U e8

are faithful,

(2)

If the

U ~8

reflect

length of

an idempotent monad, this is equivalent length

l

occurring

that is,

to

qlU ~

is the smallest ~

is smallest

~ = I+i

fl

with

(x,y)

Lemma 1.3:

f = h.g

j.g = f01 with

Let

.

~

for

~ < I

component

Proof:

~

in others,

NJ

generates

is an isomorphism. [9]).

and in fact as shown in

But

Thus the essential I s ~ ~ ~+I

with

[9].

[7] , we recall that a regular f

if

f = fl.

f01

for

a regular epic, there is a (necessarily e

is regular if for each

there is a unique

d

with

h

with

h = de

hx = hy

.

be an adjunction with monadic decomposition,

length

every object

(2)

length

.~

with

=

essential

(i)

g

A morphism

ex = ey

N :X _

with

÷ Cat

U e<

such that

UleIF l

is a regular epic component of m o r p h i s m

and whenever j

1

(see 2.9 in

[6] and Kelly

op

U e<

then so do the

so that

K

C=at .

ordinal

being an isomorphism

in some examples and

f01

for all pairs

(with

isomorphisms,

[9] , based on work of Isbell

epimorphism

unique)

N

U2 < in

a functor

then so are the

is less than or equal to the monadic

From

some

XJ and projections

(i)

The essential

= ~

(~ < 6 < <) represent

A

and of

l

U

is faithful.

Then in i.i for

of ~ = ~+i , the m o r p h i s m

F6eeSUBA

is a regular epic

e~A , and

the m o r p h i s m

elA

It is well known

(cf.

UIA = < UA, U e A >)

, and assume

is monic.

[9] and

has left adjoint

[i0] ) F1

that the c ~ p a r i s o n

functor

defined by the coequalizer

U1

diagram

169

=

FU01e 01

FT01U01

FTU01

{ FU01

T

* F1

eFU 01 It turns out that

T = Fie 01

and hence that

(GeA, eGA) = (FU01e01UIA, EFU01UIA)

where

FIe01UIA

G = FU .

c o e q u a l i z e s the p a i r

Thus

FIe01UIA

is a r e g u l a r

epimorphism. Clearly e. Ge = e.eG . W e let elA be the u n i q u e m o r p h i s m for w h i c h 1 le01ul A 1 1 e A.F = eA . Of course e is the counit of N in I.i. In the same way we see that Let

FS£d~U6A

eaA = h.g

for

g

x

and

c o e q u a l i z e r for we replace

is regular since it c o e q u a l i z e s

(x,y)

a r e g u l a r epic. y .

(GeE~A,eaGeA)

Then suppose

when

e~A.x = e~A.y

8 = ~+i .

where

g

is a

The r e m a i n d e r of the p r o o f is e s s e n t i a l l y the same if

by a family of p a i r s

(x0,y0)861

.

We m u s t find a

j

so that

jg = F ~ e a 6 U a A .

Gax~~Y

(1.4)

~A ~A

I

Gae~A

~,~.

x/l
EaGeA

,,

eaB

Since

g

is a c o e q u a l i z e r for

Let

B = Sce x = Sce y .

e B

is epic

(since

F6e~6USA . x . e B

If

U

and

is faithful, then

(2)

We show that if 1

e ~ = eS(8=e+l)

So

.

1

eaA

eeA

y .

is epic

are equal.

(cf.

.

[i0], page 88), and

and

B y the n a t u r a l i t y of and

e ~ , these

F S e a B U 6 A . eaG~A . Gey .

Now

E~A. x= seA. y

is not monic, then ~

a < I .

such that

is the first ordinal for w h i c h

B y the c o n s t r u c t i o n of

e ~ = e6(e < 6) FSeeBUSA

or, equivalently,

(8=a+l)

is an iso-

we have

eaA . x = eeA . y .

A .

is non-monic,

B = Sce x = S c e

sB

F6ee6U~A.x = F 6 e ~ 6 U 6 A . y

So it suffices to show that the c o m p o s i t e s

F 6 e ~ 6 U 6 A . eeGeA . Gex

is the first ordinal

m o r p h i s m for all objects If

.

F ~ e ~ 6 U ~ A . y . caB

F6e~6U6A . e~GeA= FSe~SUSA. G~e~A

I.I the ordinal

Let

we have o n l y to show that

eB = eeB . FeeoeUaB)

are equal to the composites use

x,y

then for some pair

If

Then by the n a t u r a l i t y of

U ee

G~x = G a e e A . Gay , this implies m o n i c -- not an isomorphism.

is faithful, we have

then

eaB

with

x~ y

is epic and

e a G e A . G~x ~ e a G a A . Gay .

eaGeA z G a e a A . Hence

x,y

a < I .

So the c o e q u a l i z e r

x. eeB~y Since

. E~B .

Gamma .

FSe~6U~A

is not

170

T h e o r e m 1.5: and

U

If

N :~

>~

is an a d j u n c t i o n with m o n a d i c d e c a m p o s i t i o n i.i

is faithful, then the d i a g r a m

(i .6)

is a r e g u l a r d e c o m p o s i t i o n of Proof:

B y 1.3

e

in (A,A~

(i) , FBe~SU B

for

B y the c o n s t r u c t i o n of i.i at limit o r d i n a l s of m o r p h i s m s G~s ~ ~ eeG ~ 1.3

F S s e S U B (~ < B < <) for

(2), s

~ < I

N

~

If

• (cf.

K

[9]) .

we have

GK

~A

£

N

when

.

B = ~+i .

a colimit of the d i a g r a m

is not m o n i c for

is e s s e n t i a l l y monadic,

a < ~ -

By

U

then the essential length

~ .

Since m o n a d i c right a d j o i n t s

right adjoints

are faithful and

1.5

U

are faithful, by 1.2 e s s e n t i a l l y rm0nadic

applies.

The regular lenghh of

is the supremum of the regular lengths of the c o m p o n e n t m o r p h i s m s

2.

ee

Also by the c o n s t r u c t i o n of i.i,

E ~ . G~E ~ = e ~ . e~G ~ ; so

N :~

regular length of

Proof:

the essential length of

is monic.

C o r o l l a r y 1.7: of

and

1

is a regular epic c o m p o n e n t of

EA

in

e

in (A,A~

~ . D

C h a r a c t e r i z a t i o n of e s s e n t i a l l y m o n a d i c a d j u n c t i o n s

This section p r e s e n t s the a n a l o g u e for e s s e n t i a l l y m o n a d i c a d j u n c t i o n s of the B e c k c h a r a c t e r i z a t i o n t h e o r e m for m o n a d i c a d j u n c t i o n s

(cf. MacLane

[i0], page 147)

avoid o b s c u r i n g the v i e w of the forest with underbrush, we p o s t p o n e to a careful r e p h r a s i n g of several f a m i l i a r definitions.

2.2

. and

To 2.4

They w i l l facilitate the p r o o f

of the following t h e o r e m and its corollaries. T h e o r e m 2.1:

The adjunction

has m o n a d i c d e c o m p o s i t i o n i.i

N :X

>A

is e s s e n t i a l l y m o n a d i c if and o n l y if

and the following

(then) e q u i v a l e n t c o n d i t i o n s are

satisfied. (A)

U

reflects extremal e p i m o r p h i s m s

(B)

U

r e f l e c t s strong epimorphisms,

(C)

U

r e f l e c t s isomorphisms.

D e f i n i t i o n 2.2: of objects of

~

and

(cf. H e r r l i e h - S t r e c k e r

is separating at an object

[5]) A

A set if

L

(.not n e c e s s a r i l y small)

171

(0)

for every p a r a l l e l

least one object extremal (a)

L

of

~

separating

m

(b) in

there

L

f,g

of m o r p h i s m s

A

if

(0)

x :L

with

A

there

fxzgx

is at

,

and

(= non-isomorphic)

L

w i t h source ÷A

monomorphism

of

L

and a m o r p h i s m

at

A

if

x :L

m

w i t h target

>A

A

there

t h a t does n o t factor

,

and strong

L

at

for e v e r y p r o p e r

is at least one o b j e c t through

pair

and a m o r p h i s m

separating

(0)

and

if for a m o r p h i s m

b

and e v e r y

÷A , morphisms

x :L

is a m o r p h i s m

for every

x

d

and a m o n o m o r p h i s m

that m a k e s

a

m

as in 2.3 there are,

for e v e r y

that m a k e the square c o ~ u t e ,

x

the d i a g r a m

2.3

commute

for the given

then

b

and

m

,

. x

L

(2.3)

c~

d

~ ' ~

m

The set A functor

is ... s e p a r a t i n g

F :X

When the case

~

~

~A

... s e p a r a t i n g

has e q u a l i z e r s

b = 1A

Definition morphisms,

is

2.4:

The f u n c t o r

strong epimorphism) strong epimorphism);

Tgt.g = Tgt.h The f u n c t o r

if

(a) implies

strong epimorphisms~

morphism,

of

if it is ... separating Im

(0)

F

.

a t e v e r y object of

is.

To see t h a t

(b) implies

(a) c o n s i d e r

imply U

U :~

~X

at an o b j e c t

and

Tgt.f = A

and

Ug z Uh

U

reflects A

if

epimorphisms

Uf

imply

f

is f a i t h f u l

at

(extremal

an e p i m o r p h i s m an e p i m o r p h i s m A

if

g ~ h

epi-

(extremal

(extremal

epi-

.

is, of course,

2.5:

Propositions:

The f o l l o w i n g

(0)

U

is faithful

faithful

and o n l y if it is faithful

(i)

U

reflects

(2)

EA

(3)

F

(4)

( when

For an a d j u n c t i o n

N :~

A

a~t e v e r y o b j e c t

and an object

are equivalent: at

A ,

epimorphisms

at

A

,

is epic is s e p a r a t i n g N

at

is m o d e l

epi-

, Sce.g = Sce.h = A ,

A . Underbrush

A

.

morphism,

and

i__n_n ~

A , and induced

by

~

is s e p a r a t i n g

at

A ;

A

of

.

172

2.6A:

The following are equivalent

(I)

(when

(2)

£A

(3)

F

(4)

(when

2.6B:

U

is faithful)

U

(under the c o n d i t i o n s specified):

r e f l e c t s extremal e p i m o r p h i s m s at

A ,

is an extremal epic is extremal separating at N

A , and

is model induced by

~ )

~

is extremal separating at

A .

The statements of 2 . 6 A are e q u i v a l e n t if "extremal" is e v e r y w h e r e re-

p l a c e d by "strong". The proofs are s t r a i g h t f o r w a r d or familiar emphasize w h y we assume (I) and

UeA

is always a regular epic

let

(.for some m o r p h i s m s a dia~onal

d

f

Since

a,b

diagonal

w

Uf

U

and m o n o m o r p h i s m b.f=m.a

d

for the square

d

(of 2.1)

m o n a d i c lengths U

l

Let

and

E

EB

Ua .

is epic

N :X

6 .

is faithful and

equivalence,

÷A

~

d

satisfying

Let

(and A) the functor

U

B = Sce a

and

C = Tgt a .

b.f = m.a

Ue

if

U

So if

U

If

eA

To show

U

is faithful).

be e s s e n t i a l l y monadic, with e s s e n t i a l and

Since

N6

is an

and hence so is c . This m a k e s e in 1.5 a g e n e r a l i z e d

hence strong

(and extremal)

reflects strong epics

epic.

Therefore, by

(and extremal epics). U .

r e f l e c t s extremal epics or strong epics, then by the U n d e r b r u s h

is faithful, and

is a c o e q u a l i z e r of

strong.

and

, we use the n a t u r a l i t y

(which will be true if

Since m o n a d i c right a d j o i n t s r e f l e c t isomorphisms, b y 1.2, so does

U

d.f= a

b • e A = m .(eC.Fw)

has a d e c o m p o s i t i o n as in 1.5.

is an isomorphism,

Conversely,

b.f = m . a

Since m o n a d i c functors are faithful, by 1.2 the right

composite of strong epimorphisms,

propositions

is one.

strong epic, and

We w i s h to show the existence of

for the square

is also a d i a g o n a l for the original square

Proof:

2.6B

m ) .

, i.e. a m o r p h i s m

Ub . Uf = U m .

and we m u s t assume that

adjoint

T g t A , Uf

split by

£A

is strong epic and right a d j o i n t s p r e s e r v e monics, there is a

is strong epic, there is a diagonal

e

(indeed, a split coequalizer, reflects strong epics, then

be any m o r p h i s m with

for the square

m.d = b .

that

To

faithful in 2 . 6 A and B w e s h o w the e q u i v a l e n c e of 2.6B

and r e g u l a r implies strong, if

For the converse,

of

[ii], page 176).

(2).

Since QUA)

U

(e.g. Schubert

UeFU

s

and

has the r e g u l a r d e c o m p o s i t i o n of 1.5.

Since

UFUe , it is r e g u l a r epic, hence e x t r e m a l and

r e f l e c t s extremal or strong epics, then

e

is one.

Then, since

FleOIu I

, (a g e n e r a l i z e d composite of strong epics) is epic, and e I . F I e O I u l =E l the natural t r a n s f o r m a t i o n e is extremal or strong. And m o n i c extremal (or strong)

e p i m o r p h i s m s are isomorphisms. Since e

_N l

is idempotent,

an isomorphism) m a k e s

N~

q

6

is an isomorphis~n.

So

an e q u i v a l e n c e of categories.

el

an i s o m o r p h i s m

(hence

173

N o w consider the case U F q i.

.U e F

= 1

U

r e f l e c t s isomorphisms.

Since

Nl

is idempotent,

and

U l e I F I U l = U I F I U l e i . UIFl~IU~ . U l e I F I U l = UIFIUle I . 1 .

Since

Ule I

is a co-

e q u a l i z e r of UleIFIU l and U I F I U l e I , this m a k e s Ule I , and hence U01Ulel , i.e. l l Ue , an isomorphism. So if U r e f l e c t s isomorphisms, then e is one. Again, 6 this m a k e s N an e q u i v a l e n c e of categories. [~

Corollary if

A

2.1a:

The c a t e g o r y

A

is e s s e n t i a l l y m o n a d i c over

c o n t a i n s an e x t r e m e l y separating set

p o s i t i o n i.i exists for Remark:

N

M. Barr in

L

m o d e l induced by

of c a r d i n a l i t y

[3] shows that w h e n

The c a t e g o r y

A

m

if and only

and the decom-

L . ~ = E ns

m

, then

any e x t r e m e l y separating set of c a r d i n a l i t y smaller t h a n Definition 2.7:

=Ensm

A

does not contain

m .

is Isbell cocomplete p r o v i d e d that it has

colimits of long chains of strong epics as well as c o l i m i t s of small diagrams. C o r o l l a r y 2.1b: L , then

A

Let

A

be Isbell c o c o m p l e t e w i t h an e x t r e m e l y separating set

is Isbell complete.

This is an i m p r o v e m e n t on H e r r l i c h - S t r e c k e r

([5], page 163) only when

A

is not

co-well-powered. D e f i n i t i o n 2.8: c l o s u r e of

L

Let

e

Suppose that

L

C o r o l l a r y 2.1d: the counit

e

at

B .

U~B

and

Then

C o r o l l a r y 2.1e: e

if and only if Suppose that

A

in

The full c o l i m i t

N

A

containing

L

Then an object

B

L

L

L .

and that

is in the full

is e x t r e m e l y separating at

is m o d e l induced by

is isomorphic to

A

L

A .

and i.i exists, that

is e x t r e m e l y separating

if and only if

USA

is isomorphic to

e _< ~ . S u p p o s e that

N

is m o d e l induced by

has an epic m o n i c factorization.

colimit c l o s u r e of I m a g e L

A .

is s u f f i c i e n t l y c o c o m p l e t e for

has an epic monic f a c t o r i z a t i o n and that

for all ordinals

and that

A

has an epic m o n i c factorization.

colimit closure of

A

be a d i s c r e t e s u b c a t e g o r y of

is the i n t e r s e c t i o n of all full s u b c a t e g o r i e s of

Corollary 2.1c: the counit

L

F

in

A

Then

X_~

L

in

A

and i.I exists

is equivalent to the full

w h i c h is the same as the full colimit closure of

A . Comment:

This m e a n s that the o p e r a t i o n s in the tower suffice.

W e explore this

in a subsequent paper. C o r o l l a r y 2.1f:

The

A-composite of e s s e n t i a l l y m o n a d i c a d j u n e t i o n s w i t h Isbell

c o c o m p l e t e targets is a g a i n e s s e n t i a l l y monadic. A - c o m p o s i t e as well).

(This is true for the g e n e r a l i z e d

174

References

i.

H. Applegate and M. Tierney,

2.

H. Applegate and M. Tierney,

3.

M. Barr, The point of the empty set, Cahier de top. et geom. diff. 13 (1972),

ematics 80,

137,

Categories with models,

(Springer-Verlag,

(Springer-Verlag,

1969),

in:

Lecture notes in Math-

156-244.

Iterated cotriples,

in:

Lecture Notes in Mathematics

1970), 56-99.

356-368. 4.

B. Day, On adjoint-functor (Springer-Verlag,

factorization,

in:

Lecture Notes in Mathematics

420,

1974), 1-19.

5.

H. Herrlich and G. Strecker,

Category theory,

6.

J. Isbell, Structure of categories,

7.

M. Kell~, Monomorphisms,

B.A.M.S.

epimorphisms

(Heldermann-Verlag,

1979).

72 (1966), 619-655.

and pullbacks.

J. Austral.

Math Soc. 9

(1969), 124-142. 8.

J. MacDonald,

Cohomology operations

in a category,

J. Pure Appl. Alg., 19 (1980),

275-297. 9.

J. MacDonald and A. Stone, The tower and regular decomposition, geom. diff., to appear. Categories

i0.

S. MacLane,

ii.

H. Schubert,

12.

A. Stone, The heights of adjoint towers, Am. Math. Soc. Notices 21 (1974), A-81.

Categories,

for the working mathematician,

Cahier de top. et.

(Springer-Verlag,

John MacDonald Mathematics Department University of British Columbia Vancouver, B.C., Canada V6T IY4

(Springer-Verlag,

1971).

1972).

Arthur Stone Mathematics Department UC Davis Davis, California

Decomposition into

John M a c D o n a l d

Considering

iterated

first d e c o m p o s i t i o n s (cf.

[6]).

regular

In the dual

decomposition (factorizing induced

both

situation

an adjoint

Eilenberg-Moore paper

constructions

E-factorizations more g e n e r a l l y

(up to duality)

by the second

need not be closed

there

only

are b r i e f l y

subsumed

2.2 below).

They will

paper w h i c h will factorizations After

images

sition

AMS

1.3

considered

Locally

composition;

Besides

1968.

property

the m e n t i o n e d in t o p o l o g y

treated

of locally

the authors

which

orthogonal

see

[131).

of a "strong

a locally

It also contains

54 C 10,

with

"On subobjects and

the notion

is p r e c i s e l y

18 C 20,

(see

got a c q u a i n t e d

and O. W y l e r

In this paper

which

in a f o r t h c o m i n g

form).

18 A 32,

orthogonal

considered

as soon as it

(instead of just morphisms;

in our terminology.

(in the dual

classification:

[13]).

the i n v e s t i g a t i o n

of f" is introduced,

]-factorization

locally orthogonal

the usually

intensively

paper by H. Ehrbar of

analogy:

monotone-light factorizations

the m a n u s c r i p t

in categories"

J-image gonal

of cones

(see also [9]).

first

range of examples

be more

include

finishing

an u n p u b l i s h e d

here as

the

new.

a few examples.

is a wide

over

diagonal-fill-in

under

anything

of the r e g u l a r

for this

from

(cf.

the unique

does not give

We can c o n s i d e r applications

arise

generalize

so far as

E

reason

[8] by

construction

out

w h i c h were

author

of m o r p h i s m s

the i t e r a t e d

as possible

pointed

the precise

systems

the n o t i o n

was

factors

paper

behaviour

as often

category)

with

[7] d e s c r i b e d

tower

has used

many

In the recent

and the adjoint

factorization

is,

Kelly

(with two factors)

E-factorizations

Isbell

into i n f i n i t e l y

the analogous

functor

gives

Tholen

factorizations

of a morphism.

and Stone

factors

and W a l t e r

dominion

of a m o r p h i s m

The present

many

of m o r p h i s m s

factorization

the first author

of m o r p h i s m s

infinitely

54 D 05

coorthoPropo-

176

I. L o c a l l y

Factorizations vestigated

factorization will ty

give

order city

property.

show

its

It h a s

all

cones

property

under

of

for

isomorphisms

and

being

called

the

used

follows

by

some

we proper-

theorems

faetorization

of

But

and

cones

for

in

simpli-

paper.

be a s u b c l a s s

closed

in-

so

diagonalization

this

E

been

the

makes

(cf.[13]).

in

, let

have of

In w h a t

the

functors

K

by use

usefulness

before

to m o r p h i s m s

a category

or sources

composition.

and

introduced

of m o r p h i s m s

necessarily

weakening

naturality

ourselves

for

of

successfully

semitopological

restrict

Throughout, containing

This

closed

been

to d e s c r i b e we

and

most

"non-compositive"

and will

examples.

authors,

classes

a

E-factorizations

of m o r p h i s m s

by many

diagonalization

orthogonal

under

of

MorK

composition

with

isomorphisms. We

recall

(e,m) it

some

of m o r p h i s m s

is c a l l e d

orthogonal

A factorization

phrases. such

that

the

composite

E-factorization

an

to a factorization

(of a m o r p h i s m

in (e,m)

me

case

e 6 E

, if,

for

exists

f)

is a p a i r

(and

f =me);

. A morphism all

is

p

commutative

(outer)

diagrams p g

has

a unique

in t h i s if for

case.

t

An

e

for

all

.

The

p 6 E

with

following

all

properties is

E-factorization

is rigid,

and

(up to

canonical

t p = eg

mt =m

can

locally

only

'

#

7

m

and

is

easily

orthogonal, that

if

is,

and

for

of

h

write

orthogonal

, if

proved:

every

every

locally

property

locally

pm

(e,m)

locally orthogonal,

an e n d o m o r p h i s m

t = I . This

isomorphisms)

i

; we

called

is c a l l e d

be

2,

mt = h

(e,m)

p 6 E ; it

E-factorization

te = e

~o

E-factorization

p ~ (e,m)

I

it

i

.-

one

~.

orthogonal

orthogonal orthogonal t

yields

p ± (1,m)

one

has

uniqueness

E-factoriza-

tions. We

examine

thogonal

Lemma

the

E-factorizations

1.1:

Since

Let

e' 6 E

relation

Then, (e'',m)

and

between

in P r o p o s i t i o n

Let every morphism

E-factorization. Proof:

precise

fe:

be

e'

in with

a locally

me' 'e = e'

one

has

K

locally

1.2 b e l o w .

orthogonal For

that

and we

or-

need

have a locally orthogonal e,e' 6 E

can hold only if

orthogonal a unique

E-factorization t

such

that

f E E. of

f.

177

e' e

commutes.

The

I ~t

i

I

:[

e''

v

diagram e

ei trivially

commutes

t f = e''

. As

I

follows

for

d = e''

(e'',m)

tm=

is

assertions If

is

M = {mEMorK

(iii)

Locally

a unique

d=

tf

. Therefore

=m

isomorphism

consists

is

and

of Lemma a

f =me''

locally

6 E

. []

the following

1.1

orthogonal

of isomorphisms

For

E-factorizations

e,e' 6 E

such

a locally

orthogonal

t

with

e''

te=

again

hence

(ii) => (iii) (e,m)

of

locally

orthogonal isomorphism.

to

f

and

by

e'e =me''

In o r d e r

rization

an

an

be

therefore,

isomorphism,

is

is

orthogonal

(i) = > (ii)

(e'',m)

such

m(tm)

I ~e : (e,m)

EN M

for

[-facto-

only.

is closed under composition.

E

6 E and

also

from

and m

but

1.2:

(ii)

Proof:

t

. Thus

, then

rization}

let

,

rigid,

i

m

Under the assumption are equivalent:

Proposition

(i)

I

i

e''

(tm) e'' = tf = e'' it

~]

m

is

m t : e'

show

that

orthogonal

by

the

composite

. By

1.1,

mC

of

Lemma E

e'e e'e

1.1

. By

(i)

exists,

. There

one

has

, m

is

an

.

E-factorization Since,

that

E-factorization

Lemma E E

are orthogonal.

the

locally

it

(e',m')

(ii),

orthogonal

suffices

e'e 6 E

to

of

m

one

has

show

[-facto-

that,

for

, necessarily a unique

e' t

that e'e

I

I

t

m'

I

e commutes. zations

From

easily

gets

is w e l l

known

Remark:

The

equivalence

for

factorizations

7.3,

now

rigidness

(iii) = > (i)

Lemma

one

the

~ of

the

the

two

locally

equations

(and

of

m

of

te ' = I

trivial).

(ii),

(iii)

cones.

orthogonal and

E-factori-

e't = I

[]

has

But

in

been that

proved case

in the

[13] proof

a

178

is e a s i e r

since

then one has

phisms whereas,

We mention tiple for

to c o n s i d e r

in the p r e s e n t

without

pushouts,

if

proof

K

its m o r p h i s m s .

that

admits

More

E

E

E

of e p i m o r -

is a r b i t r a r y .

is c l o s e d

locally

precisely,

f

only classes

context,

under

orthogonal

pushouts

and mul-

E-factorizations

if in the p u s h o u t

diagrams

A

e

le I ip

e °

.

f'

B.

~C

1

Pi we have

e 6 E

and

e. E E

f o r all

i6 I

(I a n y

set)

, we

also have

1

e' 6 E the

and

d6 E . Vice

existence

phisms.

of

This was

[13 ], C o r o l l a r y We

conclude

orthogonal stence

locally

Consider

this

pushouts

E-factorizations

generally

section

adjoints

K , and

a commutative

of t h o s e

for

~/ields

of all m o r -

factorization

of c o n e s

in

6.5(I) . by

showing

that

the

can be equivalently

to v e r y

the c a t e g o r y

of

the e x i s t e n c e

orthogonal

shown more

E-factorizations

of r i g h t

morphisms

versa,

MorK

natural

of

by the exi-

locallY

functors.

of m o r p h i s m s

a morphism

existence described

of

[u,v] : f ~ g

K in

: objects MorK

are

the

is g i v e n b y

K-diagram r f

Composition MOrEK

whose

is t a k e n objects

g

from

K . MorK

contains

are a l l m o r p h i s m s

in

(ii)

MOrEK

is coreflective in

M o r K.

MOrEK

is coreflective in

Mor K

subcategory

with coreflections

[1,m].

Proof: O n e e a s i l y p r o v e s t h a t E-factorization gives

full

1.3:

(iii)

of type

the

E

The following assertions are equivalent: admits locally orthogonal E-factorizations for all

Proposition (i) K morphisms.

*i

i

of

(i) < = > (iii).

be any coreflection.

(e,m)

f , iff

[1,m] : e - f

In o r d e r

to s h o w

Since

1 6 E

is a l o c a l l y

orthogonal

is a c o r e f l e c t i o n .

(ii) = > (iii),

the morphism

let

This

[k,m] : e ~ f

[1,f]: I ~ f

factorizes

179

as

[k,m][r,s]

Now, t h e

= [1,f]

with

[1,e],[rk,sk] are

equalized

particular chosen

e

by rk=

as a n

Consider are

all

the

kr =I.

: I - e

coreflection

I. T h e r e f o r e ,

identity.

the

(e,m)

a morphism

commutative

[ 1,m]

k

is an

, so t h e y

must

isomorphism

and

be

equal;

may

be

in

even

[]

category

pairs

, and

jr,s] : I - e ; in p a r t i c u l a r

MOrEK-morphisms

two

Fact

such

K

of

that

[r,t,s]

the

factorizations domain

: (e,m) - (p,n)

of

m

of

K

is t h e

in F a c t

K

: objects codomain

is g i v e n

of by

a

K-diagram r

el

t

ip j,.

mI

In .L.

Composition all

is

taken

from

E-factorizations

and

FactEK-MorK

:

K

(i)

. Let

FactEK

be

the

full

subcategory

of

let

, (e,m)~me

VE b e t h e composition

PropoSition

K

£

functor. The following assertions are equivalent:

1.4:

admits locally orthogonal

E-factorizations for all mor-

phisms. (ii)

VE

has a right adjoint.

(iii)

VE

has a full and faithful right adjoint.

Proof: T h e e a s y p r o o f torization

of

f

counit

VE

at

of

immediately (ii)

~

the

(iii)

: VE(e,m)

(1,f)

-

(e,m)

hs =

satisfy equal.

I.

the We

that

f)

to

is

left

consider

- f. with The

have

rk = []

is a l o c a l l y

There

to t h e

of

(i)

an arbitrary

=

and

sh =

= I,

orthogonal

From

this

So we

one

still

VE-Couniversal

[1,1],

[rk,tk,sh]

(i.e.

a

obtains have

to prove

morphism [r,t,s]

in p a r t i c u l a r

[k,h]~TE[rk,tk,sh], so t h e

f-fac-

VE-Couniversal

FactEK-morphism

[I,e,I],

[k,h]VE[1,e,1] I

(iii).

is a u n i q u e

[k,h]VE[r,t,s]

is

reader.

and

FactEK-morphisms

equation

is a n i s o m o r p h s m .

(e,m)

[1,1] : V E ( e , m ) - f

equivalence

and

[k,h]

and

iff

kr

: =

I

: (1,me) - (e,m) so t h e y

VE-Couniversal

are

morphism

180

The

third

characterization

E-factorizations pushouts (e,f)

by a d j o i n t

in K. Let

with

is g i v e n

common

RelEK

be

domain

the

and

by a c o m m u t a t i v e

of the e x i s t e n c e

functors

requires

following

e c E, and

of

locally

orthogonal

the e x i s t e n c e

category:

of

objects

a morphism

are

pairs

{u,r,t}: (e,f)~(p,g)

K-diagram

u

f

e

Composition

is

PE here

f(e)

'f

r

taken

denotes

~

K. T h e r e

Fact

the

1.5:

(i) K admits morphisms.

and

from

: RelEK

proposition

P

K,

(fixed)

is

a canonical

funetor

(e,f)~(f,f(e));

pushout

of

e along

f.

The following assertions are equivalent: locally orthogonal E-factorizations for all

(ii)

PE

has a right adjoint.

(iii)

PE

has a right adjoint with counits of type {1,1,h}.

Proof:

(i) ----~ (iii)

Let

(e,m)

be a l o c a l l y

orthogonal

let

the p u s h o u t

property

~

there

(f,u)

be

an o b j e c t

in F a c t

E-factorization

is a u n i q u e

h

such

that

turns

out

of

K

uf

. By

" < ~ - ~ _ f(e)

•

commutes.

Now [ 1 , 1 , h ]

: PE(e,f)

~

(f,u)

to

be

PE - c o u n i -

versal. (iii) one

can e a s i l y

torization

versal. Fact

~

(i) G i v e n

show

of

f

(ii)

~

that

[I,1,u]=

[j,k,h]

In p a r t i c u l a r

(iii)

and

tk=1

Their

PE-images

PE

Let

gonal

orthogonal

[j,k,h]:PE(e,g) that

{u,r,t}

one

j

and

looks

and at the

the d u a l

a class

M-factoriation

~

M

(e,m)

kt=1 two

: (1,g) by

(f,u)

are

with

u=1

E-fac-

be P E - c o u n i -

isomorphisms.

(f,u)

The

can be w r i t t e n

{u,r,t}: (1, f) ~ To p r o v e

the

as

(e,g)

equations

RelEK-morphisms --

(e,g)

[j,k,h]

concept

~

k

by a u n i q u e

jr=1

are e q u a l i z e d

We m e n t i o n For

is a l o c a l l y

(e,hf(e))

{e,1,1},{erj,rj,tk}

torizations:

arrow

[1,1,u]:PE(1,f)=(f,1)

we h a v e

rj=1

a PE-Couniversal

.

It suffices to s h o w

K-morphism

such

of

, so they locally

of m o r p h i s m s is a l o c a l l y

of

are equal.[]

orthogonal

E-fac-

K , a locally coortho-

orthogonal

M°P-factori

-

181

(m,e)

zation

property

in

K Op,

pictured

that

is, one has m C M

and the d i a g o n a l i z a t i o n

by

e

m

'T

"i

It

g

h

I i v

I

n6M 2. E x a m p l e s

We r e s t r i c t t oriz a t i o n s general:

which

ourselves

are not o r t h o g o n a l

the c o n s i d e r e d

f

and

for all

x

regular,

iff every

zation ker

and

g

(e,m)

f ~ker

with

common

y , fx= fy

of

g

e. Now,

for

E-fac-

The first one is

for all m o r p h i s m s

E

(Isbell[6],

Kelly[7]).

domain we w r i t e

implies

with

gx =gy

of any

iff

e

the class

ker

For two

f £ ker g , if,

. An e p i m o r p h i s m

ker e ~ ker g factors

is regular,

f

exist

orthogonal

and coequalizers.

Regular factorizations

2.1

of locally

a priori.

factorizations

category with kernelpairs

morphisms

to e x a m p l e s

as

g=he

is a regular

of all regular

e

is

. A factori-

epimorphism

with

epimorphisms,one

is a locally orthogonal E-factorization (hence unique up to an isomophism) and that a locally orthogonal E-facotrization is regular, if K has coequalizer~.Furthermore, if K admits regular factorizations, the following assertions are equivalent: easily

shows

that every regular factorization

In every regular factorization (e,m), m is a monomorphism. (ii) Regular epimorphisms are closed under composition. (iii) K admits orthogonal E-factorizations for all morphisms. (i)

The c a t e g o r y these

are

regular

Cat of small

locally

orthogonal

epimorphisms

the dual

of the c a t e g o r y

many

factorizations

there

is no o b j e c t i o n

zations. fibres.

to obtain

So let Then

continuous iff there factorize

E

mapping is a f

y C Y

factorizations;

since

in Cat.

The

compositions same holds

of

in

of semigroups.

have

their

to c o n s i d e r i n g orthogonal

but

be the class

Y

just

range

orthogonal

factori-

in Top with

E-factorization

connected for every

Xo,X I in X equivalent,

C in f-ly with Since

But

as long as one does

maps

: call two points

map.

[4]

of m o n o t o n e -

to all T1-spaces.

locally

orthogonal

and a c o m p o n e n t

over the q u o t i e n t

from E i l e n b e r g %

all spaces

of q u o t i e n t

a locally

f:X ~

Arising

found g e n e r a l i z a t i o n s

and e x t e n d e d

Top admits

regular

but not o r t h o g o n a l

factorizations.

authors

light

not e x p e c t

admits

need not be r e g u l a r

2.2 Monotone-light investigations

categories

Xo,X I C C, and

E is not closed under

com-

182

position,

this

analogous

results

factorization

connectedness.

are

That

is not

obtained

in this

case

may

be

x6R

,x~O}u{(O,y) :-I~y~I}~R 2

seen

singleton

from

space

In a m o r e

the

first

E

onto

complete

context

concrete

of

K , that

tains

a non-void

one

of

K

is

A

object

For m o r e to B ~ r g e r ' s

and w h i c h

considerations

[2].

are p o s s i b l e

such

admits

that

each

locally

~x,sin!) :

map

onto

a

fibre

2.3

We

of

of

K be

K

under

[5]

the

also

for

but

Let

an initisubcate-

which

epimorphic

K

Then

conimages

admits locally

is an i m m e d i a t e

conse-

1.2 above. interested

mention

E

the

is c o n t a i n e d

orthogonal

any d i f f i c u l t y

the class of all quotient maps

MA

for S t r e c k e r ' s

factorizations; for i n s t a n c e , maps

of

composition

be a c o n n e c t i o n

subcategory

investigations

thesis

A

is c l o s e d

A . Theorem

general

without

of s u b o b j e c t s .

and of P r o p o s i t i o n

recent

let

is a full

MA-factorization for

of this

succeeding

can p r o v e

and

families

with fibres in

quence

and the

under

subspace

for t e r m i n o l o g y ) :

category

of c e n t e r e d

orthogonal

~

closed

of the

Completely instead

. general

gory

and u n i o n s

p

in g e n e r a l .

connectedness

is not

projection

(cf. H e r r l i c h - S a l i c r u p - V ~ z q u e z [ 5 ] ally

orthogonal

for p a t h w i s e

that

[11] class

is r e f e r r e d

corresponding

submonotone-superlight of t h o s e

in a p a t h

not n e c e s s a r i l y

reader

quotient

component,

orthogonal

Top

E-factori-

zations. 2.3 Eilenberg-Moore factorization. and

functors

right We

we

adjoint

claim

consider

functors

that

the

and

(non-full)

its

subclass

In a c a t e g o r y subcategory M

Cat of c a t e g o r i e s

Catra

of all

of all m o n a d i c

functors.

, every right adjoint functor U:A ~X has a M-factorization, provided A and the Eilenberg-

in C a t r a

locally coorthogonal

Moore category x T ( T is a m o n a d i n d u c e d by U) have coequalizers. Proof: It is k n o w n that the e x i s t e n c e of c o e q u a l i z e r s in A makes

the

if we

consider

comparison the

functor (outer)

• XT

!

S

~

XT

have

a left

adjoint.

Now,

diagram

UT

K A

P

K:A

commutative

~X

1

'

IV I

Id

~ ,yS

US

l

R

~y

in C a t r a for any m o n a d with VK= P and uSv=

$=(s,~S,~S), then t h e r e is a u n i q u e f u n c t o r V RuT; we r e s t r i c t o u r s e l v e s to s h o w u n i q u e n e s s :

for

V(X,x)

a T-algebra

structure

y:

(X,x),

SRT -- RX,

m u s t be w r i t t e n

and the T - h o m o m o r p h i s m

as

(RX,y)

x: (TX,~X)

with ~

an S-

(X,x)

is

183

tralnsformed

into

V(TX,~X)

an

= VKFX

S-homomorphism = PFX

=

Rx:

(uSpFx

V(TX,vX)

-

(RX,y).

Since

, u$cSpFx)

t

(where

F

is

left

adjoint

y = y • SRx • SRqX (One R

notices

to

show

proof.)

a

One an

does

uniqueness has

arguments

one

not of

use V;

the

it

for

right

is,

coequalizers

(apply,

necessarily

obtains

• SRnX. adjointness

however,

V

has

instance,

used

a left

of

in

Corollary

7 of

P

the

adjoint

New factorizations from old by adjoint functors.

functor

let of

one

XT

U)

and

existence

by

adjoint

[12]

to

Let

U:

= RUT).

2.4 be

the

Since

triangle uSv

that

to

= Rx • uSeSPFX

E' all

with

be

the

left

pushout

morphisms

then

in

the the

F.

For

of

FE

pushouts

of

closure

which

obtains

analogue

adjoint

are

following

orthogonal

lifting

a class in

E

A,

of

that

a morphism theorem

is, Fe

which

A -

X-morphisms, E'

consists

with does

e E E.

not

have

case:

Let A have pushouts and let E consists of X-epimorphisms only. Then, if X admits locally orthogonal E-factor~zations, A admits locally orthogonal E°-factorizations. Theorem

2.5:

Proof:

In

order

f:

A -- B

we

form

Uf

and

then

the

to

find

the

an

locally

pushout

E'-factorization orthogonal

is

claim

that

So w e

consider

a unique

m' : C - B

(e'm')

is

with

a locally

a commutative

FUf.

upper

Fx

y: =

f

and

m'h

=

eB • Fm.

E'-factorization

of

We f.

E

b

A

X - UA,

=

Iv .,

a

eB.

m'e'

orthogonal

~ FY

D

x:

of

diagram

Fp

ul the

(e,m)

C

FX

where

A-morphism

FZ

A

There

the

diagram

Fe

FUA

of

E-factorization

----~C part

Y - UB fau

m is

~ B

a pushout

with

= bv.

Fp

eA=

eB.

with Fx

= au,

Fy.

Fp

p 6 E. W e eB • F y one

have = bv.

derives

morphisms From fx = y p .

Hence

X

184

there

is a

perty,

an

t: Y - Z

of e p i m o r p h i s m s , bp'

one

gets

Remark:

also

U

finite number

2.5 is v a l i d

This w a s

To this

limit n u m b e r

~ Sl

than

(el,C)

, if

composite

(el,C)

exists

(and

is said to be in

to a

E

if

(of a m o r p h i s m

f = m e l)

(el,C)

(el,C)

l-factorization

E . A

f)

there

is a u n i q u e letting

either

B = e +I

B

D(O1)

•

~.ee

. , , . , .

,

C(O1 )

and

mt~ =h

(E,l)-factorization

((el,C),m)

The

con-

such that

me l if in

is orthogonal diagram

•

•

.

,.

6

,

as p i c t u r e d

,

I

~I

m

transformation

is c a l l e d

hi

D E

(pl,D) ± ((el,C),1~J

and

by the d i a g r a m

tl, ,'' C(~l)

natural

mt I = h

O ~ ~< 8S I

D(yl)

~.

. We w r i t e

colimits.

such that

whenever

. ~It . .B. . . . . .t~ ,;'

is a u n i q u e

(of the

~i h

,...,t I

a limit ordinal,

,

of m o r -

((el,C),m)

m

(pl,D)

m

C(e6)

T h i S m e a n s : ther e t °o = g

'

tl,t2,..,t

t ~I~,

as for

is a f u n c t o r w i t h

if for each c o m m u t a t i v e

D(~B)

g ~. . . . . . . . t .l . .',.

set

([,l)-factorization,

, tBD(aB) = C ( a B ) t a

or

6

I

eI

sequence

to =g

to

l-composite

is a pair

Pl

and,

number

e a8 @ [ (e < l;B = ~ +I)

l-composite

((el,C),m),

gi

~

the a b o v e

and a m o r p h i s m

. It is c a l l e d an

is in

from

C : I +I ~ K

an in-

I , considered

is the

which preserves

addition

of m o r p h i s m s

the o r d i n a l

less

(and

of a c o m p o s i t e

factoriza-

end we i d e n t i f y

e~8(e < I;8 = ~ + I ) )

l-factorization

=

(in the d u a l si-

into p o s s i b l y

C(Ol) = e I

A

shown

is the c o l i m i t of the

. A pair

C(e8) = e eB)

m'sp'

of a m o r p h i s m

morphisms

sisting

so from

for r e g u l a r

first

be the u n i q u e m o r p h i s m

y6(y < K;6 = y +I)

consists

2.2.

set of o r d i n a l s

~6

E

[]

([,l)-factorizations

of factors.

Since

of e p i m o r p h i s m s ;

factorizations

and let

sv = h • Ft.

is u n i q ue.

Lemma

orthogonal

w i t h the o r d e r e d

~ 6 . Each

s

of T h e o r e m

[10],

W e next c o n s i d e r

phims

consists

is f a i t h f u l .

by M i t c h e l l

a category,

sp' = e'a,

E'

The a n a l o g u e

3. L o c a l l y

I~ I

tp = ex, m t = y, and, by the p u s h o u t p r o -

with

m ' s = b', and

tions provided tuation)

with

s: E -- C

t :D~ C

with

in this case.

loeclly orthogonal

, if

An

185

(pl,D) ± ((el,C),m)

for all

(E,1)-factorizations E-factorizations Analogously thogonal has

whenever

canonically t : C-C'

with

and

We did not i n t r o d u c e reason:

respect

Proposition

C(O1)

= e

Proof: in

and

unique

E =

1.2,

h ~ I

•

.°

.

.

.

,

m

C:

B = a + I

(e,m)

I + I ~ K

that

(Ph ' D)

be d e f i n e d

• ((e,C),m)

whenever

the

±

•

,

°

.

.

.

.

I I t8

for all

hph = meg,

•

•

P

following p

•

diagram

t

=

tBp

~8

is a

•

i tl I v

commutative I 1

Ihpl m

for

m

O < ~ < B ~ l

The o r t h o g o n a l i t y

pOl C E. Thus

~tl I

i

&

I

eg I

(pl , D)

there

I

since

oi

by

B~

I t61

.&

(1,m)

for

such that in the d i a g r a m

8 = limit ordinal.

pOl

may be r e -

((e,C),m)

I ~ a ~ 6 ~ ~.

, mt I = h, and

or

with

composition.

E-~ctorization

J

oi

Therefore,

(E,l)-factorizations

under

I

eg = t l P

are

equivalence

(E,l)-factorizations,

(E,l)-factorization

I t~l

+

(E,I)-

that e v e r y l o c a l l y o r t h o g o -

p~B =

natural

I

s h o w that,

e

rendering

first

is not c l o s e d

for

show that

I iltl

implies

of o r t h o g o n a l

by l e t t i n g

oi

g

and

(unique)

locally orthogonal

tl,t2,...,t h

I

we h a v e

locally orthogonal

(E,1)-factorizations.

£

= I

is, we m u s t

sequence P

.

orthogonal

C(a6)

We m u s t

E. T h a t

m'tl =m

3.1: An o r t h o g o n a l

any f i x e d o r d i n a l

is, one n e c e s s a r i l y in the a b o v e d i a g r a m .

I

o n l y if

g a r d e d as a (locally)

l o c a l l y or-

is of the f o r m

orthogonal

to P r o p o s i t i o n

that

of the same m o r p h i s m

is a

I

m a y be i n t e r e s t i n g

(locally c r t h o c o n a i )

that

h =m

there

We w i l l p r o v e

e admits

, and

a concept

(E,l)-factorization

K

are rigid,

((ei,C'),m')

i.e.

to = I

the

(locally orthogonall

one e a s i l y p r o v e s

, g =I

(ek,C),m),

isomorphic,

for a s i m p l e

with

of this one has that two

factorizations

if

I =1

D =C

E . The

I

to the case

As a c o n s e q u e n c e

in

m a y be i d e n t i f i e d

of S e c t i o n

(E,k)-factorizations

t =I

nal

(pl,D)

there

of

(e,m)

is a u n i q u e

tI

186

6

Let for

be

an

I < a < when

t 8 =

ordinal 6

~ =

a + I

diagram

we

have

6 + I

then

with

I < 6a

top

or when

there

is

6 ~

~

and

= t6

and

is

a limit

a

t8

a unique

suppose

inductively

= h p al

mt

ordinal

rendering

for

and the

that

a unique

1%

6 < ~.

If

following

commutative pa6

I

l

It 6

h P 6~

1

since

pa6

ty

K,

in

But

in

•

(1,m).

there

the

is

If

addition

6 <

6. T h i s

completes

reader.

[]

of

Proof: W e m u s t (p6,D)

in

I + I -

hp6.

E. T h e K

with

Since

C

all

since The

main

Theorem

orthogonal i

= ±

I

rendering

morphisms t

p

result

3.3:

6

for

proper-

for

= h p B l p 68

I S

for

can

be

6 <

all

left

to

(E,h)-factorization (E,~)-factoriza-

0 < 6 < ~

where

t 6,

= e6Pt6,

the

trivially

C

6 < ~ _< ~ < there

is

following

e

this

is

diagram

all

to

I. A s s u m e

a unique

melg

=

sequence

commutative I

66+1

I,

for

extends

I

6 < p ~ and

( (C (O6) ,C) ,m C ( 6 1 ) ) ) K

all

are

e

uniquely

immediately

I

Pt

determined

by

t6

proves

the

assertion.D

in

has a locally

is:

Suppose

that every morphism

E-factorization

and that colimits

More precisely we asst~ne that each functor

colimits

colimit

Uniqueness

with

±

6 + I -

((el,C),m)

eB6_

have

mtBp6B

step.

(p6,D)

ol

the

= too

orthogonal

D:

n66

we

the

66

t~p

6 + I.

that

D(au)

by

that

since

for each

functor

(pl = p 6 , D )

ordinal,

as a locally orthogonal

p oi

e

such

inductive

to

show

t 1 , t 2, .... t 6 , . . . , t l

But

the

((c(06),~),mc(61)I

tion

limit

A locally

3.2:

the restriction

a t6

= hpSl

may be regarded

((el,C),m)

D:

is

mt8

the

Proposition

B

a unique

of chains and

C(e,8) E E

nal, has a colimit in K; then ~ serving colimits of chains.

for

f

of chains

~: I ~ K

O ~ a <

K

such that

exist I in ~

6 < I, 6 = e + I, I

may be extended to a functor

K.

preserves a limit ordi-

C: I + i ~ K

pre-

6.

187

Then for each ordinal gonal

Proof: tion

I > 0

every morphism

f

has a locally ortho-

(E,h)-factorization.

Let

of

f.

((e1,C1~ml)

Suppose

be a l o c a l l y

inductively

that

orthogonal

for all

(E,1)-factoriza-

I < 6 < I

((e6,C6),m6)

is a l o c a l l y

orthogonal

(E,6)-factorization

of

f

whose

restriction

to a l o c a l l y

orthogonal

(E,~)-factorization

of

f

given

by P r o p o s i -

tion

((e

)

3.2

is

,C ),m

for

O < ~ < 6. D i a g r a m a t i c a l l y

f=m If

l

is of the

zation

(eBl,m)

C6

on the

and

let

C(8~)

of

of c h a i n s

6+ I m6

restriction

e I = e61e6.

= C

(6~)

for

since

each

tension

C:

morphism

such

C(1)

form

that

show

l

If

1

C with

value that

tl,t2,...,t 6

such

me6Xt6

and

then

exists

since a limit unique tisfies

(e61,m)

then

map with the

tI

required 3.3

B

equations

that

the

2 c a n be e x t e n d e d

provided

that

all

by

colimits

be the u n i q u e exists

since

is a l o c a l l y

or-

we m u s t

show

that

= melg.

From

hpl

is a u n i q u e

t6p~6

ordinal

m a p on

whenever

~ 6. If

I = 6+ I tlp 61 = e 61t6

and

colimits

~ < I. Thus of course, locally

exist.

sequence

= e~6t

mt I = h

If

1

and

is the

t I ,... ,t I

unique.

orthogonal

to l o c a l l y

colimits

=

is an ex-

E-factorization.

induced

a n d is,

words,

E. So let

that

for all

m

C

I ~ K

there

defined

and

orthogonal

C:

preserves

let

there

a limit such

is the

it is c l e a r

in that

by

a S 6

C.

just

= hp 61

tl

tlp ~I = e ~It ~

of S e c t i o n

torizations

or

a unique

is a l o c a l l y

ordinal

By T h e o r e m zations

6 = ~ + I

of

K

for

~ S I, w h i c h

In o t h e r

(pl,D)

it f o l l o w s

O S e < 6 there

all

as

E-factori-

I -

define C

E. We

colimit

hypothesis that

in

f.

l+

by h y p o t h e s i s

of the

((el,C),m)

for

ordinal

for all

((el,C~m)

C:

= e61C6(~6)

functor

Hence

of

orthogonal

a functor C(~l)

(el,C)

(pI,D)

±

a locally

= m

(E,l)-factorization

inductive

and

does.

thogonal

the

take

is a l ~ i t

mC(e6)

object

we

we

6 s ~ < I. This

I + 1 ~ K

is the

Finally

o

and d e f i n e to

we h a v e

sa-

[] E-factori-

orthogonal

More

is

precisely

(E,l)-facit is

188

sufficient C:

~ ~ K

that colimits with

ordinal

~ l, or,

zation.

In the

described

in

dually,

latter

functors.

category

X6

object

A

X~

(A)

(f)

We

Catra

Catra locally tion,

for

for that

and

the

u < 6

6

a limit

of the E i l e n b e r g - M o o r e - f a c t o r i the a d j o i n t with

ra

limit

{A }

tower

}~ < 6

construction

C ( ~ , ~ + I): X ~+I of

C

consists

for

< 6

of

~ < 6

o f the

A~

f: A ~ B

of morphisms

~ X~ 6 M

where

objects

and a morphism

÷ B

functors and

f

an

in

of

X~

X~

with

~ < 6.

or a d j o i n t

universal

shows tower

ordinal

~

properties

that

f

exactly

in

locally

dual

coorthogonal

for a f u n c t o r

M = monadic

functors

"regular

K

for

tower"

a given

U

in

to t h o s e p o s s e s s e d

or

in a c a t e g o r y

epimorphisms

the

construction,

and

(E,l)-factorization,

a morphism

E = regular

preserving O ~ ~ < ~

C(e6) : X 6 ~ X e

the p r e c e e d i n g

orthogonal

for

case

is a c o l l e c t i o n

for a g i v e n

has

case

Cat

{f~ I f~ : A s

= f

remark

for

~op ~

In t h i s

(M,l)-factorization, in

C:

= A

is a c o l l e c t i o n C(~)

in t h e

and projections

of

C(~)

exist

case we obtain

[8] w h e r e

= monadic

with

of c h a i n - c o l i m i t

C ( ~ , ~ + I) E £

by the

construc-

ordinal

K.

References

I. H . A p p l e g a t e in M a t h .

137

2. R . B ~ r g e r , Thesis

Kategorielle

compacts,

tures, 6. J.R.

7. G.M.

Sur

les

in:

Lecture

von

Zusammenhangsbegriffen,

factorization, 1974),

pp.

in:

Lecture

continues

22

296.

(1934),

3

Structure

292-

and R.V~zquez, (1979),

Notes

Light

d'espaces

J. A u s t r a l .

Math.

Soc.

9

m~triques

factorization

struc-

189 - 213.

of c a t e g o r i e s ,

Monomorphisms,

in

I -19.

Bull.

Amer.

Math.

619 - 655.

Kelly,

Notes

5 6 - 99.

transformations

E.Salicrup,

Quaest.Math.

Isbell,

(1966),

functor

Fund.Math.

5. H . H e r r l i c h ,

pp.

Beschreibungen

(Springer-Verlag,

4. S . E i l e n b e r g ,

cotriples,

1981).

On adjoint 420

Iterated

( S p r i n g e r - V e r l a g , 1970),

(Hagen

3. B . D a y , Math.

and M.Tierney,

epimorphisms, (1969),

and pull-backs,

124 - 142.

Soc.

72

189

8. J . M a c D o n a l d and A.Stone, Cahiers T o p o l o g i e Geom. 9. J . M a c D o n a l d and A.Stone,

The tower and regular decomposition, Diff~rentielle

(to appear).

E s s e n t i a l l y m o n a d i c adjunctions,

Proc. Int. Conf. C a t e g o r y Theory G u m m e r s b a c h 10. B.Mitchell,

1981

in:

(to appear).

The d o m i n i o n of Isbell, Trans.Amer.Math. Soc.

167

(1972), 319 - 331. 11. G . E . S t r e c k e r ,

C o m p o n e n t p r o p e r t i e s and factorizations, Math.

Centre Tracts 52

(1974),

123 - 1 4 0 .

12. W.Tholen, A d j u n g i e r t e Dreiecke, Math. Ann. 13. W.Tholen,

217

(1975),

C o l i m i t e s und K a n - E r w e i t e r u n g e n ,

121 - 1 2 9 .

Semi-topological

functors

I, J.Pure Appl. A l g e b r a

(1979), 53 - 7 3 .

John M a c D o n a l d

Walter Tholen

Mathematics

Fachbereich Mathematik

Department

U n i v e r s i t y of British C o l u m b i a Vancouver, Canada

B.C.

V 6 T

IY 4

Fernuniversit~t 5800 Hagen I Fed. Rep. of G e r m a n y

15

RI~4ARKS ON RADICALS IN CATEGORIES

L. Mirki and R. Wiegandt

i. Introduction In order to develop a general radical theory in a category, one of the usual ways is to deal with categories like that of rings. In such a category radicals can be defined by functors or by assignments or by properties. In this note we intend to clarify the relation between various systems of axioms inposed on the category and to c(mpare the various ways of defining radicals. Thus in section 2 we exhibit that the system of axioms introduced by Holc(mlt~ and Walker [ 5] is practically equivalent to that used by Andrttnakievi~ and Rjabuhin [2] . It turns out in section 3 that a radical functor in the sense of Carreau [ 3] is nothing but a Hoehnke radical [ 4] , and we prove that a radical functor is complete and id~tent

if and only if the corresponding Hoehnke radical is an ideal-radical.

We also exhibit the equivalence of the latter radicals with those of ~ul'ge[fer [ ii] and Amitsur [ i] , and describe radical and semisimple classes by intrinsic properties. A general reference for terminology is Mitchell [9]. As usual, monomorphisms will be denoted by >

> . For a normal monomorphism and a normal epimorphism we

shall use the symbols ~

> and

~, respectively. If

exact sequence, then the factor object

C

B ~

will be denoted by

> A

~ C

is an

A/B.

2. Axioms on categories Several systems of axioms have been imposed on categories in order to enable the development of a general radical theory. Many of these systems of axioms aim at obtaining categories similar to those of rings or

~-groups - similarity means

that one can use techniques, in particular the iscmorphism theorems in their familiar forms, like in the latter categories ([i] , [2], [3], [5], [6], [ii], [12] , [13] , etc). Some other authors, on the other hand, tried to develop general radical theory in a more general setting; accordingly they could not get too far without making further restrictions either on the category or on the radicals under consideration (for recent investigations in this direction we refer to [14] ). Here we want to give a clear picture on the present situation in ring-like categories. Almost all the systems of axicms used in them for the purposes of radical theory, are slighter modifications of that of ~ul'ge~fer [ ii] . We shall use the form presented by Andrunakievi~ and Rjabuhin [2] : it is supposed that the considered category

C

satisfies

191

(~a_) (~2) (~)

C

has zero object,

every object possesses a representative set for its normal subobjects, for each object its normal subobjects form a c(mplete sublattice under the natural partial order of subobjects,

(AR4)

every morphism has a kernel,

(~5) (~6)

evel-y morphism has a normal epimorphic image, normal epimorphisms carry normal subobjects into normal subobjects.

We reni~rk that (AR2) is i~i0osed in [ 2] in the stronger form that all subobjects of any object form a partially ordered set, but the present weaker form suffices both for the investigations in [2] and for our purposes. HoloDmbe and Walker [ 5] used a seemingly different system of axioms. Here we exhibit, however, that the two systems are equivalent. First of all, let us have a closer look at the categories considered by Fblcombe and Walker. Part of their theory makes use of the second isomorphism theorem without a clear formulation in the categories for the latter to hold. They considered a category

C

satisfying

the following axiem~: (HWI) (HW2)

C in

has a zero object, kernels and cokernels, C

intersections of arbitrary sets of normal subobjects of an

object exist, further, every object possesses a representative set for its normal subobjects. The intersection of normal subobjects is again normal ([5] Ler~m% 1.5). (HW3)

Every morphism of

C

factors through a cokernel followed by a mono-

morphism. Under these axioms

C

has inverse images of normal subobjects which are again

normal ([5] Lemma 1.4) and has epimorphic images ([5] Lemma 1.9). (HW4)

For every cokernel t (t-i (Z))=Z.

t: X---->Y and kernel

Z --> Y

we have

The first isomorphism theorem is proved to be a consequence of these axicms ([ 5] Theorem i,ii). Finally we ass~ne (HW5)

Any set of normal subobjects of any object possesses a union which is again a normal subobject.

Let us notice that in the presence of the systems of axicms ~WI)-(HW5 ) the normal subobjects of any object form a complete lattice under

n

and

u

For short, the systems of axic~s (ARI)-(AR6) and (HWI)-(HW5) will be denoted by (AR) and (HW),respectively. THEOREM i. (}~) implies (AR), and if any two s.ubobjects of objects in have an intersection, then the cenvers~e also holds.

C

Proof: In view of what has been said above, it is clear that the system (HW) implies (ARI)-(AR5). In order to prove (AR6), let kernel

ker t: Z -->X , and

N

t: X --->Y be a cokernel with

be a normal subobject of

X . By [5] Lenma i.i0

192

t(NU Z) is a normal subobject of u n i o n w e have object of

t(N).

plying that

t(Z)= OS_t(N) , by the definition of t(NUZ)

is a normal sub-

Conversely, by the definition of the image

t(N)=t(NU Z)

is a normal subobject of

Conversely, notice that case of (AR6)

Y . Since

t(N OZ)_
(AR) = (HWI)-(HW3)

t(N)
, im-

Y .

is obvious,

(HW4) is a special

and all that needs proof in (HW5) is that the lattice-theoretical

union of normal subobjects is category-theoretical union. Notice also that by (HWI)-(HW4) we already know that our category

C

has images and inverse images

of normal subobjects. Let us consider normal monomorphisms

~. :A ~--->A , iE I , 1

and let

A'

f: A --->B

be the smallest normal subobject of

A

a morphism with canonical deconi0osition

the smallest normal subobject of f (A) containing all the there is a

B:B'>~B

through which each

f(A)NB' exists, it contains all the

f~

1

f(A ),

1

containing each A i , f' A ~ f(~> > B , and

f(A)

by

C

be

f(Ai) . Suppose that

factorizes. Then, provided that and since the

f(A ) are normal

1

subobjects of

iE I ,

1

(~R6), they are normal in

f(A)NB',

too, whence

C_
A'

C wl


On the other hand, we obviously have

f(A')_>C ,

and conversely,

(f')-I(c)>A. 1

for all

iEI

conclude that izes through

, hence f(A')

(f')-I(c)>A' and

C

,

thus

C=f'(f')-l(c)>f'(A')=f(A ') . So we

are equivalent subobjects, whence

fe

also factor-

~ , and we are done.

Remark i. By [ 2] p. 397 we know that the second isomorphism theorem holds under (AR), thus it holds under (HWI)-(HWS) as well. Remark 2. Krer~oa and Terlikowska [ 6] and then Terlikowska-Os~owska [ 12] , [ 13] introduced a self-dual system of axioms which is satisfied in the categories of associative or alternative rings but not in that of not necessarily associative ones. Hence this system of axicms cannot be equivalent to (HW) or (AR), nevertheless there is a strong connection between them. Stlopose we are given a category C

satisfying the system of axic~s (AR). Consider the subcategory

of all objects of cc~positions.

C

C'

consisting

and those morphisms which are kernels or cekernels or their

(For establishing the basic results in the general radical theory

of rings, only these morphisms are really needed. ) Now it is straightforward to

193

check that

C'

satisfies the system of axic~s

AI-A6,

A6 ¢~ and A7 ~'~but not A7

of Terlikowska-Os~owsk~ [12] . Conversely, AI-A7, A6 ~, A7 ~'~of [12] inply most of (AR) but (AR4) and (AR5) only in a weaker form. In the rest of the paper we shall always work in a category

~

C

satisfying

(~). 3. Radicals In his paper [ 3] Carreau presented an elegant treatment of radicals in cate-

gories. What he did in genuine categorical terms is, expressed in the classical language, that radicals can be defined beth by means of a function and of a semisimple class. The same idea is basic in Hoehnke's earlier development [ 4] of radicals in categories of universal algebras. In the category of associative rings Michler [8]

introduced a notion of radical at the same time as Hoehnke did for

universal algebras; these two notions are equivalent for rings. Now we present Carreau's definition of a radical functor in the slightly modified but equivalent version given by Holcombe and Walker [ 5] . By the cokernel subcategory jects are the objects of

C

E(C)

of

C

we mean the subcategory whose ob-

and whose only morphisms are the cokernels of

C .

(In Carreau's terminology t h i s is a special coextensive subcategory. ) A covariant p: E(C)--->C

functor (i)

p

(ii)

is called a radical functor, if

for all

CE C ,

(iii) p(C/~(c))=o

p(C)

is a normal subobject in

for all

C6C

normal subobject

p(C)

in

C

~(p(C))
,

C ,

.

A radical in the sense of Hoehnke is a mapping

(iv)

u : E(C)--->C

is a subfunctor of the inclusion functor

p

assigning to each

CE C

a

satisfying (iii) and for any cokernel

~

frcrn C .

Theorem 2.2 of Carreau [3] states exactly that every radical functor defines a radical and conversely. The most ini0ortant radical functors are cc~plete (which means that if p(B)=B

for some normal subobject

(p(p(A))=p
for all

AEC

B

of

A , then

B _< p(A))

and ideslootent

).

By Carreau [3] Corollary 2.12 we know that complete idempotent radical functors are exactly the radicals in the sense of ~ul'ge~fer [ii]. The latter are defined, following Kuro~, as follows. Consider a function C6 C

a normal subobject

function

p

p(C)

of

C , and call

C

a

p

assigning to each

p-object, if

p(C)=C . This

is said to be a radical function, if it satisfies:

(i) any normal epimorphic image of a (2) for any

p-object is itself a

p-object,

C 6 C , if each nonzero normal epimorphic image of

zero normal subobject which is a

p-object, then

C

is a

C

has a non-

p-object.

194

The other traditional definition of radical, which goes back to Amitsur [i] even for categories,

is also defined by such a function

p

stead of (2) the following two properties are required: (~) every object is a

CE C

satisfying

(i), but in-

(iii) and

contains a unique maximal normal subobject

p(C) which

p-object.

In our category we can use the same methods as in the category of rings, and by standard reasonings we arrive at the following results. PROPOSITION 2. Radicals in the sense of ~ul'~eYfer are the same as those in the sense of Amitsur.

(This completes the discussion of Amitsur radicals in Car-

reau [ 3] . ) By Andrunakievi~ and Rjabuhin [2] V, §2, Theorem 1 we knc~ that a function satisfying

(i) and (a) defines a radical if and only if the class of all

p

p-objects

is closed under extensions, that is, (B) if

B

is a normal subobject of

then so is

A

and both

B

and

A/B

are

p-objects,

A .

PROPOSITION 3. Under (i) and (B), (~) is equivalent with the inductivity condition (y) for any

ascending chain

which are all

(Ix)

of normal subobjects of an object

p-objects, also

uI l

is a

CE C

p-object.

Next, we shall see that the radicals discussed above, have also been distinguished in Hoehnke's theory [ 4] . In fact, consider the relation B 4 A ~ 4

is a "nice"

B

is a non-zero normal subobject of

A .

M-relation in the sense of Hoehnke [6] : clearly

for non-zero objects, and if B 4 A , then there is

jE I

A

is a subdirect product of

Ai ,

4

is reflexive i6 I ,

such that the j-th canonical projection takes

and B

on-

to a non-zero normal subobject of Hoehnke [61, a radical (MI)

p(A)=O

(M2) if

and

p(B)#B

p

A. by (AR6). According to the terminology of 3 is called an 4-radical, if it satisfies

B 4A for all

imply

p(B)#B ,

B 4 A , then

PROPOSITION 4. A radical functor ponding radical satisfies condition

p

p(A)=O .

is cx~nlolete if and only if the corres-

(MI).

Proof: Suppose that (MI) holds, and let

B ~ A ,

p(B)=B , B ~ p(A) . Now we

have B''=:BI(p(A) nB)=B'=:(p(A)UB)/p(A) By (iii) ar~ (MI) it follows that morphism

~

of

(p(A) A B ) ~

>B

p(B')#B'


Alp(A).

, whereas considering the cokernel

, by (iv) we obtain that

p(B'')>_p(B)/(p(A)AB)=BI(p(A)nB)=B''

,

195

whence

p(B'')=B''

, a contradiction.

The converse iaplication is obvious.

PROPOSITION 5. If a radical functor radical satisfies

p

is idenlootent , then the corresponding

(M2).

The assertion is obvious. Moreover, under the validity of (~iI) the converse iaplication is also true. THEOPd~4 6. Every cc~plete and idemlootent radical functor defines an

4-

radical and conversely. Proof: In view of Propositions 4 and 5 all we have to prove is that the radical functor

p

defined by an 4 - r a d i c a l

idempotent. Now there is an exists a

B/p(p(A))

= B/p(p(A))

AEC

such that

. By condition

B/O(p(A)) 4 p(A)/p(p(A))

is ideml0otent.

such that B/p(p(A))4

p(A)/p(p(A))

(iii), however, we have condition

St~opose that

p(A)/p(p(A))#O

(M_l) yields

and

p

is not

. By (M2) there p(B/p(p(A))) =

p(p(A)/p(p(A)))=O

p(B/o(p(A)))#B/p(p(A))

and so for ~ a contra-

diction. As usual, to a radical IR

=

S

= [AEC

P

{A6C

: p(A)=

p

we assign two classes A}

and P

: p(A) = 0},

called the radical class and the semisir~le class of

p , respectively.

Knowing the

equivalence of the previous definitions of radicals, the connection between radical and semisir~ole classes as described in Andrunakievi~ and Rjabuhin [ 2] , V, §2, Theorem 3, yields exactly Theorem 3.10 of Holcombe and Walker [5] and its converse. (The latter is the same as [5] Theorem 3.11; in fact the sufficient condition give~l in the note after this theorem, is always satisfied in view of (AR6). ) Thus an

4-radical

p

on any object

A

can be determined both from below and from

above: u (B 4 A

: B 6 IRp) = p(A) = e (C 4 A or C = 0 : A / C E S p )

Till now we have characterized an

.

4 - radical by means of the radical assign-

merit (radical functor) and the radical class. It can also be characterized in terms of the semisimple class and by the pair of radical and semisimple classes, respectively. Such characterizations

for not necessarily associative rings or

~-groups

exist in plenty (see e.g. [ 7] and [ i0] ), and using the tools we already have in our category, their proofs can be carried out word by word in our case, too. Here we pick out just one characterization of each of the latter two types. THEORI!M 7 class of an

(~]itz [i0] Theorem 4 ). A class

4-radical

if and only if

S

S

of objects is the sen/simple

satisfies the following three conditions:

196

(a) i_ff B 4 A 6 $ , then (b)

S

B

has a non-zero factor object in

(c) for all

AEC

, ((A)S)S = (A)S

where

(A)S=N ( B 4 A

THEORI~4 8 (Mlitz [iO] Theorem 2). The classes and semisimple classes of an I~ ~, S

consists of zero objects,

(B)

A E I~

and

(C)

A E S

(D) for any

and

A/B # O S ~ A

A E C

I~

and

or S

B = O : A/B E S). are the radical

4-radical if and only if

(A)

and

$ ,

is closed under subdirect products,

inloly A/B ~ $ ,

i~ply

B~I~

,

there is a normal subobject

B

o_ff A

such that

B 6 I~

A/B ~ S .

References [ i] S. A. AMITSUR, A general theory of radicals, II, Radicals in rings and bicategories, Amer. J. Math. 76 (1954), 100-125. [ 2] V. A. ANDRUNAKIEVI~ and Ju. M. RJABUHIN, Radicals of algebras and structure theory (Russian), Nauka, Mosoow, 1979. [3] F. CARREAU, Sous-cat~gories r~flexives et la th~orie g~n~rale des radicaux, Fund. Math. 71 (1971), 223-242. [4] H.-J. HOEHNKE, Radikale in allg~meinen Algebren, Math. Nachr. 32 (1966), 347383. [5] M. HOLCOMBE and R. WALKER, Radicals in categories, Proc. Edinburgh

Math. Soc.

21 (1978), 111-128. [6] J. ~ A

and B. TERLIKOWSKA, Theory of radicals in self-dual categories,

Bull. Acad. Polon. Sci. S~r. Sci. Math. Astronom. Phys. 22 (1974), 367-373. [7] L. C. A. van ~

and R. WIEGANDT, Radicals, semisimple classes and torsion

theories, ~ t a Math. Acad. Sci. Hungar. 36 (1980), 37-47. [8] G. MICHLER, Radikale und Sockel, Math. Ann. 167 (1966), 1-48. [9] B. MITCH~.T., Theory of categories, Academic Press, 1965. [iO] R. MLITZ, Radicals and semisi~ple classes of

~-groups, Proc. Edinburgh Math.

Soc. 23 (1980), 37-41. [ii] E. G. ~UL'GE~ER, General theory of radicals in categories (Russian), Mat. Sb. 51 (1960), 487-500. [ 12] B. T E R L I K O W S K A ~ S K A ,

Category with self-dual set of axic~s, Bull. ~ a d .

Polon. Sci. S~r. Sci. Math. Astronom. Phys. 25 (1977), 1207-1214. [ 13] B. TERLIKOWSKA-OS~OWSKA, Radical and semisimple classes of objects in categories with a self-dual set of axioms, Bull. Acad. Polon. Sci. S~r. Sci. Math. Astronom. Phys. 26 (1978), 7-13. [ 14] S. VELDSMAN, A general radical theory in categories, Ph. D. Thesis, University of Port Elizabeth, S. A., 1980.

ON THE STRUCTURE OF FACTORIZATION STRUCTURES by A. Melton and G. E. Strecker

For any category on

K.

K

In particular,

we investigate the family of all factorization structures

for each such structure,

lattice of all factorization structures on

K

(E,M), we investigate the complete with left factor a subclass of

E;

this investigation is based on a Galois connection between all such structures and the lattice of all full isomorphism-closed subcategories of

K.

families are precisely all the E-reflective subcategories of

The Galois-closed

K

and all the (E,M)-

dispersed factorization structures of Herrlich, Salicrup and Vazquez.

AMS

(1980) subject classifications:

Secondary:

§0

Primary 18A20, 18A32, 18A40;

06A15, 18A22

Introduction The importance of factorization structures on categories is by now well

appreciated.

Over the years the conditions that have been considered necessary for

an "(E,M)-factorization structure" to carry that name have evolved from those requiring

E

and

M

to be sufficiently nice dual-like classes of epimorphisms

and monomorphisms such that each single morphism has an essentially unique factorization,

(E,M)-

through various stages until the current generally accepted criteria

that (among other things)

E

be a class of morphisms and

sources such that each class-indexed source has an (E,M)-factorization,

M

be a conglomerate of

(even empty or proper class indexed)

m%d, in the category,

(E,M)-diagonalization holds.

To

emphasize that we require diagonalizations as well as factorizations we call such entities "diafactorization structures." The two major references for this paper are

[HSV] and

[Ho], both of which made

significant contributions to the clarification of the nature of

(dia)factorization

structures. In [HSV] Herrlich,

Salicrup and Vazquez introduced a new type of diafactoriza-

tion structure called dispersed and proceeded to show that there is a bijection between all E-reflective subcategories of an (E,M)-category dispersed diafactorization structures on

K.

K

and all

(E,M)-

This was a generalization of the

result that for nice categories such a correspondence exists between the epireflective subcategories of [Sl],

[S2] , [S4]).

K

and all perfect factorizations

(cf. [Hel],[He2],[Na],[Ne],

It also ~aproved and put into the proper context much of the

earlier work on quotient reflective subcategories,

connectedness properties,

corresponding factorizations

[SV2],

(cf. [C], [P], [SVI],

and

[$3]).

In §i, via a modification of the main result of Hoffmann

[Ho] (cf. also Harvey

198

[Ha]), we show that the development classes of a category

K

precisely those classes,

such that

E, for which there exists an

diafactorization structure on problem of

[HSV].

any E

(E,M)

are

(E,M)

is a

This answers the outstanding open

diafactorization structure,

(cf. [HS2] , [T]).

(E,M)-category

E

must be a class of

As a by-product of this theorem we also have, for

K, an internal characterization of all those

for which there exists a

structure

(Th.l.3).

(see [Ne])

The proof of Theorem 1.3 also provides an alternative proof of

the fact that for any epimorphisms

K

M

(Th.l.9).

D

such that

(C,D)

C

contained in

is a dispersed diafactorization

It is interesting to note that such classes are (to within

existence of the colimits) the "standard" classes of E-morphisms introduced in [SI] and investigated further in [$2]. In §2 we describe and investigate a Galois connection that makes precise the nature of the bijection discovered in [HSV]. of an (E,M)-category

K

Namely, the E-reflective subcategorles

and the (E,M)-dispersed diafactorization structures are

precisely the Galois-closed classes and are complete lattices

(in a suitably large

universe)

General Galois

that are anti-isomorphic with each other

(2.6(2)).

results, as well as special properties involved,

are used to investigate in more

detail the structure of the complete lattice

of all diafactorization structures

(C,D)

on

K

with

C

a subclass of

E.

Q

In particular,

partitioned into a family of complete lattices Q

it is shown that

(called levels)

can also be viewed as a union of complete lattices

§i.

Characterization of Diafactorization Structures Definitions and Notation

(i) In all that follows

K

is

(called images) all of which

have a point in common and none of which meets any level non-trivially

i.i

Q

(2.6(i) (i)) and that

will denote a category, and

Mot K, Iso K

(2.6(i) (ii)).

and

Epi

will denote the classes of all morphisms, all isomorphisms and all epimorphisms of

K.

All subcategories will be assumed to be full and isomorphism-

closed. (2) A K__-source with domain

X

empty and possibly proper) domain (3) K

is a pair

i

in

where I

fi

I

is a class

(possibly

is a K-morphism with

X.

is called an (E,M)-cate~or~ and

ture on

(X,(fi)i)

and for each

K

provided that

E

tion with K__-isomorphisms and

(E,M)

is called a diafactorization struc-

is a class of K-morphisms closed under composiM

is a conglomerate of K-sources closed under

composition with isomorphisms such that: (a)

K

has the (E,M)-factorization property;

has a factorization (Z,(mi) I) (b)

K

belongs to

i.e., every K--source

X~Y.

= x--~Z ~-i~Y. l l M, and

has the (E,M)-diagonalization property;

K__-morphisms and

(X,(mi) I)

and

(Z,(hi) I)

where

e

(X, (fi) I)

belongs to

i.e., whenever

e

E

and

are K--sources such that

and

f

are e

199

is in

E,

(X,(mi)i)

is in

and for each

M

there exists a unique m o r p h i s m i

in

d:Z---~X

i

in

I, h.e = m.f, then 1 1 f = de, and for each

such that

I, h. = m.d. l l

(*)

Y--~-~1 Z

I

d''1

l

X -------9-W. m. 1 1 [If only on (4)

(a) is satisfied,

(E,M)

is called a factorization

structure

K° ]

([HSV])

If

(C,D)

on

A

K

of

K K

is an

such that

E;i.e.,c:X---~Y f:X--+A

(E,M)-category,

is called C

is precisely

is in

with

A

C

in

then a diafactorization

(E,M)-dis~ersed

iff

A

c

structure

iff there exists a subcategory

all the A-extendible

is in

E

morphisms

in

and for each K--morphism

there is some K - m o r p h i s m

g:Y---~A

such that

f = gc. (5)

Let

E C

Mor K

(a)

~(E)

then:

will denote the conglomerate

the property e (b)

in

A(E)

E, then

X fi~y = X ~ Z ~Y. • i is an isomorphism.

e

will denote the conglomerate

the property that if square

i

a K__-object e:X--~Z

in

morphism (d)

~

e

(*) c o ~ u t e s ,

for each (c)

of all sources

that if

in

I

is in

1

of all sources

E

then there exists a unique (*) commutes.

(cf.

[S I]

is called an E-injective

E

and each K - m o r p h i s m such that

having

(X,(mi) I)

and if for each

Y

g:Z---~Y

(X,(fi) I)

is a factorization

i

in

d:Z--~X

and

with

having

I

the

such that

[$4])

object iff for each

f:X---~Y, there exists a K--

f = ge.

is the category whose objects

are members of

E

and whose ^

morphisms

(6)

(e)

A0:KE

Let

C

>K

and

E

hOmK_ (e,e)

of

h = gf.

(a)

iso--com~ositive

(b)

left cancellative

(c)

(f,g)

is the functor defined by be subclasses

K_-morphisms for which

belongs to

are pairs

iff

h

w.r.t.

E, then

f

Mor K Then

C

belongs to E

where

~0(f,g) and let

ge = ef. = f.

(cf.

f, g

and

[Ho]). h

be any

is said to be: C

whenever

iff whenever

must belong to

h

{f,g} C

belongs to

C U Iso K; C

and

f

C;

pushout p r o n e iff (i) every K--source pushout

X

(X, (c i) i )

ci > Y.

di ; Z

d (ii) every 2-indexed K--source, out

with each with

d

(X,(k,c)),

ci in

in

C

has a multiple

C; and

with

c

in

C

has a push-

200 X-c

k

~Y

t

I

Z •

(d) a development

c

with

class

(ef.

[Ne] (t))

c

in

C.

iff

(i) C~__ Epi K, (ii) C

is iso-compositive,

(iii) C

(e) an E - s t a n d a r d (i) C

class

(cf.

is a development

(ii) C

1.2 Remark.

and

is pushout prone; IS I],

[S 2] (tt))

iff

class of E-morphisms,

is left cancellative

w.r.t.

and

E.

The following are some w e l l - k n o w n properties

structure

(E,M)

on

K

that we will use in the sequel.

(i)

E

is iso-compositive.

(2)

E

and

M

of any diafactorization

determine

each other;

in fact

M = A(E).

We next obtain an improved version of the main t h e o r e m of Hoffmann that no conditions w h a t s o e v e r

morphisms steps in

E.

are put on the category

Some major steps of the proof,

however,

K

[Ho]

in

or the class of K-

closely

follow analogous

[Ho].

1.3 Characterization For any category

T h e o r e m for Diafactorization K

and any class

E

Structures

of K-morphisms,

the following are

equivalent: (i) There exists a conglomerate

ization (2) E

structure on

is a development

M

of K-sources

for w h i c h

(E,M)

is a diafactor-

K. class.

(3)

(E,A(E))

is a factorization

(4)

(E,~(E))

is a diafactorization

(5) The following hold:

structure on

(a)

E

(b)

A0: ~

K.

structure on

K.

is iso-compositive; ~K

is a topological

functor (%f%).

(t)

In [Ne] Nel d e f i n e d development classes ulation avoids his smallness condition.

(tt)

In [Sl] and [S_] standard classes of epimorphisms are defined more genz erally, without the requirement of the existence of (multiple) pushouts in what corresponds to (6)(c).

(tt+)

A functor

F:A---~X

has a factorization

is called topological

somewhat less generally.

iff each F-source

(X

gi

Our form-

~FAi) I

(X gi y FA.) = (X r F A Ffl ~r FA.) where r is an Xl l -isomorphism and (A--~i~A.) is an F-initial A-source -- or, equivalently, l every F-sink has an (F-final A__-sink, isomorphism)-factorization. (cf. [He3]).

201

Proof: (i)

(4) ---~(i) and ~(5).

Ao-sOurce. tion

(i) <

~{3).

(5) "

>(2).

E

(X

i ) A0ei)i

be a

S.

(Initiality

comes from

functor must be faithful

it is easily shown that

E ~Epi

K.

(E,M)-diagonalization.)

(see [He3]) , and since It remains to be shown

is pushout prone.

Let

(A0(ei)

of

Since a topological

(X,(ei)i)

be a K-morphism.

be a nonempty K--source with each If for each i

fi ) Y)I"

Let

in

A0(e i)

isomorphism)-factorization.

I we let

A0(gi,hi)

ei

A0(e) " r ~ Y

Then for each

in

E

and let

f:X---~Y

fi = f' then we have the A0-sink

i

in

I

be its

(final -~K-sink,

the d i a g r a m

f = rg. l> y

X

eii

I er-1

Z.

l

commutes;

h

>W

l

and by the finality of the sink

((hi) I k3{j},W) E

S =

Then (X,(e.f.)_) is a K-source which by (i) has an (E,M)-factorizami i i ± -~ Zi) I. Then X - - ~ ~0e ~ 0 ( f i ' m i ) ~ A0e i is the (isomorphism,

is topological,

that

f

To show 5(b) let

(X e > Y

initial source)-factorization

40

Clear.

5(a) follows from 1.2(1).

is a colimit,

where

((gi,hi)i,e),_

h. = er 3

it folloWS_l that

By 5(a),

er

is in

E.

Thus

is pushout prone.

(2) =--.-~ (4).

Suppose t h a t

the K--source

(X, (ej)j)

f a c t o r s of each

(X, ( f i ) i )

is any K-source.

Since

consisting of all those E-morphisms

f . , has a m u l t i p l e pushout

E

is pushout-prone,

e. 3

which are first

l

e. X- 3 ) ~ y .

d. 3>Z

e with

e

in

fi = m.e.l

E.

Thus for each

Since

E

i

in

is iso-compositive

Thus we have

(E,e(E))-factorizations.

e(E), and

and

f

pushout-prone,

I

there exists some and

m. such that l E C_ Epi _K, (Z, (m i) i ) is in e(E).

Suppose that

h. are such that for each i in I l ^ the pushout fe = ef can be formed with

a family of K_-morphisms

It is unique since e

>

~

ml

is in

E,

(mi) I

is in

h.e = m.f. Since E is 1 l e in E. Thus there is

k. such that for each i the following d i a g r a m con~nutes. l e(E), e must be an isomorphism, so that ~-i~ is the needed

By the definition of diagonal.

e

k.

lhi

e

is an epimorphism.

202

1.4

Corollary

([HS2] , [T])

epimorphisms in

1.5

Corollary

structures on

(cf. [Ho]) K

Corollar~.

there exists a

The conglomerate of all left factors of diafactorization

For any (E,M)-category D

such that

(C,D)

the development subclasses of

1.7

must consist of

is closed under arbitrary nonempty intersections.

a largest member, it is a complete lattice

1.6

E

is an (E,M)-category, then

If

K.

Corollary.

For any

Thus if it has

(under the inclusion order).

K

the subclasses

C

of

E

for which

is a diafactorization structure are precisely

E.

(E,M)-category

K

the conglomerate of all subclasses of E

which are left factors of diafactorization structures on

K

is a complete lattice

(under the inclusion order).

1.8

Remarks.

(a)

Corollary 1.4 shows

that the requirement that the diagonal be

unique can be deleted from the definition of diafactorization structure (1.1(3)). (b) Corollary 1.6 answers the open problem 2.13 of [HSV]. (c) It should be noted that Theorem 3(3) of [S4] is a forerunner of part of the following characterization theorem.

1.9

Characterization Theorem for Dispersed Diafactorization Structures For any

(E,M)-category

K

and

C C E, the following are equivalent:

(i) There exists a conglomerate of K-sources

D

for which

(C,D)

is an (E,M)-

dispersed diafactorization structure. (2) C

is an E-standard class of morphisms.

(3) Every (4) C

(C-injective)-extendible E-morphism belongs to

is the class of A-extendible morphisms in

E

C.

for some subcategory

A

of

(5) The following hold: (a) The subcategory of C-injective objects is C-reflective. (b) C

is left cancellative w.r.t.

E.

(6) The following hold: (a) C

is iso-compositive

(b) C

is left cancellative w.r.t.

(c) ~ 0 : ~

Proof:

~

E.

is topological.

The equivalence of all but (2) and (6) is shown in [HSV]

(1) and (5) ~ ( 2 ) .

By (i) and Theorem 1.3, C

is left cancellative w.r.t.

E; and, thus, C

(Theorem 2.11).

is a development class; by (5) it

is E-standard

(i.i(6) (e)).

(2)4~----~>(6). Immediate from Theorem 1.3 and Definitions i.i(6) (c), (d) and

(e).

K.

203

(2)~-~(5).

By Theorem 1.3 we know that there exists some

a diafactorization structure.

For any object

X, let

factorization of the source of all morphisms from

X

X

D

such that

c ~ ~

(C,D)

to C-injective Objects.

^

Diagonalization shows that

X

is

di ~Y: be the (C,D)^

is C-injective.

Thus, since

C C Epi K, X~ c ~ X

is

the C-reflection.

1.10 Corollary.

For any (E,M)-category the conglomerate of all subclasses of

which are left factors of an 6E,M)-dispersed diafactorization structure is a complete lattice

i.ii Remark.

§2

(under the inclusion order).

In 2.6(2) an "external" proof of Corollary i.i0 is obtained.

The Structure of All Diafactorization Structures In this section,

for any (E,M)-category

K, we investigate the structure of the

conglomerate of all diafactorization structures

2.1 Definitions and Notation. and Q

F:P--~Q c _~Fa

For each lattice,

and

iff b

G:~--~p

((P,~),

G[~], G-l[b]

for each

c

in ~

(P,~)

and

with

C C E.

(Q,_mO are partially ordered classes

are functions such that for each

a ~ Gc, then

in

If

(C,D)

(Q,~), F, G)

is called the b-level of F [P] = {Fa/~ cla e P} c

Since for any diafactorization structure

a

in

P

and

c

in

is called a Galois connection. ~, and if

(Q,~

is a

is called the c-image of

(C,D), D = A(C)

P in ~.

(1.2(2)), then by

Corollary 1.6 for us to investigate the structure of all diafactorization structures with

C ~E

it is equivalent for us to investigate the structure of the conglomer-

ate of all development subclasses of

E.

either of these conglomerates by

When thought of as development classes

Q.

Throughout this section we will denote Q

will be ordered by inclusion and when thought of as diafactorization structures it will be ordered by inclusion on the first elements of the pairs also let

P

tions

and

F

denote the conglomerate of all subcategories of G

ble morphisms in

as follows:

For each subcategory

E, and for any

erated by all C-injective objects. concentrated morphisms and in structure

in

Q:

N.B.

[Ho] GC

A

of

K:

FA = all A__-extendi-

GC = the full subcategory of

In

[HSV] FA

We will

K

gen-

is called the class of A--

is called the 9erm of the diafactorization

(C,A(C)).

2.2 Theorem.

(t)

C

(C,D).

K, and define func-

((P,C3,

Thus each pair structure of

(Q,_~, F, G)

(C,A), where

is a Galois connection on complete lattices.

GC = ~

and

K, in the sense of Maranda

FA = C, is a regular injective [Ma].

(t)

204

Proof:

(P,~)

For any that FA

A

FA

is c l e a r l y a complete lattice and

in

P, since

E

G:Q--~P

and class

Thus, b y T h e o r e m 1.3, F A [HSV]

b e l o n g s to

(Th 2.3).)

Hence

Q.

(That

F:P--~Q.

is clear, and one i m m e d i a t e l y sees that for any s u b c a t e g o r y

C

of K - m o r p h i s m s ,

C

A

of

is c o n t a i n e d in the class of all A--extendible E-

morphisms iff each A_-object is C-injective.

2.3 Remark.

is one by C o r o l l a r y 1.7.

is a d e v e l o p m e n t class, it is s t r a i g h t f o r w a r d to show

is a d e v e l o p m e n t class.

is a left factor has also b e e n shown in

That

(Q,C)

Thus the d e f i n i t i o n is satisfied.

In v i e w of the Galois c o n n e c t i o n of T h e o r e m 2.2 the general p r o p e r t i e s

of Galois c o n n e c t i o n s can be i n t e r p r e t e d as corollaries.

B e f o r e w e do this we w i s h

to first e s t a b l i s h some special p r o p e r t i e s of the Galois c o n n e c t i o n at hand.

2.4 Proposition.

For the Galois c o n n e c t i o n of T h e o r e m 2.2:

(i) F[P] = all E - s t a n d a r d classes of d i a f a c t o r i z a t i o n structures of

K

(or, equivalently,

K).

smallest E - s t a n d a r d class c o n t a i n i n g it. zation structure has a smallest larger than it. hull")

all

(E,M)-dispersed

Each d e v e l o p m e n t class of m o r p h i s m s has a Equivalently,

each

(E,M)-diafactori-

(E,M)-dispersed d i a f a c t o r i z a t i o n structure

The p r o c e s s of o b t a i n i n g the " E - s t a n d a r d hull"

is the Galois closure o p e r a t o r

(or "dispersed

FG.

(2) G[Q] = all E - r e f l e c t i v e s u b c a t e g o r i e s of

K.

Each s u b c a t e g o r y of

K

has an

E - r e f l e c t i v e hull and the process of o b t a i n i n g E - r e f l e c t i v e hulls is the Galois closure o p e r a t o r (3) The K - l e v e l of m e m b e r of

Q

has only one member, n a m e l y

and is in every image, Fc[P], of

(4) E a c h level in

Proof:

GF. Q

Q

(i) That each

FA

in

is in left c a n c e l l a t i v e w.r.t.

Q.

general Galois theory. X

is inunediate.

(Th 1.9).

Thus

(E,M)-

The remainder follows from

([MS]).

let

rX:X~-~X

all C - m o r p h i s m s w i t h domain E; thus

E

(I.I(6) (e)) and so the left factor of an

d i s p e r s e d d i a f a c t o r i z a t i o n structure

m o r p h i s m in

P

has a smallest m e m b e r and is a c o m p l e t e lattice.

each is an E - s t a n d a r d class

(2) For any K - o b j e c t

Iso K, and this is the smallest

GC

X.

be the m u l t i p l e p u s h o u t of the source of

With r e s p e c t to

is E-reflective.

GC, r x

Conversely,

is a r e f l e c t i o n suppose that

B

is E-

r e f l e c t i v e in K and Y is a GFB-object. Since the B_-reflective E - m a p ^ ry:Y---~Y is in FB, there is a g such that gry = iy. Thus ry is an isomorphism,

and

is in for any

Y

G[Q]. A

in

belongs to So

B.

Hence

GFB < B.

Since, as always, B ~ GFB,

G[Q] = all E - r e f l e c t i v e s u b c a t e g o r i e s of

P, GFA

is the s m a l l e s t m e m b e r of

G[Q]

K, and since

containing

A, it m u s t

b e the E - r e f l e c t i v e hull. (3) C l e a r l y

Iso K

is the smallest d e v e l o p m e n t class

(1.1(6)(d)) and

--

S u p p o s e that

GC = K.

Then every m e m b e r of

C

m u s t be a section.

F K = Iso K. C--But since

205

C C E ~ Epi K (Th. 1.3), we have that

C c Iso K.

(4) For any E-reflective subeategory B_-reflection map. is in

C

to it.

B

and any object

X

in

K

let

rx

Then by the construction of the reflection maps,

for each

C

in the B-level.

be its

(2), r x

Thus the meet of the B-level belongs

So the B-level is complete.

2.5 Remarks.

(a) It should be mentioned that a Galois connection in a more restric-

tive setting ([He2], [SI]

and

[S2]) has previously been used to obtain epire-

flective hulls as Galois closures. (b) That each B-level has a smallest member has also essentially been shown by Hoffmann

[ H o ] (Prop. 2.5).

member for each level.

Notice that general Galois theory gives a largest

This is Th. 2.3 of

[Ho].

A summary of some of the properties that follow from the results of this section as well as the general Galois theory ([MS]) follows.

2.6 Summary and Sample Results.

(i) Q

is a complete lattice that can be viewed as

a union of complete lattices in two ways: (i) as the disjoint union of complete lattices, called levels such that in each level the join or meet of any nonempty family is its join or meet in Q, and such that the level with the smallest member of

Q

is a singleton.

(ii) as the union of complete lattices, called images, all of which have the smallest member of

Q

in common and are such that when they intersect they

coincide from any common point on down.

Furthermore all non-empty meets in

images are the same as the corresponding meets in

Q.

(2) The conglomerates of all E-reflective subcategories of persed diafactorization structures on For each development class

[HSV]

and all (E,M)-dis-

are anti-isomorphic complete lattices.

C C-__E, we have C-relativizations of all above

results (C-reflective hulls, etc.). from

K

K

Thus, in particular, we have the result

that, the E-reflective subcategories of

K

are in one-to-one

correspondence with the (E,M)-dispersed diafactorization structures. (3) Each subcategory of

K

has an E-reflective hull, each development class has

an E-standard hull, each (E,M)-diafactorization structure has an (E,M)-dispersed hull, and each conglomerate of development subclasses of velopment hull (the join of it in

E

has a de-

Q).

(4) The dispersed hull of any (C,D) in Q is the largest (E,M)-diafactorization ^^ ^ structure (C,D) for which the C-injective objects are the C-injective objects. (5) Let

H

be the family of all subcategories of

which are

Then

H U

and smallest member

~H.

(6) If

C

and

B.

C

{~H}

K

the E-reflective hulls of

is a complete lattice with largest member

are development classes with

C ~ C, then there is a unique C-

standard class, C*, such that the C-injective objects and the C*-injective

7Ut)

objects coincide. (7) In

Q

the B - l e v e l and the C-image intersect iff

c a t e g o r y of

K.

is a C - r e f l e c t i v e sub-

If they do intersect, their i n t e r s e c t i o n is a singleton

w h o s e member is

2.7 Remark.

B

(C,A(C))-dispersed.

F i n a l l y we p r o v i d e a sketch of w h a t the complete lattice

like a c c o r d i n g to some of its p r o p e r t i e s that we have obtained. a v e r y large lattice indeed. legitimate

([HSV] Th 3.2).

In the case

K = To_T_g]~ and

Q

m i g h t look

Note that it can be

E = Epi K, it is non-

All of the n e a r l y - h o r i z o n t a l

lines r e p r e s e n t levels;

these are o r d e r e d by t h e i r top points, w h i c h represent the d i s p e r s e d d i a f a c t o r i z a tion structures sent images.

(or E - s t a n d a r d classes).

All of the n e a r l y - v e r t i c a l lines repre-

Images that m e e t in a p o i n t are identical from that p o i n t down and

they all m e e t at the K-level,

the lowest level, w h i c h is a singleton.

Points of

only o c c u r at the i n t e r s e c t i o n s of levels and images, and each such intersection has at m o s t one point.

C-imag%

F[P] $ kE

C

B_leve~~l ~'~~ K--level

Q

207

REFERENCES [C]

P . J . Collins, Concordant mappings and the concordant-dissonant factorization of an arbitrary continuous function, Proc. Amer. Math. Soc. 27 (1971), 587-591.

[Ha]

J. M. Harvey, Topological functors from factorization (Proc. Int. Conf. Berlin 1978), Springer Lecture Notes in Math. 719 (1979) 102-111.

[He I] H. Herrlich, A generalization of perfect maps (Proc. Third Prague Topological Symposium. 1971), General Topology and Its Relations to Modern Analysis and Algebra III, Academia, Prague (1972) 187-191. [He 2]

, Perfect subcategories and factorizations (Proc. Colloq. Karzthely, Topics in Topology). Colloq. Math. Soc. Janos Bolyai, 8, North Holland, Amsterdam (1974) 387-403.

[He3]

, Topological

functors,

General Topology and Appl. 4 (1974) 125-142.

[HSV] H. Herrlich, G. Salicrup, and R. V~zquez, Can. J. Math. 31 (1979) 1059-1071. [HS I] H. Herrlich and G. E. Strecker, Verlag 1979. [HS 2] completions,

Dispersed

Category Theory,

factorization

structures,

2nd ed., Berlin:

Heldermann-

, Semi-universal maps and universal Pacific J. Math. 82 (1979) 407-728. Factorization

[Ho]

R.-E. Hoffmann,

[Ma]

J. M. Maranda,

[Me]

A. Melton, Which dispersed diafactorization structures on Top are hereditary?, General Topology and Modern Analysis, Academic Press, New York (1981) 281-290.

[MS]

A. Melton and G. E. Strecker,

[Na]

R. Nakagawa, Relations between two reflections, Sect. A, 12 (1973) 80-88.

[Ne]

L. D. Nel, Development classes: An approach to perfectness, reflectiveness and extension problems (Proc. Second Pittsburgh Internat. Conf., TOPO 72, General Topology and its Applications), Springer Lecture Notes in Math. 378 (1974) 322-340.

[P]

G. Preuss, On factorization

Injective

of cones, Math. Nachr.

initial

structures,

Trans. Amer. Math. Soc. ii0

Structures of Galois connections,

categories,

categories,

Reflexividad y coconexidad 14 (1976) 159-230.

, Reflectivity

[SV 2 ]

(1964) 98-135.

preprint.

Sci. Rep. Tokyo Kyoiku Daigaku

of maps in topological

[SV I] G. Salicrup and R. V~zquez, Univ. Nac. Autdnoma Mexico,

87 (1979) 221-238.

preprint.

en Top, An. Inst. Mat.

and connectivity

in topological

preprint.

[S1 ]

G. E. Strecker, Epireflection operators vs perfect morphisms and closed classes of epimorphisms, Bull. Austral. Math. Soc. 7 (1972) 359-366.

[S2 ]

, On characterizations of perfect morphisms and epireflective hulls (Proc. Second Pittsburgh Internat. Conf., TOPO 72, General Topology and its Applications), Springer Lecture Notes in Math. 378 (1974) 468-500.

[S3]

, Component properties and factorizations, Topological Structures, Math. Centre Tract 52, Mathematisch Centrum, Amsterdam (1974) 123-140.

208

[S4 ]

, P e r f e c t sources (Proc. First C a t e g o r i c a l T o p o l o g y Symposium), S p r i n g e r L e c t u r e Notes in Math. 540 (1976) 605-624.

[T]

W. Tholen, S e m i - t o p o l o g i c a l functors I, J. P u r e Appl. A l g e b r a 15 53-73.

A. M e l t o n D e p a r t m e n t of C o m p u t e r Science W i c h i t a State U n i v e r s i t y Wichita, Kansas 67208 U.S.A.

G. E. S t r e c k e r D e p a r t m e n t of M a t h e m a t i c s Kansas State U n i v e r s i t y Manhattan, Kansas 66506 U.S.A.

(1979)

A

REMARK

ON

SCATTERED

Adam M y s i o r ,

Gda~sk,

SPACES

Poland

1. I n t r o d u o t i o n A topological a relatively

is h e r e d i t a r i l y oonneoted whether

is c a l l e d point.

This

problem that

solved

discrete

Is every

there

a single

scattered

topological It

turns

out

that

into

Therefore

~ome

one

space

the answer

problem

zerodimensional. [10]

. It

if and

is w e l l

only

power

if it of a

ger~eralizat[on

regular

space

E

open

is

has

space

of

:

completely

regular

subsets

any non-trivial

topological

a natural

the following

the

space

is z e r o d i m e n s i o n a l

scattered

of

a long

by R.C.Solomon

completely

power

raised

its

scattered

contain

negatively

embedded

is

not

of

avery

regular

space.

problem

it d o e s [9]

space

be homeomor~hioally

two-point

i.e.

if each that

completely

a topological

Semadeni's

is c l e a r

Z.Semadeni

scattered was

scattered

It

disconnected

subspaoes.

every

known

oan

space

isolated

can be

space

E

embedded

such

into

that

some

?

[s n e g a t i v e .

Namely,

we

get

the

following

THEOREM.

For

every

regular

space

which

of hereditarily

2.

cardinal cannot

I/~ be

disconnected

there

embedded spaces

is a s c a t t e r e d any

into

completely

topological

of cardinality

~ ~

product

.

Proof

Let We

dA/v

be an arbitrary

construct

containin~

an

example

two distinct

for

every

hereditarily

and

every

continuous

X

ditarily

disconnected

The

be

construction

whi oh was

proved

in

points

p

and

disconnected function

a space

cannot

cardinal. of a scattered

f

embedded spaces

of

the

[7]

.

completely

q

such

space

: X--~E

into

any

E . It

space

X

that

is b a s e d

f(p)

space

: f(q)

of cardinality is c l e a r

topological

of cardinality

regular

that

product

~

4/M/

such of here-

~ A4~.

on

the following

]emma

X

210

Lemma. 2 ~.

M

Let

Then (i) M

and

if f ~ n e r (ii) M

exists M

the

open

compact

~

same

sets)

is z e r o d l m e u s L o n a l

for

space

a space

have

(= m o r e

a countable (lii)

be a metrizable

there

clopen

a~y

~,BCM

that

it f o l l o w s

with

such

weight

underlying then

d~

and

oardinality

that

the

sets

and

topology

and moreover,

each

the

of

H

point

topology

of

, of

M

has

neighborhood,

, if

o1~Anel~

=~

, then

[OIMAnOIMB

I < 2~

.

A

Observe and

completely

To

a,ld

the

~o

space

= 2~

an ortho,lormal

base

oardinality

have

and

oardinality

oardinality for

the

Choose Define

less now

a new

subsets

The

of

than

space

X

is

and

space

M

on in

by

~

oarmot

is s c a t t e r e d

of the be

such

space

of

by e set

latter

is v e r y

dLsconneoted

that

H

with

with

subsets

be disconnected

p,qqH

generated

. Denote

the r e q u £ r e d

the

open

dN space

H of

much

like

by a set

of

.

H

space

cardinal

a Hilbert

non-empty

proof

points

~

continuous

and

44/ . It is a m e t r i c

p!ai1e c a n n o t

completely

disconnected

Denote

H

2M e

topology

an arbitrary

the

an arbitrary

all

9

2 ~ . The

Euclidean

open

take

2~

two distinct

M

is s c a t t e r e d tarily

than

that

oardinelity

that

[IO] 6 h . 8 . 1 5 )

of oard[nality

2 4d/ a n d

less

fact

X

(see

weight

~P

(ii)

regular.

oolstruct

/~/~d~

from

Let

cardinality

N = H -

sets

open

space

closed

H X

It

is

easy

be

an arbitrary

~ /p~/ a n d that

{p,q}

in

by

E

We prove

of a l l

put

obtained

example.

function.

family

by ell

the

regular. of

and

and

let

subspaces

f

that

X

heredi-

: X----E

= f(q)

Z

all

.

to c h e c k

f(p)

.

of

be

.

X

such

that I x - z l < 2 Observe

that

cardinal£ty

2 ~.

f-1(f(p)) E ~

The our

(@)

It

If

in

He:~ce

X

P,qE

property

Z

that

of

of

neighborhoods

the

for

every

~E ~

q~ f-1(f(p))

space

p

X

is

or

q

have

. l~e p r o v e

and

f(p)

that

= f(q)

the crucial

point

.

of

~en.

Z E ~

a~id

U

is a o l o p e n

in

Z

neighborhood

of

p

then

.

is c l e a r

that

IX - U I < 2 4~ . S i n c e Since

open

. It f o l l o w s

following

argumentat

u E ~

all

the

set

U

IX - Z I < 2 4#

oI~(U(~M)(~ cI~(Zg~M

- U)

is c l o s e d

in

it s u f f i c e s = ~

. We

prove

to p r o v e

X

that

it f o l l o w s

from

that IZ - U I < 2 4~.

Lamina t h a t

211

]ClM(U~M)~clM(Z~M

- U)I<

On the o t h e r h a n d cardinality cannot

- since

2~

2 ~.

- ClHUVClH(Z

be d i s c o n n e c t e d

Hence

- U)

b y a set

ciHU=H

[ClHU~OIH(Z

every non-empty

a.d

- U)]<

open subset

= cl~

= H . But

of o a r d i n a l i t y

less

2 4~.

of

H

has

the s p a c e than

2

H

. Hence

~]<2 ~

Iz -

or ClH(Z The

latter

topology such

- U)

= H

is i m p o s s i b l e

of

that

X

that

V~Z(

I V ~ Z I= 2 e

U

and

and

IU[< 2 ~ .

- it f o l l o w s

there

from

is a n o p e n

; since

IV I= 2 @

IU]= 2 ~ . H e n c e

in

and

the d e f i n i t i o n H

IX - Z I < 2 ~

IZ - U ] <

of the

neighborhood

2#

and

V

of

p

we h a v e

the p r o p e r t y

(~)

is p r o v e d .

Define now, by transfinite of c l o s e d

subsp.ces

E~ = N{U

: f(P)6 U

of

E

reoursion,

such

%hat

a family

E

= E

<~÷}

{E~ : ~

and

for every

~ < /M~*

o

Since

IZ[~ /~A/ a n d

an ordinal

~o<

¢~o~ ~
that

such

that

that from

. Since I~[
of s o m e

with

(~) that

=

= u6

ends

Denote

by

Top

it is c l o s e d

t[ve

is

for

every

ordinal

ordinals

~ <~

is a f a m i l y and

~ . It f o l l o w s

f-1(U)6

~

}

Iul<+

that f - ! ( E o ) ~ ~ .

from

that

of s u b s e t s

every

of

the c o n t i n u i t y Uqe~

E

is a c l o p e n

U6~

for every

E

A4~+ .

. We p r o v e ~

with of

it

~<

It is o b v i o u s

: U~)

Every

of a l l

(see e.g.

respect

suboate~ory

subeate~ory,

there

known

subeategory

with

of

f

. Since

w+ h a v e

.

of

Top

E

space

[5]

Ch.X. 37)

E

~ such

spaces that

and a

is e p i - r e f l e c t i v e

is c o n t a i n e d

subcate~ory

is a s i n g l e

topological

to the f o r m a t i o n

its e p i - r e f l e c t i v e

&n e p l - r e f l e o t i v e that

f-~(f(p))62

the c a t e g o r y It

functions.

objects.

~

conclusions

isomorphism-closed if

ordinal

is

the p r o o f .

3. C a t e G o r i c a l

nuous

for every

there

to prove that

there

~<

is d s o r e a s i n ~

disconnectedness

for all

, E~ = N{U

o£ s o m e E~ w i t h ~ < Z ~ .

the h e r e d i t a r y

induotiOno

IEI< 44~

E~

: ~ <44~ +}

f-1(E~) 6 ~

by t r a n s f i n i t e

subset

E~ = E~o

. Therefore

that

the p r o p e r t y

This

{E~

such

£-I(E~)6~

~

that

subset and

/M~+

to s h o w

f-1(E~)6

is a c l o p e n

" It f o l l o w s

We p r o c e e d Assume

U

the f a m i l y

E~o = If(p)l

suffices

and

h u ll,

if and o n l y and

in a s m a l l e s t which

is s a i d that

of p r o d u c t

we denote

to be s i m p l e A = ToPiE } .

conti-

full

sub-

epi-refleoby

Top~

provided

.

212

Among simple suboategories topolo6ical spaces,

are for example

the categories

To-spaces , completely regular spaces,

sional spaces and the category

Top developable

spaces

of all

zerodime~-

( [3], [8]).

On the other hand there is a large number of theorems stating that various

suboategor[es

Our ~IEOREM Namely,

can

are

the non-simplicity

completely regular spaces] spaces}

(see e.g.

be considered as a new result

it yields

any suboate~ory

not simple

[1], [21, [41 , [61 , [81 )o in this direction.

of the category

Top{scattered

and - moreover - the n o n - s i m p l i c i t y

lying between

Top{scattered

of

completely regular

and the category of all h e r e d i t a r i l y disconnected

spaces.

References I.

D.W. Hajek and A. Mysior, On n o n - s i m p l i c i t y of topological categories, Prec. Categorical Topology Conf. Berlin 1978, Springer Lecture Notes in Math. 719 (1979) 84-03.

2. D.W. HaJek and R.G. Wilson, The n o n - s i m p l i c l t y of certain categories of topological spaces, Math. Z. 131 (1973) 357-359. 3. N.C. Heldermann, The category of ~ - c o m p l e t e l y regular spaces is simple, Trans. ,~ner. Math. See. 262 (1980) 437-446. 4. H. Herrlioh, Warm sind alle stetigen Abbildungen Math. Z. 90 (1965) 152-154. 5. H. H e r r l i o h and G.E. Streoker, 6. S. Mr6wka, 479-481.

On universal

spaces,

Category theory, Bull.

Aoad.

in Y konstant Boston

Polon.

1973.

Sci. 4 (1956)

7- A. Mysior, The category of all zerodimensional realoompaot is not simple, Gen. Topology Appl. 8 (1978) 259-264. 8. A. Mysior,

Two remarks on D-regular spaces,

?,

Glasnik Mat.

spaces

15 (1980)

15~-156. 9. Z. Semadeni, Sur les ensembles clairlsemes, 19 (1959) 1-39. 10. W. Sierpidski,

Cardinal and ordinal numbers,

Dissertationes Warsaw 1965.

11. R.C. Solomon, A scattered space that is not 0-dimensional, London Math. Soc. 8 (1976) 239-240.

Institute of Mathematics U n i v e r s i t y of Gdadsk 80952 Gdadsk Poland

Math.

Bull.

Bornological Ll-functors as Kan extensions and Riesz-like representations L.D. Nel

The setting for this study is formed by the following C = (complete bornological

vector

spaces,

closed categories:

bounded linear maps),

B = (Banach spaces,

bounded linear maps),

A = (Banach spaces,

bounded linear maps with norm at most i).

X will be a measure functor LI(X,-)

space which is the union of subspaces with finite measure.

The

: A ÷ A was shown recently [7] to have an A - r i g h t adjoint M(X,-).

Our m a i n purpose here is to extend this to a C - a d j u n c t i o n

(*)

[LI(X,E),

F] = [E, M(X,F)]

thus to obtain a far reaching generalization for operators

on Ll-spaces.

better known Banach space L

(E,F in C),

of the classical Riesz representation

For orientation we m e n t i o n that M(X,B)

reduces

to the

(X,B) if and only if the Banach space B has the Radon-

Nikodym property [7].

Our stepping interest. C ÷ C

stones toward the above goal evolved into results of independent

We establish

that every functor ~:B + C has a left Kan extension Lan~

along the inclusion T:B + C. Surprisingly,

B - r i g h t adjoint ~ preserves

:C ÷ C..

F ~

The promised adjunction

~:B + B

recently

of [~7] to the C - a d j u n c t i o n /I(V,-)

in [6]

(here V is a bornological

Using

implies a C-adjunction

(*) becomes a special case.

We also obtain a related C - a d j u n c t i o n /I(X,-) the procedures

:

with

the Lan of every such B - r i g h t adjoint composite.

these facts, we show that every B - a d j u n c t i o n LanTF ~ LanT~

the Lan of a composite T~

q m(X,-)

~ /(V,-)

:C + C

by applying

:C ÷ C that we derived

set).

About C and categories of completant disks For an introduction to complete bornological their importance

in Functional

Analysis,

was shown in [6] to be autonomously category Bo (bornological

sets).

spaces

(cbv-spaces)

and

The category C of these spaces

algebraic over the cartesian closed topological

As such, C is complete,

cotensored over Bo, has well behaved carries a Bo-related

see [3].

vector

cocomplete,

tensored and

(regular epi, m o n o ) - f a c t o r i z a t i o n s

closed symmetric monoidal

structure

( [-,-],

8, K,

and it ... )

[2]

214

2 w h e r e the m o n o i d a l u n i t K is of course the scalar field

(real or complex).

Every B a n a c h space B has an u n d e r l y i n g cbv-space TB.

The obvious a s s o c i a t e d

functor

T:B

is full and faithful; moreover, ture.

÷

C

it p r e s e r v e s and reflects the closed m o n o i d a l struc-

One thinks of B as a full subcategory of C, closed under formation of hom-

objects and tensor p r o d u c t s

(i.e. p r o j e c t i v e norm completed tensor products).

Recall [3] that every cbv-space E comes equipped w i t h a specified family E - ~ 6 ~ of

comple~nt disks

i.e. disks D contained in E such that the v e c t o r subspace E D

spanned b y D becomes a B a n a c h space w h e n normed with the g a u g e then the b a s i c b o u n d e d sets of E. Let us turn E - D ~

of D.

D as o b j e c t s and all inclusion maps D ÷ D' among them as m o r p h i s m s . (in terms of elementary facts [3]) to v e r i f y that E - ~

preorder

(definitions in [5]).

is

It is easy

8mallj filtered

and a

The functor

E(_) : E - P ~ k

is d e f i n e d to carry

These D are

into a c a t e g o r y by taking the

÷

C

(D ÷ D') to the inclusion

(TED ÷ TED,).

The i m p o r t a n t k n o w n fact

that every c b v - s p a c e E is the "inductive limit" of its a s s o c i a t e d B a n a e h spaces ED, can now be r e s t a t e d in the following technically p r e c i s e manner:

I. Remark.

2. Lemma.

The inclusion maps

Suppose A : P ÷ Q

is a functor such that P is f i l t e r e d and for every ob-

ject Q in Q the comma c a t e g o r y exists Q ÷ AP'

(Q+A) is a n o n - e m p t y p r e o r d e r

Then

(in the sense of [5]).

R o u t i n e verification.

3. temma.

The functor

Span: E- Q~6 k

d e f i n e d to carry functor.

(i.e. for some P' there

and for every P in F there is at m o s t one m o r p h i s m Q ÷ AP).

is a final functor

Proof.

TE D ÷ E b u i l d a colimit cone for the functor E(_).

(D + D') to

÷

(T+E)

(TED ÷ TED' -~ E)

(all i n c l u s i o n maps), is a final

A

215

3 Proof.

An object

(f:TA ÷ E) of

f a c t o r i z a t i o n [5],

say T A ÷ E' ÷ E.

so does its image E' we can

(T+E) has, qua C - m o r p h i s m ,

Since the d o m a i n of f u n d e r l i e s a B a n a c h space,

(proposition 3:1(2)

in [3]).

rewrite the above f a c t o r i z a t i o n as

closed u n i t disks of A and A'. (f:TA ÷ E) + SpanD'.

Thus

a (regular epi, m o n o ) -

Accordingly,

T E D ÷ TED, + E,

E' = TA'

(say) and

w h e r e D and D' are the

This f a c t o r i z a t i o n r e p r e s e n t s a (T%E)-morphism

(f%Span)

is non-empty and it is c l e a r l y a preorder. L e m m a

2 ends the proof.

L e f t Kan Extensions The left Kan extensions to be w r i t e Lan~ for L a ~ . calls for ding

c o n s i d e r e d will always be along T, so we will

For b a c k g r o u n d we refer to [5] and [13.

The theorem to follow

C - e n r i c h m e n t for its o w n sake; however, for the p r e s e n t p u r p o s e of exten-

a B - a d j u n c t i o n to a C-adjunction,

the g i v e n S ~ t - b a s e d v e r s i o n is a d e q u a t e

b e c a u s e C - a d j o i n t functors are a u t o m a t i c a l l y C-functors.

4. Theorem.

Every functor ~:B + C

has a left Kan e x t e n s i o n

Lan% such that

(Lan¢)T = ¢ .

Proof. Once existence is proved, the e q u a t i o n

(Lang)T = ¢ rather than extension u p to

natural i s o m o r p h i s m will hold b e c a u s e T is full and faithful.

For existence it is

enough to show that for every E in C the composite T (T+E)

E ~ B

> C

has a c o l i m i t w h e r e T E is the u n d e r l y i n g functor Span is final has a c o l i m i t .

5. Remark.

But the latter is clear b e c a u s e

(TA + TB).

E-Di6k

Since #T E Span

is small and C is cocomplete.

since the composite T

E-Di]Jk

is just

(TA ÷ TB ÷ E) ~

(lemma 3), this can be a c h i e v e d b y showing that the composite

E(_),

span

(T+E)

T E

B

> C

w e have

L a n C E = c o l i m ~ T E = colim¢E(_).

This c o l i m i t cone can be o b t a i n e d by applying L a n } to the cone of inclusions TE D + E (D in

E-D,O~k). since right adjoints do not u s u a l l y p r e s e r v e colimits or left Kan extensions,

216 4 the following useful fact comes as a p l e a s a n t surprise.

6. Ihe0rem. preserves

The left Kan extension A = LanT~, w h e r e ~ is a B - r i g h t a d j o i n t the left Kan extension of T~ for every B - r i g h t a d j o i n t

~ : B ÷ B

B ÷ B i.e.

ALanT~ = LanATQ.

In particular, for every B a n a c h space B and every cbv-space E we have

(6a)

[TB, El = colim[TB, E(_)]

(6b)

[TB, LanT~-]

Proof.

La~TB,

=

We first establish just the special case

E - D / s k ÷ [TB, E ] - D / S k

space ~ B , T E D 3 .

and

T~-].

(6a).

D e f i n e the functor

[B,-)

by putting

[B,D) = {h £ [TB, E] ! h(x) E D w h e n e v e r

That the d i s k [B,D)

(cf. remark 5):

Ix[ <

i).

is c o m p l e t a n t is clear frcm the fact that it spans the B a n a c h

As c b v - s p a c e [TB, E] carries the natural b o r n o l o g y [33, for w h i c h

the disks [B,D) form a base, by definition. admits at l e a s t one m o r p h i s m Q ÷ [B,D).

This m e a n s every object Q in [ T B , E I - D ~ k

Hence, by lemma 2, [B,D) is a final functor.

O b s e r v e now that w e have a c o m m u t a t i v e d i a g r a m

E-D/s k

E (-)

x C

[B,-) ]

[TB,-)

[TB, E ] - D / ~ k

C

[TB, E] (-)

U s i n g r e m a r k 1 and f i n a l i t y of [B,-) w e conclude that [TB, E] = colim[TB, El(_) colim[TB, EI(_)o[B~-) special case

(6c)

(6b).

= colim[TB,-] o E(_).

Using

Thus we h a v e

(6a).

=

L e t us now p u r s u e

(6a) and r e m a r k 5, we conclude

[TB, -] = Lan[TB,T-].

C h o o s e r so that BEFA, B3 = B[A, ~B] holds in B, natural in A and B. (the m o n o i d a l unit) we have ~ = BEFK,-],

hence

Then for A = K

217

5 (6d)

T~

We are now able to calculate

[TB, LanT~-]

= [TeK, T-].

as follows:

= [TB, Lan[TFK,

= [TB, [TFK,

This gives us is naturally the general

(6b).

(6d)

-]]]

(6c)

= [TB ® TFK, - ]

(exponential

= I T ( B @ FK), -]

(good nature of T)

= L a n [ T ( B @ FK), T-]

(6c)

= Lan[TB,

[TFK, T]]

(retracing above steps)

= Lan[TB,

T~-]

(6d).

Finally,

isomorphic

T-I]

we observe

law)

that every functor of the form A = LanT~

to a functor of the form [TB, -],

by

(6d) and

statement of the theorem follows from the special case

(6c).

Thus

(6b).

Extending B-adjunctions to C-adjunctions 7. Theorem.

If F:B ÷ B is B-left

adjoint to ~, then

then LanTF:C ÷ C is C-left adjoint to LanTQ.

Proof.

By hypothesis

we have for every A in B isomorphic

functors

[TFA, T-] = [TA, T~-].

Taking left Kan extensions

Lan[TFA,

and using theorem (6b), we obtain

T-] = [TFA, LanT-]

Lan[TA,

Hence

natural

= [TFA, -3

T~-] = [TA, LanTF-].

[TFA, F] = [TA,

for A in B and F in C.

and

(LanTF)F]

since the functors

[-, F] and [-,

(LanT~)F]

are

218

6 left adjoint functors,

they preserve

left Kan extensions.

[ (LanTF)E,

natural

for E and F in C,

El = [E,

It follows that

(LanT~)F]

w h i c h is the desired C-adjunction.

Let us now turn to the special C - a d j u n c t i o n

promised at the beginning.

we

define the functors

LI(X,-)

respectively

: C + C

M(X,-)

as the left Kan extensions LI(X,-) B

8. Theorem.

Proof.

and

and

of LI(X,-)

does,

down to an A - m o r p h i s m

M(X,-) C

F] = [E, M(X,F)]

The A - a d j u n c t i o n

as any A - a d j u n c t i o n

of the composites

T ~ B ~

[LI(X,E),

: C ÷ C

B

in C,

B

naturally

and M(X,-)

to a B-adjunction,

T >

) C

in E and F.

established

in [7] extends trivially,

since every B - m o r p h i s m

and scaled up to itself again.

can be scaled

This enables us to deduce the

result at once from theorem 7.

In [7] we gave an explicit realization

of the space M(X,B)

an explicit formula for the natural isomorphism was shown

to generalize

forward manner

the classical

to the present setting,

Banach spaces.

The space LI(X,E)

[LI(X,A) , B] = [A, M(X,B)]

Riesz formula. since everything

introduced above,

colimit of the Banach spaces LI(X,ED)

and also , which

These extend in a straight can ultimately be reduced to

for example,

(cf. remark i).

tered colimit of the Banach spaces M(X,FD)

(B in B)

Similarly,

is the filtered M(X,F)

is the fil-

and [LI(X,E) , F] that of the Banach spaces

[LI(X,E D), FD,]-

In view of the characterization ving ordinary right adjoints

of enriched right adjoints as cotensor-preser-

[4] and dually,

the following

is an immediate con-

sequence of theorem 8.

9.

Corollary.

naturally

LI(X,

E 8 F) : LI(X,E) 8 F

and

M(X,

[E,F])

= [E, M(X,F)]

in C,

in E and F.

This corollary gives a further illustration is inherent in enriched category theory.

of the useful functorial

calculus w h i c h

219 7

Another Riesz-like representation It was shown in [7] that LI(X,A) of a system of s p a c e s / ~ , A ) ,

can be represented as a filtered colimit in A

where Q varies in a certain category T~y3

tions arising from the measure space X, while M(X,B) associated system of spaces / ( Q , B ) . /

(V,F) which generalize

of tessella-

is a projective limit of an

In [6] we introduced cbv-spaces /I(V,E)

the Banach spaces /I(S,A)

and

and / (S,B); actually, /I(V,E)

is the tensor product of the bornological set V with E in C while /

(V,E) is the

cotensor of such V with E, so that

[/I(V,E), F] : [E,/ (V,F)]

naturally for E and F in C.

By departing from this natural isomorphism and making

V vary through the tessellations of [7] , considered as indiscrete bornological

sets,

the procedure of[7] can be carried out mutatis mutandis in the present setting. Accordingly, one defines /I(X,E)

as the filtered colimit of the /I(Q,E)

as the associated projective limit of the Z ( Q , F ) .

and

m(X,F)

In this way one arrives at

the following result.

I0. Theorem.

[/I(X,E),F] ~ [E, m(X,F)]

naturally for E and F in C.

The spaces /I(X,E)

consists, as does LI(X,E), of (equivalence classes) of

integrable functions X ÷ E; but unlike LI(X,E) , these functions can attain essentially only countably many values.

This comes about because, as a vector space, /I(X,E)

is just the union of spaces of the form /I(Q,E).

Thus the filtered colimit of the

system /I(Q,A), when formed in A gives us LI(X,A)

and when formed in C gives us the

rather different cbv-space /I(X,A) which is not a Banach space even though A is. Thus theorem i0 also illustrates that not every C-left adjoint is the extension of a B-left adjoint.

References 1

E. Dubuc, Kan extensions in Enriched Category Theory, Springer Lecture Notes

2

S. Eilenberg and G.M. Kelly, Closed Categories,

in Math., 145 (1970).

bra, La Jolla 1965, Springer, Berlin

Proc. Conf. Categorical Alge-

(1966) 421 - 562.

3

H. Hogbe-Nlend,

Bornologies and Functional Analysis, North Holland, Amsterdam

4

G.M. Kelly, Adjunction for Enriched Categories,

(1977). Reports of the Midwest

Category Seminar III, Springer Lecture Notes in Math. 106 (1969) 166-177. 5

S. Mac Lane, Categories for the Working Mathematician,

Springer, New York

220

8

S. Mac Lane, Categories for the Working Mathematician,

Springer, New York

(1971). L.D. Nel, Enriched algebraic categories with applications

in Functional

Analysis, Proc. Conf. Categorical Aspects of Topology and Analysis, Carleton Univ. 1980

(to appear).

L.D. Nel, Riesz-like representations methods, Advances in Math.

Carleton University, Ottawa, Canada. NSERC aided.

for operators on L 1 by categorical

(to appear).

EXACTNF.SS _AND PRCLTECTIVITY* S. B. Niefield Union College Schenectadv, New York 12308

i.

Introduction Let

functor

V

be a syr~netric monoidal category.

M@-:V---~V

An object

M

of

V

is exact if the

has a left adjoint.

In [6], we obtain the following characterization of exact R-modules and cc~rmtative R-algebras over a cc~nutative ring R.

A module

M

is exact if and only if

it is finitely generated and projective if and only if it is finitely presented and flat.

In particular, one can show that the left adjoint to

HC~R(M,-) . An algebra

A

M®-

is given by

is exact if and only if the underlying module is exact.

The construction of the left adjoint is obtained by applying the following general le~ma with

and

A

the categories of modules and algebras, respectively,

forgetful functor,

S

the sy~netric algebra functor,

Lemma 1

M

F = A@-,

and

T

the

G = TA@-.

Consider the following diagram of categories T S

where

TF = GT,

and

S

lar e p i m o r p h i ~ for all Proof

is left adjoint to A.

If

G

functor of

A, for every object is a regular epi.

EA'

A'

with the counit

~A:STA--~A

has a left adjoint, then so does

It suffices to show that for every

since

T

of

But, if

M A

in

M , A(SM,FA)

F.

is a representable

admits a presentation

G'---~G,

a regu-

gM---~SN-

~A'

we have

A_(SM,FA) & M_(M,TFA) & M(M,G~fA) ~ ~_~(G'M,TA) ~ A(SG'M,A) and the desired result follows. Let ioc denote the category whose objects are locales (i.e. complete lattices A

satisfying the distributive law

S c A) ,

a A ~ / S = ~[a A s [ s

6 S]

for all

a E A

and

and morphisms are functions which preserve finite meets and arbitrary sups.

In 1978, ~ r t i n

Hyland [3] showed that a locale

continuous lattice.

A

is exact if and only if it is a

What follows is an outgrowth an attempt to obtain an abstract

construction of the left adjoint to

A®-

for a continuous locale

A, that is, a con-

struction like the one for co~nutative algebras (Hyland's construction uses theories) . Now, Joyal and Tierney have obtained many results for locales by considering the category S1 of sup lattices

(i.e. complete lattices and sup preserving maps) as

abelian groups, and locales as cc~nutative rings.

After several discussions we de-

cided t/lat one should be able to obtain Hyland's theorem using a method much like * This research was supported by a Killam Postdoctoral Fellowship at Dalhousie University.

222

that of the module/algebra results.

Unfortunately,

this doesn't quite work.

Sup-

lattices behave like modules, but locales are too different from algebras to apply lerana i.

2.

But there is a way around this problem, as we shall see in section four.

Exact sup lattices We begin with some basic facts about sup lattices.

a *-autoncraous category in the sense of Barr [i]. Hom(M,N) = S I(M,N),

where the sup of a family [fi:M--~N}

( V f i) (m) = V[fim}, opposite lattice

M °.

the unit

I

is defined by

is complete lattice ~, and the dual of

As usual, Hom(M,N) = HQm(N°,~I°)

that we shall reserve the notation If

First, we note that S1 is

The internal horn is given by

M*

and

M is the 00 = Hom(M,N ) . Note

M@N

for the dual Hom(M,I) .

[Mi] is a family of sup lattices, their product ~[Mi in S_l_is given by the

cartesian product with pointwise sup.

Now, ~ M

is also the coproduct of the

M

1

S1 since ( )0 is a ccmplete duality. as a quotient

~X-~e M, where

identification of and coproduct of Theorem 2 .i

2x X

Moreover, every sum lattice can be expressed

is a set, namely take

with the power set copies of

~(X) .

X = M

Note that

a)

M@-

Horn(M,-)

c)

_H is flat

d)

M

e)

MO-

f)

M*

g)

M* ® -

h)

Hom(M,-)--~ M ® -

and ~X

e =V,

using the

is both the product

2.

The following are equivalent for a sup lattice

b)

Proof

X

in 1

M

has a left adjoint has a right adjoint

is projective ~ Hom(H*,-) is projective and

M** ~ M

~ Horn(M,-)

We shall show that

a) ~ c )

~d)

~ e) ~ f) ~ g) ~ d), e) ~ h) ~ a),

and

h) ~ b) ~ d) . a) @ c)

If

M®-

has a left adjoint, then

M

is flat since every right adjoint

preserves monomorphis~s. c) ~ d)

If

_H is flat, then Horn(M,-) preserves epimorphisms since

monomorphisms, Hom(H, -) : (M @ -°) ° , and d) ~ e)

If

( )

M = ~X, then the canonical map

0

M®-

preserves

interchanges monics and epis.

M@N--~Hem(M*,N)

is an isomorphism for

it is the composite of the isomorphisms O

Hom(2X,N O) ~ Note that M

~X

[Hom(~,NO)X] O

~ (NOX)o

~ _ _ ~ H o m ( 2 . , N ) X --~ Hcm (~X ,N)---~Hmm(~ X* ,N)

is projective since it is the free sup lattice on the set

is any projective sup lattice, then

M

is a retract of

2X

for some

X.

Now if

X,

and

the desired result follows. e) @ f)

Clearly,

M** ~ M.

being a left adjoint.

Also,

M*

is projective since

M®-

preserves epis

223

f) ~ g)

If

M*

is projective, then applying

d) # e)

we have

M* (9- ---Horn(M**,-) ,

and the latter is isomorphic to Horn(M,-) . g) = d)

Clearly,

e) ~ h )

If

M*®-

M

~ Horn(M,-)

h) 9 a)

and

b) ~ d)

M

is projective since

M(9- -~ HC~(M*,-), by

then

M*®-

M*®---gM®-.

preserves epis. But, we also know that

e) 9 g) .

h) ~ b)

are clear'.

is clearly projective since Hom(M,-) has a right adjoint.

After seeing this theorem, ~ k e

Barr remarked that the equivalence of

is valid in any

*-autonomous category.

Proposition 2.2

The following are equivalent for an object

a) and e)

In fact, we have the following proposition M

in a *-autonomous

category a)

M(9-

b)

M* ® - = Horn(M,-)

c)

Horn(M,-)

d)

M@-

e)

Hom(M,-)---4M ® -

Proof

has a left adjoint

has a right adjoint

~ Hom(M*,-)

a) ~ b)

If

L--J_M@-,

then

Hom(M,N) ° -_-M(gN ° -~ H o m ( I , M ® N o) _--Hom(LI,N °) = (LI(gN) ° and (LI)° ~ Hom(I,(LI) °) ~ Hom(LI,I °) _= H o m ( l , M @ l °) -_-M @ I ° ~ Hem(M,l] ° Therefore, Hom(M,-) & L I ® - ~ M* (9 -. b) ~ c)

is clear

c) ~ d)

If

R

denotes the right adjoint to M~N

Horn(M,-),

then

~ Hom(M,N°) ° ~ Hom(l,Hom(M,N°) °) ---

Hom(Hom(M,NO), o) & Hom(NO,R(IO)) & Hom(R(i o)o,N ) and R(I °) ~ Hom(l,R(l°)) ~ Hom(Hc~n(M,l ).I°) ~ Hem(M*,I °) ~ Hom(I,M *°) ~ M *° Therefore, d) ~ a)

and

b) ~ e)

If

shown that

3.

~8-

---IIom(R(I°)°, -) & Horn(M*,-)

e) ~ a M*@-

are clear

---Hom(M,-) , t/qen Hom(M,-)--~Hem(M*,-) .

b) @ d) .

Therefore, Hom(M*,-) ~ M ® - ,

Now, we have already

and the proof is complete. R

Projective sup lattice One is tempted at this point (in analogy with the module/algebra theorem) to

attempt to prove that an exact locale is flat or projective as a sup lattice. is not the case.

This

We know (from Hyland's theorem) that the exact locales are the

continuous lattices, and a more careful analysis

of projective sup lattices will

show that t]]ese lattices are completely distributive, whereas there are clearly

224

continuous lattice which are not completely distributive, e.g. the opens of the reals. Recall that a complete lattice family

M

is completely distributive if for every

[mijli E l,j E S(i) ] we have A i £ I (VjEJ(i) mij) = Vf:i~<]iJ(i ) (AiEzmif(i)) over I

A complete lattice m : V [ n In < m], m ~ V S , S _cM, subsets

S

Theorem 3

then

of

ous lattice.

M

where

M

is completely continuous if for all

n < m

n m s,

(read for some

we have

Note that if we require that the

The following are equivalent for a sup lattice

M

is projective

M

is completely distributive

c)

~4 is completely continuous a) # b)

Suppose

adjoint right inverse.

M

is projective.

Indeed,

VM

m|--~[nln s s, for some

preserves all sups and infs. b) ~ c)

s E S.

m E M

m) if whenever

be directed, then we have precisely the definition of a continu-

a)

the map

is completely below

Thus, a completely continuous lattice is necessarily continuous.

b)

Proof

n

Suppose

m E M.

Let

Then the map V~:P(M)--*M

has a right inverse

s E i m]

Thus,

M

admits a left

i, and replacing

gives the desired map.

~ Vs}, J(S) : S,

and

by

Therefore, V M

~{ is completely distributive since

I : [S q ~ I m

i

~(M)

m~j = j.

is. Then

by complete distributivity we have A s E I V j E s ms, j :

Vf:i__+<) s ( A s E I over I

Now, the left-hand side is clearly equal to m, for f:I--~jS over I, d~en SEI n ~ms,f(S) = f(S) E S. c) 9 a)

~m.

It suffices to show that V~{:~(M)--~M

is clearly projective. M

n = AsEimS,f(s)

If

is cc~pletely continuous.

m E M, define To see that

ccrnpletely continuous lattice) if V s = V [ t l t ~ s,

for some

[m] E I. If

We claim that if

m s Vs, then

admits a right inverse, for

i(m) = [nln ~ m]. i

ms,f(s ))

s

~(M)

Then, Z { i = ~{, since

is sup preserving we note that (in a

n ~ V S , then

n ~ s

for some

s E S,

since

s E S]. I

Remarks i. There is an analogue of

b) ~ c)

for continuous lattices [2;p.58]

which complete distributivity is replaced by a

in

weaker notion, i.e. the sets

[mijlJ ES(i) ] are reQ~]Jred to be directed. 2.

Note that projective locales (i.e. projective in Loc) are necessarily projective

as sup lattices, (but not conversely) for a local the map VA:~(A)--~A to show that

A

A

is projective if and only if

admits a right inverse in Lo___cc. ~oreover, it is not difficult

is projective in Loc if and onlv if

A

is completely continuous,

225

the completely below relation preserves finite meets and the top of lattice

A.

2 × 2.

It is clearly projective in

sup lattices), but

4.

1 ~ i,

An example of a locale that is projective in

(i,i) ~ (i,i)

since

S1

S1

where

1

denotes

but not Loc is the

(it is the coproduct of projective

(i,i) = (i,0) V (0,i).

Exact locales Now we return to the original probl~n, i.e. an abstract proof of Hyland's

theorem.

At first, it se6~s that the locale/commutative algebra analogy breaks down,

but Andr~ Joyal observed that this is not really the case.

He noticed that, in the

proof of le~na i, we do not use the full power of the existence of the left adjoint to

G, we only use the fact that the functor

M(~{,GT-): A - ) S e t s

is representable.

To apply (this amended form of) le~na i, we need to know that the forgetful functor Loc--~Sl locale

2X

has a left adjoint.

on a set

X,

But, every locale is a quotient of a free

and

SI(2X,A) ~ Sets(X,A) ~ A-slat(K(X),A)

a Loc(+CI(K(X)),A)

where A-slat denotes the category of meet semi-lattices and meet preserving maps, denotes the free meet semi-lattice functor, and locale functor, i.e. + C I M Theorem 4

is the locale of d~4nward closed subsets of

The following are equivalent for a locale

a)

A

is exact in Loc

b)

A

is a continuous lattice

c)

A ~ -: Loc--~SI

preserves monomorphisms

d)

A@-:Loc---~sl

preserves equalizers

e)

A@-:Loc---~SI

is representable.

Remark

The following proof of

troduction to [4].

a) ~ b)

K

+Cl:A-slat--~Loc denotes the free M.

A

is due to Joyal and is outlined in the in-

Any attempt to improve upon this beautiful a r g ~ n e n t w o u l d be

futile. Proof of theorem 4 L

a) ~ b)

If

preserves projectives, since

L:Loc---~Loc denotes the left adjoint to A ® - preserves epimorphismas.

AO-,

then

In particular, if

S

denotes the Sierpinski locale (i.e. the free locale on the sup lattice 2), then L(S) is projective.

But, the lattice Loc(P,2) of points of a projective locale is nezes-

sarily continuous, and Loc(L(S),2) ~ Loc(S,A@2) Therefore, b) ~ c)

A

~ Loc(S,A) ~ SI(2,A) ~ A

is continuous.

To see that

A@-:Loc--*SI

preserves mono~orphisms, it suffices to sho~

that given a diagram

A

C ° fo

B°

226

where

g

is sup preserving and

h:A-~C ° If

such that A

f:B--~C

is a monomorphism of locales, there exists

f°h = g.

is continuous, then the m o r p h i s m o f

locales

V:IclLA---#A admits a sup

preserving right inverse, where Idl A denotes the locale of downward closed (upward) directed ideals of

A.

But Idl A (and hence any retract) in S l c l e a r l y satisfies the

desired property since the map SI(IclLA ,M)--->Lat(A,M)

given by

(f:IdlA--,M)~--~(A+seg)IdlA-~fM), is an isomorphima, where Lat denotes the category of distributive lattices and finite meet and join preserving maps.

In particular, given a diagram Idl A

Co

~ B° fo

fg(+seg):A--~C °

is finite meet and join preserving, and hence induces the desired

fill-in I d l A - + C

°.

c) ~ d)

This is J/anediate since every monomorphism in Sl is an equalizer, and

A @ -:Loc--~SI d) ~ e)

If

takes equalizers to monomorphimns. A ® - preserves equalizers, then

preserves products, in any case.

satisfies the solution set condition If

f:A---~B °

A @ - preserves all limits, since it

Thus, it suffices to show that

is an element of

A 8 B = H~n(A,B°) ° , then

finite meets, and hence induces a locale morphism f + seg = f. and so

[+CI(A°)] provides a solution set for

g°:B°--gC°

1 ® g:A @ C---*A ® B

A ® B---*A ® C,

A@-.

takes

But, if

+ seg to

preserves

such that

A seg~+Cl(A°) O g:C--~B

is described as follows.

whose left adjoint is 1 ~ g.

it is not difficult to show that 1 ® f e) @ a)

takes

induces a sup preserving map Hom(A,B°)--~Hom(A,C°),

preserving map

f:A~--*B

f:+CI(A°)---*B

We claim that i ® f:A®+CI(A°)---~A®B

morphis~n of locales,

A ® -:!mc---+Sl

[5;pi18].

to

f,

is any

Composition with and hence an inf-

with this description,

f.

By (the amended version of) lem~a i, it suffices to show that the functor S I(M,A ® -):Loc--~SI

is representable, for every M. M = 2x general

and

L(~)

represents

M, we express

ate quotient of

M

Now, A®-,

e) then

takes care of the case where @ML(£)

as a quotient of "2 x,

represents and take

M = ~.

S I(2X,A@-).

If

For a

L(M) to be the appropri-

L(2X).

In conclusion, I would like to thank Andr6 Joyal for many helpful observations and suggestions. References i.

M. Barr, ",-Autonc~ous Categories", Springer Lecture Notes 752, (1979).

2.

G. Gierz, K. H. Hofmann, et. al., A Compendium of Continuous Lattices, SpringerVerlag, 1980.

227

3.

J.M.E. Hyland, "Function spaces in the category of locales", Continuous Lattices, Springer Lecture Notes in Mathematics, to appear.

4.

P.T. Johnstone and A. Joyal, "Continuous categories and exponentiable toposes", to appear.

5.

S. MacLane, Categories for the [~orking Math6~atician, Springer-Verlag, 1971.

6.

S.B. Niefield, "Cartesianness:topological spaces, uniform spaces, and affine schemes", J. Pure Appl. Alg., to ap~ar.

Constructive

Arithmetics

M. P f e n d e r

R. R e i t e r

M. S a r t o r i u s

Introduction .We found e s s e n t i a l sive Universe'

which

parts of A r i t h m e t i c

is w e a k e r

It just allows

formalization

can be a s s u m e d

to be consistent,

basic u n i v e r s e

has a n o t i o n of equality,

can be d e v e l o p p e d consistently

relative

i.e. n o t to collapse.

numbers

on c o n s i s t e n c y

of

'Primitive

recursion

and h e n c e

of the n a t u r a l

and that e q u a l i t y and e x t e r n a l l y

whose

equality to d e v e l o p

consistency

of the schema of p r i m i t i v e

numbers

can be e x t e n d e d

defined

this shows that it is p o s s i b l e in a f r a m e w o r k

Recur-

Object.

It turns out t h a t this

that A r i t h m e t i c

to this e q u a l i t y

Altogether

of the n a t u r a l

exclusively

of the schema of p r i m i t i v e

in such a w a y that i n t e r n a l l y

b e t w e e n m a p s coincide. Arithmetic

on the c o n c e p t

than Topos T h e o r y w i t h N a t u r a l N u m b e r s

recursion

is b u i l t

formalized

m a p theoretically.

As far as things

are n o w we still d e p e n d

the integer p a r t of a root. We first within

introduce

the g e n e r a l

this f r a m e the basic

the logical

of this d e p e n d e n c y

frame

algebraic

is for further

'Primitive R e c u r s i v e

structure

of taking

Universe'

of the natural

study.

and d e v e l o p

numbers

and

some of

structure.

i. A l g e b r a i c

definition

of P r i m i t i v e R e c u r s i v e

A Primitive Recursive minal

Elimination

on a quite e v i d e n t p r o p e r t y

object

Universe

is a c a t e g o r y U

i ('one-element-set'),

binary

O b j e c t N in the sense of

[Fr~, propos.

m i t i v e recursion' : T h e r e

are m a p s O:

and to a n y g i v e n m a p s f: A - - ~ B associated

a m a p g~f: A × N - - ~ B

satisfying

the equations

i--~ N

('iteration',

-:

g~ f

B

Axs

(of

'sets'

(cartesian)

5.21,

and

product

'maps')

('zero')

and

s: N - ~ N

g: B--~B

intuitively:

g* f

>

B

Numbers

'scheme of pri-

('successor')

('step function')

g~f(a,n)

= gn(f(a))),

> AxN

-:

w i t h a ter-

and a N a t u r a l

i.e. w i t h the f o l l o w i n g

('initialisation'),

A×N

A

Universes

(1)

given is

229

and w h i c h is u n i q u e in t h i s regard, then h -= g~f. As

seen

i.e.

if h: A x N - - + B

(! : A - ~ i is the u n i q u e m a p

a b o v e for the N N O - s t r u c t u r e ,

defined algebraically,

i.e.

(left projection) , A , B ~ - + r :

i n s t e a d of g~f

satisfies

(I),

into the t e r m i n a l o b j e c t i) . the other p a r t s of the s t r u c t u r e too a r e

in terms of o p e r a t i o n s

(e.g. A , B ~ - ~ A × B ,

A , B ~-+l: A x B - - ~ A

A x B - ~ B, f: C--7 A, g: C--+ A ~-~ (f,g) : C - - ~ A x B

(induced

map) ) and e q u a t i o n s and - for the N N O - s t r u c t u r e - a n i m p l i c a t i o n of equations.

2. G e n e r a l For reads:

scheme of p r i m i t i v e r e c u r s i o n

the c l a s s i c a l n a t u r a l n u m b e r s the To m a p s g: A - ~ B

(general)

('initialisation')

f: A x N - - + B such that f(a,O)

= g(a)

s c h e m e of p r i m i t i v e r e c u r s i o n

and h: A x N x B ~

and f(a,y+l)

In a P r i m i t i v e R e c u r s i v e U n i v e r s e this w o u l d read: is a s s o c i a t e d a u n i q u e f: A x N - ~ B Axs

A×N Freyd

verses

>A×N

f

> B =- A × N

shows this for topoi, ([Fr],

such that A

((l,r) ,f)

(A,O!) ; A x N h

~ (AxN) x B

'the'

like

(2) l i t t e r a l l y a s

B e c a u s e the n o t i o n of P r i m i t i v e R e c u r s i v e U n i v e r s e to g i v e a p r i m i t i v e r e c u r s i v e a l g o r i t h m w h i c h p r o d u c e s of the a b s o l u t e l y f r e e

~ t r u c t u r e E of type of P.R.

sets of sets ~nd of m a p s and w h i c h p r o d u c e s

Recursive Universe

(the L i n d e n b a u m - A l g e b r a ) ,

sets of sets and of maps.

(f: A--+ 9,g

in =

it is p o s s i b l e

at the same time a c o n g r u e n c e on 'the'

(initial)

Primitive

i.e. the f r e e one over the two e m p t y

and the m a p s of E

this s t r u c t u r e

(_E,~)

(Herbrand-model). T e r m s of the (map terms),

t h e o r e m s a r e all ('unigersal p r o -

since it is p r i m i t i v e recursive.

in [La] by w r i t i n g it up as a p r i m i t i v e

o u t - G O - T O - p r o g r a m in the sense of

(and

the sets and the m a p s

(equations) .The c o u n t i n g a l g o r i t h m

is f o r m a l i z a b l e in our system,

fact is shown in d e t a i l

(counts)

By t h e n a t u r e of f r e e construction,

(set terms)

: A-+B)

and

in the c l a s s i c a l frame.

is a l g e b r a i c ,

it into

c o n s t i t u t e s our t h e o r y as well as its c a n o n i c a l m o d e l t h e o r y a r e the sets of E

>B

U n i v e r s e over the two e m p t y

(counts)

that s t r u c t u r e w h i c h - t a k e n as e q u a l i t y - - m a k e s

This l a t t e r

r e c u r s i v e PL - w i t h -

[ B - L].

s t r u c t u r e of the N a t u r a l N u m b e r s O b j e c t N

introducing

shows f i r s t

g

B~A

his proof u s e s o n l y m e a n s of P r i m i t i v e R e c u r s i v e U n i -

Primitive Recursive Universe

After

( AxN)×B-+ B

5.22). V i e w i n g v a r i a b l e s as p r o j e c t i o n s a l l o w s us to state

3. C o n s t r u c t i o n of

4. S ~ n i r i n g

f

h:

> B.

assertions

gram')

(2)

T o any g: A--~B,

p a r t i a l l y a l s o to prove)

pairs

B is a s s o c i a t e d a u n i q u e

= h(a,y,f(a,y)) .

O+x ~ x

+: NxN --+ N and

s(x+y)

and

° : N x N --+N

_= sx+y

as u s u a l by p r i m i t i v e r e c u r s i o n one

and f r o m this that

(N,O,I,+,')

is a u n i t a r y

cc~nu ta tive semir ing.

5. L o g i c a l Freyd

s t r u c t u r e of P r i m i t i v e R e c u r s i v e U n i v e r s e s

shows in

[Fr],

5.11

that

i ~

O

s N ~--

N

c o n s t i t u t e s a sum. W e g e n e r a l i z e

230

(A,O!) this

slightly

constitutes

to the f a c t

a sum,

(flg) : A x N - - e C

i.e.

:_= (A f

which

Let us N--+N A ~+

N = A ~

f

truth values. i.e.

the

6. F u r t h e r

We d e f i n e (OIN),

showed

sum.

maps

f,g:

~ f

and

O with

h(A~s) (A,O!)

~ N

-= g

all

(O1 i) :

of t h e f o r m

predicates

junctors.

arguments

same method

:=

or

((O10!) I(O[I!)i)~

:~ N × N

logical

sg

1. M a p s

for A = I nullary

the o t h e r

of

A - ~ C. T h e m a p

signum-fu~ction

greater

and

N instead

=_ g(a) .

by truth-table

By t h e

=_ g. We u s e

The

using

boolean

uniqueness

tautologies

N,

of

on predicates

showed.

p(O)

p: N - ~ N

_= O, p(sx)

as

~ x, a n d

induced

o u t of t h e

the truncated

sum

O

1

subtraction

~ N L

s

N

by

by

p~N

N×N

> N

O'-x _= O,

:=_ N x N

> N,

i.e.

x = O ~ x, x-'sy = p(x'-y) . F o r

sx-'sy = x-'y ( c o m p e n s a t i o n ) ,

(x+y)-" (y+z) properties induction

~ x-z, of

x-'x _= O, p(x'-y)

(x-'y)-'z = x-(y+z)

equality

and

order

(association).

this

_= pxay, These

o n N to be i n t r o d u c e d

operation

we

(x+y)'-y -= x

laws

below

are

show

(absorption)

important

a n d for

for

the proof

of

pr inc ipl es.

Exponentiation

above

similarly

are

NxN

A×N

is a u n i q u e

on N

-

der.

N--+N,

there

h(A,O!)

number

:=_ (I IO!):

the predecessor

i.e.

natural

neg

the

Given

of N: T h e

predicates,

can be

Algebra

structure

A~s ~ AxN ~

(flg) (Axs)

h(a,l)

are called

junctors

sequel

and

A

g: A ~ N - - + C

and

N

(f Ig) o u t of

in t h e

_= f

=_ f(a)

every

'truth-tables', the

f: A - - > C ,

has the property

h(a,O)

identifies

Define

for

induced

p:-=

g--> C)

the 2-1ike

sg

N

by their

equations

used

A

study further

set A

2 for c a s e - d i s t i n c t i o n :

in p a r t i c u l a r

intuitively

arbitrary (fl g) (A,O!)

object

C I AxN ~

gives

for

such that

a not yet available h

that for an arbitrary

The

is d e f i n e d

equations

lemmata.

be d e f i n e d We n o w

by p r i m i t i v e

of t h e s e

Theathe

operations

notions

of

recursion, necessary

'prime

number'

likewise

in t h e and

division

sequel

are

'greatest

with

remain-

shown using

common

divisor'

the can

as usual. introduce

We will discuss

equality

on N a n d d e r i v e

an important

extension

Peano's

axioms

f~om

of P e a n o - i n d u c t i o n ,

the

primitive

recursion.

'diagonal

in-

duction' .

7. P r e d i c a t e

of

The distance

equality between

equality

by

N~N

= ~N

Equality

is r e f l e x i v e

Peano' s a x i o m s Natural

numbers

read

o n N and natural :-~ N X N

and

the proof

numbers dist>

symmetric

N and

in o u r f r a m e w o r k

are arrows

x:

i--+N.

of P e a n o ' s

is d e f i n e d neg)

N

equality

by

axioms dist(x,y)

(neg m e a n s

equality

on N implies

as follows:

:=

(x'-y)+(y±x), with

logical

O).

equivalence.

231

P1 P2

O: I--+N (x=y)

is a natural number

impl

(sx=sy) = i!

w h e r e infix n o t a t i o n is used: x = y instead of =(x,y), a impl b instead of impl(a,b), impl being the i m p l i c a t i o n junctor. P 2 m e a n s that the successor is well defined, crucial P3

(sx=sy)

i.e. that the 'map'

s has this

(internal) p r o p e r t y of a mapping. impl

(x=y) =_ l!

i.e. s is injective as a mapping. P4

neg(sx=O!)

-= I!

i.e. O is not a successor. P5

induction,

here in f i r s t order form:

If for a p r e d i c a t e (i)

X(a,O)

(ii) x(a,x)

-: i!

(this stands for 'property')

2t : A × N ~ N

and

impl ~((a,sx) -m i!

then x (a,x) =- I!

i.e.

is overall true.

Specializing to A = i g i v e s the c l a s s i c a l P5. T h e p r o o f s of P1 to P4 are straight forward by the use of tautologies,

the

lemmata a b o v e and the cited p r o p e r t i e s of equality. For proving P5, show that

) N

N

1

(X,N)

-(i,O~>

(X,N) NxN

commutes, u s i n g

_= (or (l,Xsr) , sr)

NXN

or(l,~(sr) = and(or (l,~sr),impl()f,)/s)) . T h e n c o n c l u d e by

u n i q u e n e s s of the iteration m a p above. Sometimes we u s e a sharper induction scheme, the / ~ - i n d u c t i o n : If

X(a,O)

Herein

_= i! and ( i=O ~(a,i)) impl )~(a,sx) ~ I! then x i/__kO~(a,i) is g i v e n by iteration of and: NxN--~N.

)£(a,x) -= l!

T h i s induction p r i n c i p l e is proved by a p p l i c a t i o n of the foregoing principle.

8. D i a g o n a l induction O n l y in special cases the induction p r i n c i p l e s so far d i s c u s s e d are a p p r o p r i a t e for showing g e n e r a l t r u t h of n - a r y p r e d i c a t e s on N. In particular t h e y d o not suffice for E l ~ n e n t a r y Arithmetics.

We need another

induction principle:

'diagonal induction' ,

232

i.e.

induction

along

If a p r e d i c a t e

the direction

of t h e d i a g o n a l

of N x N w h i c h

of t w o N N O - v a r i a b l e s

76: A x ( N x N ) --~N

~ (a, (x,O!))

-= I!

i.e.

X is t r u e on t h e x - a x i s ,

(ii)

~i(a, (O!,y))

=

i.e.

X

(iii)

~i (a, (x,y))

is t r u e o n t h e y - a x i s ,

]!(a, (sx,sy))

impl

-= i!

as f o l l o w s :

satisfies

(i)

i!

reads

i.e.

truth

is o v e r a l l

true.

and

is i n h e r i t e d

in p a r a l l e l

to

the diagonal, then

X ( a , (x,y))

Geometrically, that NxN

i.e.

this principle

an

to p r o v e

isomorphis~n

the rationals

it.

~

is o b v i o u s .

is w h a t w e u n d e r s t a n d

far developped define

-= I!

B u t s i n c e we c a n n o t

geometrically

In o r d e r

count:

to r e d u c e

N--*NxN

be

sure at the moment

by it, we h a v e to u s e the two dimensions

in a n a l o g y

our A r i t h n e t i c s to one,

so

we f i r s t

to t h e u s u a l m e a n d e r - c o u n t i n g

of

by

s

N

~

o/ ~

, I

=-

1 Ic ° u n t

=

N

NxN

I

NxN

(N×N, nc~ l)

;~

Icount I

(N×N)x N

(pxsl ( s r , 0 ! ) l)

-~ NxN

I (N×N) x s (N~N)x N

i.e. b y count(O)

-= (0,0) if 1 c o u n t (n) _= 0

(s r c o u n t ( n ) ,0) c o u n t (sn) =_ (i~ 1 c o u n t ( n ) , s

and

t r y to

show that count count

where

h a l f (0)

First

we

For

:= O,

-i

h a s as a n i n v e r s e (x,y)

h a l f (sx)

show that count

this commutativitiy

-i

of

r count(n))

count

-i

: N×N--+N

d ef ined b y

:m h a l f ((x+y) • (x+sy)) +y

:- h a l f (x) + o d d (x) a n d Odd (0)

.

is a r e t r a c t i o n (~)

else

in

for c o u n t .

:- O, o d d ( s ~ )

:_= neg odd(x)

233 N

)

~count

~

0

Icount

.-=

> N×N ~ e o u n ~ N-1 ~ N , n e g i)

(N×N) xN ((p×s l(sr,O!)l)

~

N

> NxN count -i

(~)

N

~

N

is sufficient by u n i q u e n e s s of the iterated map. B y pulling b a c k the sum the upper edge of of

(s,O!)

and

(~)

(NxN,O!)

N~N

-> (NxN)xN <

(NxN) x s

(NxN) xN

b e c o m e s the induced out of the sum

along

(0! ,N)

N

(NxN,neg i)

> N~N ~

sxN

NxN

((O],s) Isxs), the latter being an induced m a p out of N x N d e c o m p o s e d

the same way. N e x t it is shown that the upper part of of

half((sx). (ssx))

and h a l f ( ( s x + s y ) . s ( s x + s y ) ) + s y . half ( (x. sx) +sx) Corresponding

(~e)

and the induced out of the

is the induced out of the sum a b o v e (same)

sum of

half((sx)- (ssx))+sx

S i m i l a r l y one shows that the lower p a t h is induced by

and the induced of

half ( (sx. ssx) +sx)

and

half ((ssx+y) • (ssx+sy)) +sy.

inducing c o m p o n e n t s of upper and lower p a t h a r e equal: this is immediate

for c o m p o n e n t s two and t h r e e and v e r i f i e d by d e f i n i t i o n of half and d i s t r i b u t i v i t y of '. ' over '+' for the c o m p o n e n t one. It would be sufficient to show that count

-I

is a section by giving a n a r b i t r a r y

r e t r a c t i o n of it. A proof using count seems not to w o r k v e r y well. We f o l l o w instead [Pe], [H-B] and

IDa] by u s i n g the integer

ted by root(x). A b b r e v i a t e

q(z)

(part of) r o o t of a natural number, d e s i g n a -

:= half (root (8z+l ) +l) "-I

q' (z) := 2z'- (q(z)) 2 Then, as in the l i t e r a t u r e cited count(z)

:= (q(z) '-half(q' (z)'-q(z)) ,half(q' (z)-'q(z)))

is the c a n d i t a t e for the wanted r e t r a c t i o n of c o u n t proof is the e q u a l i t y

q(count-1(x,y))

-i

. The c r u c i a l point in Davis'

= half (root(2x+2y+l) 2+8y) +1) "-I = x+y.

If we can show this e q u a l i t y in P r i m i t i v e R e c u r s i v e U n i v e r s e s we are done,

because

from this the r e t r a c t i o n p r o p e r t y of count is straight forward. The wanted e q u a l i t y is further r e d u c i b l e to

root( (2x+2y+1)

2+8y)

=

I 2x+2y+1

if

[ 2x +2y+2

otherwise

2y ~- 2x+l

T h i s is c l a s s i c a l l y ~evident' ,and we have a c l a s s i c a l proof for it using nested induction.

234

Diagram

f o r m of the last statement:

((x,y) , sg (2y'- (2x+l))

N×N

(2x+2y+l] (2x+2y+2)l)

(NxN)x N

> N

/

\ r o o t ((2x+2y+1) 2+8y)

of this the

We will call fulfillment

'root property'

for a P r i m i t i v e R e c u r s i v e

Universe. For the d e f i n i t i o n [H-B]: The

of r o o t there a r e at least two p o s s i b i l i t i e s .

root(O)

- O,

second one - used

which

is c o m b i n e d

root(sx)

in our proof

- O,

rrem(sx)

-~

(categorical

translation

So, f r o m t h e

'root property'

By the

induction

T h e proof of

taking

statement

the integer

- is the f o l l o w i n g

p a r t of the root:

~ O

s root(x)

if

root(x)

otherwise

rrem(x)

O

if

s rrem(x)

otherwise

by using

The first f o l l o w s

root(x))2"-sx)

rrem(x)

= 2.root(x)

= 2.root(x)

sums)

follows

N ~= N×N. F r o m

this we p r o v e the p r i n c i p l e

of

as follows:

/k-induction X-= I[

by

rrem(O)

r o o t (sx) =

yields

of the c l a s s i c a l

w i t h the r e m a i n d e r

root(O)

diagonal

=_ r o o t ( x ) + n e g ( ( s

principle

we s h o w

)~(Axcount)

_= l! r count being a n i s o m o r p h i s m

w h a t we h a v e to show.

)6(A×count)

_= I!

turns out to be quite a d i f f i c u l t

one.

It u s e s the

u n i q u e n e s s of the induced m a p o u t of the sum -i count (O! ,N) count -I (SxN) N -> N
of

~

By diagonal

and

special

induction

e q u a l i t y to all o b j e c t s symmetry, verify

transitivity

its p r o p e r t i e s

9. E x t e n s i o n

we n o w s h o w t r a n s i t i v i t y of our

of a linear

Furthermore

of

'the' u n i v e r s e

sians p r o d u c t s

induction

E

and p r o v e we d e f i n e

its properties: t h e usual order

to all objects,

reflexivity, on N and

substitutivity

definition.

. From

(iterated)

e q u a l i t y c a n b e extended

- substitutivity,

in the u n i v e r s e

of e q u a l i t y on N. T h e n w e expaned

a r e of the form i or an

is trivial,

by c o m p o n e n t w i s e

for n-ary predicates.

ordering.

of the n o t i o n of e q u a l i t y

All o b j e c t s

f: A - - + B

initial u n i v e r s e

and others.

of i and N. E q u a l i t y onl

by d i a g o n a l

induction principles

i.e.

We s h o w r e f l e x i v i t y (a=b)

substitutivity

impl

product

of e q u a l i t y and -

(f(a)=f(b))

follows

cartesian

f r o m i and N to c a r t e -

~ I!

transitivity.

for all Symmetry

holds

235

for this equality. So our

'equality'

has all the p r o p e r t i e s of a n equality.

iO. Order o n N We define

~_: NxN --+N

by

x_Zy :=_ neg (x'-y)

and show - using d i a g o n a l induction

and several arithmetic laws

(again proved by d i a g o n a l induction)

antisymmetry,

l i n e a r i t y and trichotomy.

transitivity,

- reflexivity,

ii. R e l a t i o n s h i p of arithmetic o p e r a t i o n s v e r s u s eqrlality and v e r s u s order W e show the a d d i t i v e and m u l t i p l i c a t i v e s i m p l i f i c a t i o n rule, d i s t r i b u t i v i t y , a d d i t i v e and m u l t i p l i c a t i v e m o n o t o n y and a b s e n c e of zero divisors. 12. E q u a l i t y e x t e r n a l l y and i n t e r n a l l y f _= g vity'

implies

(f=g) -: I[

(reflexivity of equality). The converse,

seems u s not to hold a priori. We 'force'

a n extended notion of external equality:

it b y introducing

f ~ g

iff

(f=g) ~ I!

'coreflexi-

into the u n i v e r s e w h i c h is c o m p a t i b l e

w i t h all the d e f i n i n g o p e r a t i o n s of a P r i m i t i v e R e c u r s i v e Universe. We show consistency of this notion of e q u a l i t y b y proving:

If

O ~ sO

then

0 _= sO.

A P r i m i t i v e R e c u r s i v e U n i v e r s e such a s the above, provided w i t h a n e q u a l i t y notion satisfying all the p r o p e r t i e s r ~ t i o n e d

a b o v e inclusive coreflexivity,

P r i m i t i v e R e c u r s i v e U n i v e r s e w i t h Equality.

is called

In such u n i v e r s e s the d i a g o n a l is an

equalizer of equality and i!, injective m a p s f

(i.e.

(fa=fb)

impl

(a=b) =_ i!) are

monic. 13. I n t r o d u c t i o n of formal q u o t i e n t s by e q u i v a l e n c e p r e d i c a t e s and of extensions of p r e d i c a t e s W e n o w introduce into our u n i v e r s e s formal q u o t i e n t s by e q u i v a l e n c e p r e d i c a t e s and formal extensions of predicates. Sets of the extended u n i v e r s e are t r i p l e s p, ~

and f

a n equivalence,

(A, A × A - ~

i.e. reflexive,

N, A

~ ~N) w i t h p r e d i c a t e s

symmetric and transitive, all of these

c o m p o n e n t s in a P r i m i t i v e R e c u r s i v e U n i v e r s e U w i t h Equality. M a p s from A ~ = to B%~ a r e U--maps f: A - ~ B

satisfying

N o t i o n of equality:

iff

f ~ g

ffimpl ~(f×f)

9/ impl

and ~ i m p l

(A,f,X)

~f.

(fZg) -: i[

W e have to show: 1) The n e w u n i v e r s e V is a P r i m i t i v e R e c u r s i v e U n i v e r s e w i t h Equality. 2) 3)

(A, A×A ~ (N, N × N

N, A

l!

, N)

is the quotient

(neg Isg l) } N, N

i!) N)

A/p

of A byff.

r e p r e s e n t s the t w o - e l e m e n t

u s u a l properties. 4) V _ h a s e x t e n s i o n s { X : A }

of p r e d i c a t e s

~(: A ~ 2 .

set 2 w i t h its

236

This V has finite limits, monos are injective and

_V constitutes a good framework

for Aritl~netics of the integers and of the rationals. For details of this paper see [Pf ] , IRe2],

[Sa] and [Rel] .

References

b-L]

W.S. Brainerd, L.H. Landweber, Theory of Computation, New York 1974

[De]

M. Davis, Computability and Unsolvability, New York, Toronto, London 1958

[Fr]

P. Freyd, Aspects of Topoi, (1972),

[GeJ

Bulletin of the Australian Math~natical Society 7

1-76

G. Gentzen, Die Widerspruchsfreiheit der reinen Zahlentheorie, 117

Math.Annalen

(1940) , 493-565

K. G6del, 0ber formal unentscheidbare S~tze der Principia Math~natica und verwandter Systeme I, Monatshefte fiir Mathematik und Physik 38

(1931) ,

173-1 98

[H-B] [La]

D. Hilbert, P. Bernays, Grundlagen der Mathe[natik, M. LaBmann, G6dels Nichtableitbarkeitstheoreme

Berlin 1934

und Arithmetische Universen,

Diplomarbeit TU Berlin 1981

[Pe]

R. P%ter, Recursive functions, New York 1967

[Pf]

M. Pfender, Algebraische Mengenlehre, Vorlesungsskript TU Berlin 1979

[Rel]

R. Reiter, Mengentheoretische Konstruktionen in Arithmetischen Universen, Diplomarbeit TU Berlin 1980

IRe2]

R. Reiter, Grundlagen einer algebraisch konstruktiven Fundierung der Arithmetik

[Sa]

(in Vorbereitung)

M. Sartorius, Kategorielle Arithmetik, Diplomarbeit TU Berlin 1981

ADJOINT

DIAGONALS

TOPOLOGICAL

COMPLETIONS

Hans-E.

Abstract

:

Given

two t o p o l o g i c a l concrete

a commutative

(i.e.

category

initial

square.

construction

are d i s c u s s e d ,

Quite

recently

adjoint category C

of

We w i l l

Q

Our g e n e r a l

of some of this

as a d i a g o n a l

and a p p l i c a t i o n s

of this

too. [4]

has g i v e n

defined

show that this

of a far m o r e g e n e r a l

characterizations

diagonal

of this d i a g o n a l

a topological

of q u a s i s p a c e s

T.

instance

between

formed by

completions

or c o a r s e s t

Properties

E. J. Dubuc

situation

of functors

or final)

an a d j o i n t

in a r e l a t e d

Porst

square

and a finest

d i a g r a m we c o n s t r u c t

FOR

T

by a s u i t a b l e

construction

one u s i n g

of t o p o l o g i c a l

a construction

category

of an

and a

subcategory

is a special

the w e l l k n o w n

functors

setting w i l l be a c o m m u t a t i v e

(cp. [2] , [3] , square

external [7] ,

[9]).

of functors

C

(I)

where

Al ~

the f u n c t o r s

topological

Ti

~

(i=i,2)

and a m n e s t i c

full e m b e d d i n g s . as of a c o n c r e t e logical

q

and the

H e n c e we m i g h t category

completions

of

C.

C

A

2

may be t h o u g h t functors

Ei

to be (i=i,2)

think of this b a s i c

over

X

In general,

(proper)

together with however,

to be

situation two topo-

we w i l l

not

make use of all these a s s u m p t i o n s . Any diagonal a coarsest [3])

q:A2

> AI

of this s q u a r e

(there e x i s t at least

and a f i n e s t one w h i c h w i l l be d i f f e r e n t

defines

a new square

of f u n c t o r s

in g e n e r a l

238

A2 A:

(II)

This

square

functor always

now

£

admits

such

assume

~

It is n o w

our m a i n

first

is full,

adjoint

to look

which

1 :

£q < T2 1

T2-coarsest

purpose q.

Z : AI-~A2,

and

Zq : i

contained

Proposition

= T~

then

for

proposition

implicitly

T2Z

to be the

q

+ ~ A2

a quasidiagonal

that

addition

a left

Z

•

(ep.

if in

[2]).

for c o n d i t i o n s

which

condition

is a s l i g h t

generalization

Let be g i v e n

a

shall

quasidiagonal;

A necessary

in [2, Thm.

i.e. We

is g i v e n

make

£

by our

of a r e s u l t

2.6].

a commutative

square

A2

(II)

AI /

Z

TI~ where a

the

left

functor

adjoint

quasidiagonal Proof (~)

:

of

Tz q

To s h o w

T2B h ~ T 2 ~ A

: Z

X i~. ~TT2

is t o p o l o g i c a l . s.t.

of t h a t

f\~4 A2

T2oZ

= TI

If t h e n then

,%

,% :Af is the

) A2

is

coarsest

diagram. is c o n s t r u c t e d

by

initial

lifts,

i.e.

= T 2 (B--~ iA)

iff (~)

T2B

h>T2£A T2f~T2 B' = T2(B hf~B')

for

all

f:A---~qB'

where

(5)

(~)

feat

(A,qB')

is the T 2 - i n i t i a l

lift of

(T1f) • Hence

assume

and

hence

f6A2(ZA,B')

239

By

Tholen's

[ i0] w e has

generalization

can a s s u m e

underlying

that

[8] of

Wyler's

the u n i t

~

taut

of the

lift

theorem

adjunction

Z~-+q

maps TI~ A = ITI A.

Hence we have T1f

= T2f.

(with the

for m a p s

Moreover above

(by a d j u n c t i o n ) holds

f

by

and

f

corresponding

faithfulness

notations),

and given

an

some

t h e n we h a v e

~

for

f :A

of

% T2

acts and

Proposition

on m o r p h i s m s

the

2

T2

any

adjunction

we have

f =

g : ZA---+B '

>qB'.

If n o w

it is (~)

in p a r t i c u l a r

h = T2B h ) T z ~ A - T Z ~ A ) T ~ A That

of

by

condition

: Given

= T 2 ( B h--~A)

as c l a i m e d Tz~

comes

= T I.

a commutative

from

q.e.d. faithfulness

[]

square

C

(I)

with

A! +

EI

Assume

codense

that

q

adjoint

Proof:

quasidiagonal

of

dense)

(I),

of the

p r e s e r v e s init[ality. of

A2

T1-finally

q , any d i a g o n a l

the c o a r s e s t Then

(i.e.

q

and

exists.

corresponding

Consequently

T2 Let

topological ~

square

then

be (II).

is a l e f t -

q.

Given

and morphisms

a Tz-initial

source

hi:B--+ q A i a n d TIB k > T 1 q A

(A,Pi:A--+ A i)

k:TiB ÷ TIqA

T l q P i ~ T 1 q A i = Tlh i

s.t.

in

Az ,

240

we have to look for a unique qpioh

= h.~

~:~q ÷ 1

for each

i.

h:B + q A

Consider

with

the following

is given by the condition

a Tl-final

T1h = k

£q < 1

sink given by codenseness

of

and

diagram where

and

(mj,B)j

is

E1

T2P i = Tlqp i T2A

I

~

> T2A i

k

T2E2C j

1 = T2EAi

> TIB

~ T2A i

T2£m j = Tim.J Here

r.]

(for each

T2-initiality provides

T2Zh i = Tlh i

of

j 6 J)

is the lift of

(A,Pi).

a unique

h:B ÷

Finality qA

with

k~Tlm j

of the sink Tlh = k

and

due to

(mj,B) hom.

now

= qr..

J

To get

qPi°h

= hi

By the following Tlq(Pi°rj)

it is sufficient

proposition

= Tlq(~ij)

Pi°r'J = m..ij, hence

To prove

uniqueness

we have

Tlh = k

we have

to prove h.om. 1

j

By faithfulness qpiohom.j

of

h,

and

= qPi°qrj

assume

qpi°h

= h.1

will have the same final lift by

of

J

qPioh~ m. = h.om..

= q(~..)

and h~nce

1j

T2

we get

= h'°m''l ]

that for some for each (m.,B)

as

[:B +

qA

i.

Then

Tlh

k,

by a similar

J

argument Theorem T2°~

as above,

hence

[8] we conclude

= T I.

h = h. that

q

Now apply proposition

Again by Tholen's

taut

has a left adjoint

~

1.

construction

We go on establishing hypothesis

mentioned

this proposition

covers

in the introduction.

some conditions

in the preceding

with

[]

We will point out later to what extent Dubuc's

lift

equivalent

proposition

that

to the crucial

£E 1 = E 2 .

241

Proposition

3 :

corresponding lowing

(i

q

coarsest

conditions

~

(ii

Let

are

T2h (iv)

diagonal

of d i a g r a m for e a c h

(i)

faithful. the

h:EIC--+qB

C c

and

(iii)

underlying

(i) =

(iv)

:

(~B)

from

the

Hence

we

ob C

and

(ii) and

are

(iv)

functors

Given

(iv) ~ phisms ce

get to

(i) :

the

the

fol-

h:E2C

some

÷B

with

which

1

are

equivalent

are e q u i v a l e n t

Ti

Zq <

the m a p s

= AI (EIC,qB)

obviously

natural

T2(ZBOlh)

since

in a d d i t i o n

surjective.

since

T2

q

commutes

are

faithful.

consider with

T2~ B = 1

transformation

iq--+l

arising

[2]).

= T1h

where

~BOZh:ZEIC

= E2C--+B

(i) .

As

in the p r e c e d i n g

Zc:ZqE2C

ob C.

exists

B E ob A2

h:EIC--~qB

is the

condition

according

Then

Z

(I)

there

ZB:IqB---+B (where

(II).

and

C 6 ob C

q:A2(E2C,B)----~AI (qE2C,qB) :

(I)

= T1h

for all

Proof

of

of

equivalent:

is a d i a g o n a l

for e v e r y

any

quasidiagonal

ZEI (C) = E2 (C)

(iii

be

Hence

= iEIC--+E2C it r e m a i n s

]: T 2 E 2 C - - + T 2 1 E I C By d e f i n i t i o n

of ~ T2~ T2E2 c 1 } T 2 ~ E I C _ _ _ ~ T ~ is the T 2 - i n i t i a l

p r o o f we h a v e

with

to s h o w

= T2(Jc)

T2(~c) that

for some

natural

= 1

for e a c h Jc:E2C

for e a c h C ~EIC.

be the c a s e p r o v i d e d k' B = T2 ( E 2 C _ _ ~ B ) w h e r e the f a m i l y

this w o u l d

lift

(T1k = TIEIC--+ TIqB) .

mor-

of the

family

of all m a p s

(~)

is with

242

NOW

T2~ol

= T1k

T2qk'

=

= T~k'

by

(iv).

[]

Remark s (i)

The e q u i v a l e n t ly f u l f i l l e d

conditions

if the

(initially)

dense

of p r o p o s i t i o n

diagonal

q

(which w i l l

3 are

is f u l l

not be

obvious-

or if

the

case

E2

is

in

general). (ii)

The e q u i v a l e n t in the (a) (al)

conditions

following

X = Set, ~i

C = CompT2

= Span

Here

~i

(compact

(Spanier-spaces),

fulfilled

Hausdorff ~2

spaces)

= Top

spaces)

= Span

(i)

3 are

cases

(topological (a~

of p r o p o s i t i o n

A 2 = Unif

is c h e c k e d

easily

if

q

(Uniform is the

spaces)

finest

diagonal

(cp [4]). (b)

(iii)

The

X = ] , C

any p a r t i a l l y

a concrete

category

ordered

over

any

join-dense

completion

of

A2

any m e e t - d e n s e

completion

of

matically

of p r o p o s i t i o n

as is s h o w n

setting

of r e m a r k

Let

be

C

by

the

as

1

~1

conditions

set c o n s i d e r e d

3 are

not

following

C

(cp [5]). fulfilled

example

auto-

in the

(iib).

the p a r t i a l l y a

ordered

b

set

c

\o/ d

and take the [ 5 ].

AI

largest If the

the M a c N e i l l e final q

completion

completion is c h o s e n

_C -I

as the

coarsest

gets q(x)

= infl{y

(x) = i n f 2 { y

of

C

and

in the n o t a t i o n

• C l x ~< y} 6 A., Ix ~< q ( y ) } .

diagonal

A2 of one

243

From

these

graphs (iv)

The

of

[5])

in the

setting

iff

£x < y

iff

qy < x

are

following

(ii,

(cp

incomparable

relations

proved

a)

(ii, b)

satisfy

y < Zx

between

to be u s e f u l

(e.g.

in p r o p o s i t i o n

Z:AI--* A2

moreover

easily

using

the

= d ( ~ a) .

of r e m a r k s

x < qy

x,qy

already

Z(a)

established

q:A2 --+ At,

and which

it f o l l o w s

that

adjointness

maps

The

equations

to a p a i r

which

the

iff the

2 specializes

map

C

of m o n o t o n e identically

following

relations

y,£x

incomparable.

are

functors

in the s p e c i a l

£

and

instances

q

are

of r e m a r k

[4]).

Proposition

4 : Given

assumptions

with

is a c o a r s e s t

any

q:A2

diagram

+ ~i

diagonal

(I)

fulfilling

an a r b i t r a r y

of

(II)

the

the b a s i c

diagonal.

following

If t h e n

statements

hold: (i)

qZq

(ii)

T~ q

£qZ < T Z

Proof:

(i)

natural

transformation

By c o n s t r u c t i o n

U:iq ÷ 1 Hence (ii)

with

q~:q~q

With UZ:£q£

Proposition following

and

÷ q

U

T2U

of

£

there

with

5 : Under

assertions

(i)

£q~

=

(ii)

q£q

= q.

Proof

: (i)

(ii)

By a d j u n c t i o n

canonical

= i. T1 (q~)

= T2~

as in the p r e c e d i n g ÷ Z

is the

T2(~)

the

= I.

p r o o f we h a v e

= i.

assumptions

[]

of p r o p o s i t i o n

2 the

hold

is an a p p l i c a t i o n we h a v e

of

[2, T h m 2.3

a natural

(ii)]

transformation

~:i

÷ q~

244

with

T I (~) = i.

~q:q ÷ qZq

Hence

Together The

with

above

proposition

relations

of a new c o m m o n Let

K

be the

objects full

is n a t u r a l

between

subcategory

~i

: Assume

the

Then

ob K = Z(ob At)

tions

of

K

~

q

q

the AI

of

A2

and

serve

K

is c o r e f l e x i v e

and

L

of p r o p o s i t i o n

as an i s o m o r p h i s m

A2:

of t h o s e

let

of all q - i m a g e s

ob L = q ( o b A2) ,

and

[]

introduction

consisting

ZqB = B,

assumptions and

result.

allow

consisting

~

coreflexion

map

as r e f l e x i o n

If we

call

spaces Under

be

of

the

K.

2.

and the

restric-

of c a t e g o r i e s

one

can

the

A1

Cot.

and

L

~2

with

is r e f l e x i v e

(as in

[4]) the

rephrase

assumptions spaces

and

in

in

the

counit

AI

with

e

as

the u n i t

map.

K

generated

the

last

same

coreflexive

of m o d e l

results

of p r o p o s i t i o n

is at the

a full

category

as f o l l o w s :

2 the

time

generated

category

a full

subcategory

of m o d e l

reflexive of

A2-

subcategory (cp. [4,

2.3]).

Finally

we w a n t

Proposition T~

and

the

q < T1qiq.

i.e.

L.

Moreover

of

£

= 1

of the c a t e g o r i e s

the e q u a t i o n

of

TI (~ q)

this g i v e s

subcategory

satisfying

Corollary

5(i)

subcategory full

with

to d e s c r i b e

6 : Assume

is t o p o l o g i c a l

following (i) (ii)

Since

Then

K

internally.

of p r o p o s i t i o n for e a c h

2, w h e r e

A ~ ob A2

the

are e a u i v a l e n t

A e ob K the

sink

(EzC f,A)

To s h o w the s i n k

A

that

of all m o r p h i s m s

is

(i) i m p l i e s

(ii)

assume

of all such

sink

£k+lqA)

assumption,

the

of the

(IEIC

with

domain

in

C

T2-final.

(Eic--~k qA)

by the d u a l (ii) .

assumptions

in a d d i t i o n .

statements

and codomain Proof:

the

the c a t e g o r y

(generalized)

=

taut

k

A = lqA. is

(E2C Zk~A) lift

T2-final is

theorem.

by

T2-final This

implies

245

To prove

the converse

suffices

to show that

I:T2A ÷ T 2 ~ q A which

~f

for each

f

that

= T2(A-~qA

by d e f i n i t i o n T2EzC

assume

of

Z

(E2C f-~A)

and our assumption

of the above

sink

of all maps

and each

TIh:TIqA

is e q u i v a l e n t

h

of the initial

where

f

above

lift

+ TIqA'

= TI (ho f)

is as d e s c r i b e d

to

fh~A')

Now the above c o m p o s i t e is equal to TI~ 1 T1h TIEIC ~TiqA ~TIqA +TIgA' h

It

)

1 T2h ,T2A--+T2ZqA- . ~T2 A' = T2(E2C

of the source

is T2-final.

and

corresponds

to

f

by

adjunction. Now c o n d i t i o n

(iii)

of p r o p o s i t i o n

3 gives

With notations

and assumptions

Corollary

1 :

K = A2

iff

Corollary

2 :

Let

be at the same

subcategory

C

of t o p o l o g i c a l

that the e q u i v a l e n t is

(isomorphic

Corollary U:A + X

to)

3 : The M a c N e i l l e

completion

Using

the fact that

dual

a reflective

one gets

of

[]

as above we get in p a r t i c u l a r is finally

categories

conditions

is r e f l e c t i v e l y

gical

E2

the result.

dense.

time

a full

~i

and

of p r o p o s i t i o n subcategory

completion

contained

finally

~2

3 hold.

of

dense

and assume Then

~2

~l"

of any concrete

in any finally

dense

category topolo-

A. the concept

obvious

of t o p o l o g i c a l

dualizations

functors

of the p r e c e d i n g

is self-

results

as

for example: C o r o l l a r Y 3 °p : The M a c N e i l l e c o m p l e t i o n of any concrete category U:A + X is a coreflective s u b c a t e g o r y of any i n i t i a l l y dense topological

completion

of

A.

Examples (i)

Assume

in the general

the c a t e g o r y

setting

of q u a s i s p a c e s

that

X = Set

determined

by

and

AI

is

246

U: = T2E2: ~ ÷ Se__tt (and a G r o t h e n d i e c k - t o p o l o g y in the sense

of [4].

(X, a d ( C , X ) c E o b set of maps admissible (~)

where

C + X

where

maps

An A l - m o r p h i s m

The e m b e d d i n g

The

finest

diagonal

=

calculation.

Hence

apply

situation

to this

condition

(iii)

identification we r e f e r (ii)

The M a c N e i l l e

category the

whenever

is a map

e 6 ad(C,X).

(UC,C(C',C)c,eo b C ) (using

+ ~i

arising formula

(~)) .

from this

as is shown by

a straightforward

2, 5, and it's

since m o r e o v e r

corollary

very obviously

3 is fulfilled. K

situa-

[4]

arising

For the

in this

setting

2.13]. of

CompT2

category

which

dense

known

is a c o r e f l e x i v e

P r o x of p r o x i m i t y

is an i n i t i a l l y

is w e l l

completion

spaces of

subsince

CompT2

[6]) .

This e x a m p l e s may be used ferent

of the c a t e g o r y

latter

(cp.

unhandy

to the c o n d i t i o n

dense

of c a t e g o r i e s

to be a s o m e w h a t

of

EIC =

of p r o p o s i t i o n

completion

is a

(ad(C,X)) C

(Y, ad(C,Y) C)

propositions

to [4, Ex 2.12,

are pairs ad(C,X)

with

q:~2

(T2A,A2 (E2C,A) c)

shows h o w the t e c h n i q u e s

developed

in this note

in o r d e r to get a d d i t i o n a l

relations

between

topological

completions

of a c o n c r e t e

dif-

category.

Acknowled@ement The a u t h o r comments

is i n d e b t e d

attention

to G.C.L.

led in p a r t i c u l a r

proposition

2.

Thanks

to [8].

also

C)

finally

t i o n is then g i v e n by D u b u c ' s q(A)

on

~ ~f e ad(D,X).

ad(C,Y)

El: ~ + A

c h e c k e d to be

~i

subject

(X, ad(C,X) C) + fee

of

is set and

the family

f 6 C(D,C)

such that

is e a s i l y

objects X

is among others

~ 6 ad(C,X),

f:X + Y

Hence

C)

J

BrHmmer whose most valuable

to the p r e s e n t to W.

general

form of

T h o l e n w h o drew the author's

247

Remarks i.

added in proof:

If one drops

amnesticity

the d e f i n i t i o n through.

of a t o p o l o g i c a l

(e.g.

the c o n s t r u c t i o n

finest

one is often

or c o a r s e s t

construction). to a given

goes

equations

q~q = q);cp. [9] for

concerned

- diagonals

changes

in

in this case.

It can be p r o v e d q

everything

the obvious

T2o~ ~ TI,

of diagonals

In a p p l i c a t i o n s i.e.

functor,

One only has to replace

by e q u i v a l e n c e s

2.

and the p r o p e r n e s s - c o n d i t i o n

with

extremal

-

(cp [2 ] or Dubuc's

that

the c o n s t r u c t i o n

of

the order.

REFERENCES [1 ]

Adamek,

J.,

largest

initial

20 [2]

Herrlich,

H.,

Strecker,

completions,

G.E.,

Comment.

Math.

Least

and

Univ.

Carolinae

(1979) , 43-77

Br[immer,

G.C.L.,

functors,

Topological

Springer

Lecture

functors

Notes

and structure

in M a t h

540

(1976),

109-135 [3]

Br~mmer,

G.C.L.,

rization

of topological

in Math. [4]

Dubuc,

540

E.J.,

Springer

[51

Hoffmann,

(1976), Concrete

Lecture

Herrlich,

H.,

Notes Initial

R.-E.,

An e x t e r n a l

functors,

Springer

characte-

Lecture

Notes

136-151 quasitopoi, in Math.

Proc.

753

completions,

Durham

(1979), Math.

Conference,

239-254

Z. 150

(1976),

i01-ii0 [6]

Herrlich,

H.,Strecker,

universal

initial

G.E.,

completions,

Semi-universal Pacific

maps

J. Math.

82

and (1979),

407-428 [7]

Porst,

H.-E.,

and t o p o l o g i c a l (1978),

201-210

Characterizations functors,

Bull.

of M a c N e i l l e Austral.

Math.

completions Soc.

18

248

[8]

Tholen, W.,

On Wyler's taut lift theorem, General Topol.

and its Appl. [9]

Tholen, W.,

8 (1978), 197-206 Wischnewsky, M.B., Semitopological functors II,

J. Pure and Appl. Alg. i0 ] Wyler, O.,

15

(1979),

75-92

On the categories of general topology and

topological algebra, Arch. Math.

Fachbereich Mathematik Universit~t Bremen 2800 BREMEN Fed. Rep. of Germany

(Basel)

22

(1971), 7-17

INTERNAL CATEGORIES AND CROSSED MODULES Timothy Porter School of Mathematics and Computer Science University College of North Wales Bangor,

Gwynedd, Wales(U.K.)

This note is an attempt to indicate how one might initiate a combinatorial of presentations

in algebraic

categories

attention on one construction, entation.

other than that of groups.

in tile presentation

(Brown-Huebschmann

resolution used by Huebschmann to other categories

ated crossed modules

presentation

Lie algebras, monoids,

is in doubt.

in essentially

Thus we could have assigned

algebraic categories

to give as neat as possible a description

The existence of free

of such a construction.

In the case of small

such a construction was

For another large family of algebraic categories,

those monadie over the category of groups explicit construction,

to each

is known, hence it only remains

(over a fixed object set), and hence of monoids, [4].

small categories

in the category of groups, namely that this cat-

to that of crossed modules.

given and used by Mitchell

this construction

To be able to handle such cases we note a well

an internal category and have used this instead.

internal categories

categories

Generalising

In fact in the last two cases the existence of associ-

known result on internal categories egory is equivalent

to a pres-

between relations

[I]) as well as forming a part of the crossed

[3] in group cohomology.

of interest e.g. algebras,

etc., poses certain problems.

It will concentrate

namely that of the crossed module associated

This construction occurs in the study of the identities

study

which generalises

categories

of groups with operations,

categories

still holds,

(i.e. groups with operations), that in the group case.

the equivalence

we give an

In many of these

of crossed modules and internal

so one can replace the internal category by a crossed module

which is smaller and hence easier to study. The plan of the paper is as follows. of crossed modules other settings.

In section

(in Groups) and indicate how one may define analogous

This is followed by an account of Mitchell's

3, we prove the main result on the construction categories

I, we briefly review the theory

of "groups with operations";

objects in

construction.

of free internal

In section

categories within

in this the important point is not their exist-

ence, but the simple and explicit nature of the construction.

Finally we illustrate

this with a brief discussion of the situation in associative k-algebras

for

k

a

commutative ring.

I.

Presentations

of sroups

We consider a presentation a set and

v : R---+ UF(X)

free group on

of a group

G

to be a triple

(X; R, v)

is a function taking values in the underlying

X , such that the eokernel of the adjoint map

where

X

set of the

is

250

: F(R) ---'+ F(X) is

G • Classically

this restriction the relations extended

one had

an inclusion but as

is essentially

over a "formal

A crossed module action of

v

is misleading.

B

on

A

$

will usually

In fact the problem of studying

that of calculating

normal

closure"

consists

of

the kernel

F(R)

along

(a, b) ~-+ a.b)

~

amongst

or rather

~ .

of a group homomorphism

(written

of

not be a monomorphism, identities

O : A---+ B

together with an

such that the following

two properties

hold:

(i)

for all

a ¢ A , b E B

(ii)

for all

a I , a2 ~ A ,

0(a.b)

= b-10(a)b -I = a 2 ala 2

al.0(a2) For future reference

we note the following

(the Peiffer

simple way of writing

identity)

these two conditions:

L ai~.a

0 : A---+ B

is a crossed module

if and only if the following

diagram c o m u t e s

____+

1

) A

) A ~< A "+==--A - = - +

=I I ---+

I

A

where

B N B

the three rows are split exact,

on itself by conjugation, unlabelled

vertical

The proof There

maps

of

"---+

i I

B "'~

B

on

......

(I)

I

......

(2)

1

......

(3)

Io

" ) B ~< A ~ - - - - B

ol l --'-~

1

=

(

'B

"--~

thus representing A

and of

are the obvious maps

B

respectively

the action of

on itself by conjugation;

induced by

e

A

the central

on the semi-direct products.

is routine.

is clearly

to the category,

a category

of crossed modules

(Groups) ~ , of group morphisms.

and a forgetful

funetor

We need to construct

from that

a left adjoint

to this functor. Suppose phism on

f : H ~--+ G

H

by taking

(a)

HI =

(b)

fl

is a group homomorphism,

then we can form the free

~ H with isomorphisms ~ : H ~ H and with the natural g~U(G) g g g obtained by permuting the indices of the coproduct, : HI

G-mor-

:

• ) G , given by

fl(hg)

= g-lf(eg(hg))g

if

hg

G-action,

Hg

Now it should be clear how to form the free crossed module. Any element

of the form h-lk-lh(k.f1(h))

is in

Ker fl "

normal

and G-invariant.

it has a natural

Moreover

G-action

h , k E H!

the subgroup,

P , of

Hence on forming

HI

....

the quotient

and that there is an induced : C ---+ G

(4)

generated group

by such elements C = HI/P

G-morphism

is

one finds

that

251

satisfying axiom (ii)

: for all

In the study of identities, of the form of the presentation. of understanding

the identities,

c] , c 2 ~ C , cl.f(c 2) = c21cic2 elements

.

such as in (4) are always present regardless

As such they have little use in the initial stages and one loses little on dividing out by them.

elements are often called Peiffer elements and the subgroup

These

P , the Peiffer group of

f . Although the lemma clearly allows one to generalise module to other algebraic categories crossed modules on morphisms

than groups,

in such categories

of defining a crossed module suggests but it is not obvious this.

To circumvent

is not immediate.

The equational way

that such an existence theorem should be true,

initially how to provide a general categorical these difficulties,

construction

for

we replace the category of crossed modules

(in Groups) by that of internal categories these two categories

the definition of a crossed

the important existence of free

again in Groups.

is fairly well known (Brown-Spencer

The equivalence between

[2]) but we give a sketch of

the proof as it will be useful later. Given any group homomorphism form the semidirect product,

0 : A---+ B

and a right B-action on

A , one can

B m A , and define two functions

do)

B~A-~IB where

d0(b, a) = b , d1(b , a) = bS(a)

phism if and only if

0

the natural splitting on "composable

.

do

is a B-morphism, s : B ----+ B ~ A

is a homomorphism but

(i.e. satisfies

dl

(i) above).

is a homomorThere is also

and one can attempt to define a composition

pairs" by (b, a) o (D0(a), c) = (b, ac) .....

This is associative

and with the

s(b)

(5)

as identity elements one gets a category.

However this category will not be internal unless

o

is a group homomorphism

and this

holds if and only if (ii) holds. Conversely take

given

(C, B; do, dl, s, o) , an internal category in Groups, one can

A = Ker d o , 0 = d]IA

C , to get a crossed module

and the B-action on

It is easily checked that in many categories ponding approximately

to Orzech's

However for small categories not get an equivalence

"categories

induced by

conjugation within

of "groups with operations",

of interest",

only an embedding of crossed modules

corres-

[5], a similar result holds.

(with fixed object set) and thus for monoids,

In another family of important categories, commutative

A

e : A---+ B .

one does

into internal categories.

including those of unitary algebras over

rings~, the formation of kernels takes one out of the category concerned,

so the construction

of crossed modules from internal categories

breaks down completely.

In the next section we consider the first of these cases and the solution given by Mitchell

to this basic problem in [4].

252

2.

Presentations Let

¢

relations

of categories and 2-categories

be a small category with

in

¢ , that is,

R

0

its set of objects and let

is a set of pairs

(a, b)

R

be a set of

of elements of

C

with the

same domain and codomain. Suppose

A , B : p

(of length one) from

) q

A

to

in B

¢

we say that a symbol

(x, a, b, y)

is a path

if A = x ay x by = B ,

and with

(a, b)

Let and let

~(p, q)

or

(b, a) £ R .

be the set of all paths of length one between elements of

~0(x, a, b, y) = x a y

, el(X , a, b, y) = x b y

.

(~0 (R)(p, q) , ~(P, q); c O , e I , be the free category on the directed graph,

We call

i

For each

C(p, q)

p, q ~ 0 , let

; i)

(F(p, q), ¢(p, q); g0' e|) . the "vertical composition". There is also a possible "horizontal

compos-

ition": Suppose we write = (x I . . . . . .

(~, ~, ~, I)

x n) etc. and

for an element of

~0(R)(p,

e0(~, ~, ~, E) = x I a| Yl = A

q)

where

say,

~i(~, ~, ~, I) = x n b n Yn = B ; similarly

(2, ~, !, !)

in

g0(Z, ~, ~, X) = C , Sl(~, c, !, X) = D .

Define the "horizontal

~0(R)(q, r)

two elements by (~, ~, ~, X) • (2, ~, ~, X) = (~, ~, ~, ~C) i (B~, ~, d, X) , where

xC = (y|C, y2 C, ..., yn C)

etc.

One can represent this schematically as follows: A path of length one

(x, a, b, y)

is

A

rx

~X

Vertical composition

a

~

A

b

Y ~

B

~

is the obvious

:

~

i

A ~

B

= C C

"Horizontal

composition"

is A

C AC lC

lB

f3

B

D

BD

This looks fine until one checks the "Godement interchange

law"

with

composite" of these

253

which will make composition

into a functor and the

~0(R)

into a 2-category - it fails.

The problem is that there is no reason why one should have the equality

1

1

B

~

1

=

1

~

This is the analogue of the Peiffer identity in this setting and one should probably call such pairs Peiffer pairs.

On dividing out by the congruence

generated by these

Peiffer pairs one obtains a 2-category. Remark The analogy between Peiffer pairs and Peiffer elements

seems very strong, but it

will need more study to be certain that the roles played by them are as close as it seems.

Mitchell,

[4], uses a notion of degenerate

closed path to define the congruence

relation.

This notion is the analogue of that of a primary identity sequence in com-

binatorial

group theory.

The relationship between these latter and the Peiffer elements

is explored in [I].

3.

Presentations

and internal cate$ories

As suggested above, ated to presentations of presentation

in some cases internal categories may be more easily associ-

than crossed modules.

in a general situation,

In fact considering

the obvious definition

this is fairly clear.

U) Suppose given presentation of within

D

C

¢ +-#-- D

, F

relative

to

left adjoint to D

U

and an object

should consist of an object,

C

of

¢ .

A

X , of generators

and a pair of maps

do °

R---~UF(X) d|

~0

such that the adjoint diagram F(R) ~

F(X) dI

has as equaliser an epimorphism Thus a presentation

~ : F(X)

is essentially

~ C .

an internal directed graph.

To construct a

free internal category on such a directed graph, one can mimic the construction in Sets, taking into account the difficulty need not commute.

This construction works in many algebraic categories

antees that the basic program suggested and interesting

class of categories

used

that in general coproduct and pullbacks

in this note is feasible.

one can say more;

and so guar-

However in a large

one can control the construction

of the free category much as one could the construction of the free crossed module in Groups. Initially

let us suppose the category

C

has coproducts.

In this case one can

254

form, on any internal directed graph, an internal "directed graph with identities" D.G.I.,

that is one with distinguished vertex loops.

Given

The construction

or

is simple.

do in

C 1 ----+ C O dI we form

DG(¢)

do )

c o II c I

in

co

DGI(~)

S

with the

do

and

dI

extended in the natural way and with

This provides a left adjoint for the forgetful Now we must impose extra conditions functor,

U', from

C

preserves

pullbacks,

on

s

functor from

C .

the inclusion morphism.

DGI(C)

to

CG(C)

.

We shall assume there is a forgetful

to Groups, which preserves

"internal categories".

this suffices so, for instance,

if

U'

(If

U'

has a left adjoint,

the

condition is satisfied.) In groups and hence in

C , the composition

by the group multiplication.

in internal categories

More precisely the interchange

is determined

law,

(a o b) . (c o d) = (a . c) o (b . d) which expresses

the fact that

o

is a group homomorphism readily implies

(of. Brown-

Spencer [2]) that a o b = a s (d0(b))-Ib Thus if

b : y

~ z

and

c : ~

.....

(6)

~ m , the interchange

law implies that in any

internal category the equations s(y)-Ibcs(m) -I = c s ( m -I) s(y)-Ib

.....

(7)

hold. Thus given D $ we may form a congruence on equations

D

~ CO

in

DGI(C)

by relating the two sides of (7) plus any other

coming from the interchange

laws for possible other operations.

Dividing

by this congruence gives a new d.g.i. _._.+

D/~ ~ and now defining nal category.

o

Co

by (6), one shows that this is the underlying d.g.i, of an inter-

The universal

property is easily checked.

To sun~narise one has: If

C

has a left exact "forgetful"

functor to Groups then in the adjunction

Cat(c) ~---DGI(C) the unit is naturally an epimorphism with kernel generated by "generalised elements"

Peiffer

such as [s(y)-Ib cs(m)-l][es(m) -I s(y)-Ib]-I

and similarly for any other operations

existing on objects of

C .

It is perhaps better to think of this as a list of instructions

rather than as a

255

theorem.

To illustrate

k-algebras

4.

for

k

it, we will examine the case of a category of unitary

a commutative

ring.

Example Let

C = k-algebras

general principle holds.

(with I),

U : ~ ---+ A b c

We consider

Groups

presentations

adjoint pair

preserves

relative

pullbacks

so our

to Sets i.e. for the

U k-alg ~F-- Sets

A presentation

(X; R; do, dl)

leads

to a d.g.i. do

k[X] ~ k[R] ~

k[X] S

and hence to Peiffer elements as

+

is commutative.

for

That for

+ x

and

x .

reduced

The Peiffer

identity for

(-b c + s(y)(e - s(m) + d)) = ( a d - (a + b - s(y)) Dividing out by this congruence forgetting and

the unit elements,

c ~ Ker d I , (8) gives

b e ffi b S(dl(e))

.

e(a' a) = a'0(a)

(ii)

e(a)a'

earlier and so

As for any

= Sdl(e)b

for

(without

.

occurs

If

~ Ker d I

if,

b ~ A ,

Remembering h o w the A0-action on

for

this crossed module directly

I) of polynomials

over

A0

There is a map

on elements

e(a)a'

- a a'

of t~e form

A = A~[R]/P

Consequences

, 0 = dllA

e , e - s d1(e)

A

a E A , a' ~ A 0

of the diagram

term.

then

(8)

(I) - (2) - (3) considered

is a crossed module.

R , having zero constant

elements,

......

Simplification

a , a' ~ A .

exactly to commutativity 0

.

, e(a a') = e(a)a'

= a a' = ae(a')

One can construct k-algebra

cb

s(m))

category.

A = Ker d o , A 0 = k[X] .

is trivial

s , this gives us equations

(i)

These correspond

we put

Similarly

comes from the splitting

leaves an internal

b c = 0

+

to the expression

and

e

etc.

as follows.

Let

in the (non-commuting)

A~[R]

be the

indeterminates,

d o - d I : A;[R] ---+ A 0 , which Let

P

is zero

be the ideal generated by such

is the induced map.

of this construction

will be explored

in another paper.

References I.

R. B r o w n and J. Huebschmann, Identities among relations, to appear in: Low Dimensional Topology, Ed. R. Brown and T.L. Thickstun, London Math. Soc. Lecture Notes, C.U.P.

2.

R. Brown and C.B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proe. Kon. Ned. Akad. v. Wet., 19 (1976) 296-302.

3.

J. Huebsehmann, Crossed n-fold extensions (1980) 302-314.

4.

B. Mitchell,

5.

G. Orzech, Obstruction Theory in Algebraic Algebra, 2 (1972) 287-314, 315-340.

Rings with Several

Objects,

and cohomology, Adv. Math., Categories

Comm. Math. Helv.,

8 (1972)

55

1-161.

(I, II), Journ.

Pure Appl.

SUBDIRECT

IRIIEDUCIBILITY

Ale§ Pultr,

Subdirect algebras

AND CONGI~UENCES

Prague

irreducibility

was originally

for each monomorphism (1)

P i ~ :A--~Ai Pi~ A

is subdirectly

~E i

is

When extending we should follow explicitly

what

(perhaps

is

not

lowing

the in

simple

relations)

examples (X,R)

a subobject section

of

tegory,

all

On t h e

other

ible

in

the

of

° BxB

the

o£ n e v e r

one has in mind

above

Consider (sets

mappings

congruences is

a subobject

there

with

is

folbinary

A m since

. But,

on A t i n

obviously is

of (2)

the

tak$

irreducible,

R x2 ~[(OpO)~)

congruences)

definition

extension

moreover,

of graphs

3 x3 ~{(0,1)~) being

and,

directions,

subdirectly

an

such that the factor-

(for a formal

preserving

B = (2,

are

C = (3,

speaking,

category

of non-trivial

equivalences sense

not

where

any s y s t e m hand,

"primeness"

A is

and tak-

setting we are~

is that thus understood

: In the

products

to tell what we mean by a con-

in question

any o f

namely obtained

to this basic construct-

set of an object

and r e l a t i o n

= (R,{(0,1),(l,0)})

of forming

: intuitively

The trouble

categories~

(I). It expresses

really means,

(3) in a more general

in the category

such that

concrete

form

with respect

with the problem

with

b

trivial,

be non-trivially

the constructions

an i m p l i c a t i o n

on is

Ei

irreducibility

on the underlying

equivalent

the

are those which cannot

But this is easy

remains

even

of congruences

("prime objects"

encountered

i£

less elegant)

the subdirect

see 2.I below). not

of if

such that all the

irreducible

some o f

ions arsenal), Regarding the statement

object

i

irreducible

at least, one of the morphisms

the notion to more general

from others when using ing subobjects

{Ei~

trivial,

the

that s.i. objects

equivalence

~:A--~A

are onto,

in any system

(2)

first,

for varieties

is an isomorphism.

Equivalently,

gruence.

defined

([i~). An object was said to be subdirectly

A is

an i n t e r this

ca-

non-trivial.

subdirectly

o£ a p r o d u c t

irreducunless

con-

257

rained

as a subobject

i n some o f t h e

a s y s t e m 0£ n o n - t r i v i a l The a i m o£ t h i s

congruences

note

subdirectly

irreducible

tegories

terms

[4]

in

to present

objects

o£ t h e

behavior

however,

in the

a characterization general

one h a s

trivial

one.

o£ f i n i t e

concrete

of congruences

(Theorem 3°3

was p r o v e d f o r

the

special

ca; Cfo case

of

of graphs).

the

restriction

( t o be a b l e due m a i n l y clear

here,

in reasonably

where such a characterization

classes

ty

is

factors; intersecting

to finite

to apply to the fact

that

what the natural

(as the

replaced

is,

analysis

besides

finite

case

o£ t h e

in

the

[6], ~ ] ) o

tecnnical

[5] w i t h o u t it

infinite

E.go,

of some of the

reasons

modifications), is

is

reasonably

subdirect case

irreducibilithis

question

(I) with "monomorphism"

is not the same as "if

~ A i , it is a subobject definition

the

while

(cf.

by "subobject"

suitable

for

from

generalization

"primeness")

needs further

objects,

Theorem 3.3

Ai"

A

is a subobject

of

which may be a more

and may be not.

§I. Definitions l.l.

A concrete

category

with

a fixed

faithful

said

t o be a s u b o b J e c t whenever ~:C--~A

~cf. e . g . Dually, (in ~ , U )

[2],

(in

with

together ~:A--~B

is

i£ a

~:C--~B

,

there

is

a

£:B--PA

is

said

t o be a q u o t i e n t

if

~:A--~C

£.U6 with

resp.

= UT U~

onto,

for a

~:B--~C

, there is a

= f • ~

is a subobject

resp.

and that in everyday-life

are what one intuitively

dings of subspaces, the quotients.

)

for

an e p i m o r p h i s m )

a category-~

UT = £ .

(One sees easily that if objects

(£~,U)

U/~o £ = U~

is

. A monomorphism

~1).

whenever

is one-one

(J~,U)

U:~( --~ ~et

of induced

~n varieties

understands

subgraphs,

o£ algebras,

a quotient, categories

under

the term

etc. Similarly

U~

sub: embed-

it is with

all monomorphisms

are sub-

objects.) An o b j e c t (3~,U)

if

such that

for all

A

every the

Pi~

is

said

t o be s u b d i r e c t l y

irreducible

in

o n e o£ t h e

Pi~

subobject :A-'~Ai

are

onto,

at

least

258 is an i s o m o r p h i s m .

1~3.

C~,U)

Let

a preordered

class

be a c o n c r e t e

(sometimes called

category.

For a sat

the

over

fibre

X

define

X )

-Kux = ({A ; U A = X ~ , ~ ) with

A~B

we w i l l

iff

there

write

1.4.

is

~:A~B

categories

list.

The r e a d e r

commonly s a t i s f i e d

will

the

certainly

U preserves

(ii)

If

X

is is

the

and

(ii),

of J(

and

a

if

X

respect in ~

there

(iv) make

of algebras,

mapping,

is

then is.

to pro-

.

a decomposition

a quotient. sure

X--~y ~UX

hand,

in a c a t e g o r y

with heredity

each

X ~UA

is a s u b o b j e c t

there

JN.UX a r e m e e t - s e m i l a t t i c e s Obviously,

if

a monomorphism and in

and

a bijection

with

reflective

~:A-~B

a subobject (iii)

closed

ts

are

that

induces

~CUX

is a po-

an i s o m o r p h i s m

J(UY .

=

3.

an i n v e r t i b l e

is a set and it is finite

For every morphism L

2. In c a t e g o r i e s

the

f:X--~UA

ffi 1 x

and s u b o b j e c t s

now

and c o c o m p l e t e .

UT

subcategory

: I.

complete

and

ducts

to

conditions

an i s o m o r p h i s m

Every

Remarks

we w i l l

.

(v)

According

these

be p r o v e d f o r (vi)

= f

(iv)

/~

-

U~

T = Ia • Each JCUX

with

t

T with

I£

1.5.

is

(in such a case

categories.

an i s o m o r p h i s m

~:A--~B

(i)

that

limits~is

a set

= 1x

below will

conditions

observe

(iii)

(vi)

U~

for J~U(UA)

in everyday-life

(i)

there

with ~UA

theorem 3.3

satisfying

quite

~UX

~:A--~B We write

The c h a r a c t e r i z a t i o n

concrete

set.

an

).

a category

(i.e. where

( s e e ~5

onto,

satisfying

(vi)

~

then all

;1.~

with

A and

Uf~:X~UA),

all

),

c a n be w r i t t e n

/~ is

On the other

for each object

~:B--~A

a coequalizer UE

are antichains.

as~o£

an isomorphism.

coequalizers

are

with/~

Consequently,

carried

by onto

maps. 1.6.

Convention

if it is a m a x i m a l

meet-irreducible of

algebras

all

if the

: An o b j e c t

element

it

ls

objects

of ~ U A

such in

A

is

said

t o be m a x i m a l

. Similarly,

~UA.

are maximal).

(Note

A

that

(in

~6, U))

is said to be

in categories

259

§R. C o n A r g e n c e s 2.1....~.

congruences

and c r i t i c a l

A ~-equivalence

is

a reflexive

symmetric

transitive

re-

lation £L in

a category

£ :E--~X~X if

U

~

(some may p r e f e r

with

pi ~

preserves

the

ffi 4 i

the more usual

)° I t

is

coequalizer

said

of

representation

t o be a ( J C , U ) - c o n ~ r u e n c e

~1 ~ &2 '

ice,,

i £ we h a v e

for

= coequ(~l•~ 2) U~-(x) 2o2.

Nemarks

theoretical the

other

hand,

tical

equivalence

second

second

(the

In varieties

condition

condition

is

once consider algebras

given

identifying condition)

algebra by a

being with

heredity

that

automatically.

(see

1°5°2),

the

where

category

each

with

elements

three

bob = c, b

x,y

o£

both

implication

a setsuf-

set-theorewhile

with b

conditions

the

c

U

= x ~

See what happens

c

but

preserves

(~,U)-congruences

x = y )

and t h e m u l t i p l i -

(which violates

with

play

groupoids

x.x

a,b,c

= b otherwise.

but not

the

o£ n o n - i d e m p o t e n t

the

a n a when i d e n t i f y i n g

is

not

the

with

limits, always

when

first a (which

the

inter-

a (~,U)-con-

gruence. 2.3.

Convention

(~I,E2):E--~A Then we h a v e for will

(in

categories

isomorphism

be d e n o t e d

and t h e

satisfying

be a c o n g r u e n c e , ~:E---AKA a set-theoretical equivalence

~ = coequ(~l,E2)'A---pA"

Take the

an o b v i o u s •

~:A--~A

fl

with

(i)-(vi))

bijection U~

:

defined by E ' = UE(UE)

Let

Pi E = ~i ° on UA and

f:UA °

~UA/E+.

= £ ° The o b j e c t

A"

by

coequ(~l,~ 2)

~--~Al(el,e 2) .

°

On t h e

as a ~C-equivalence,

satisfying

with

o£ a s y s t e m

fact

as a X-equivalence

satisfied

violates the second condition). 2 . One c h e c k s e a s i l y that i£ section

the

o£ a c a t e g o r y

at

cation

o£ a l g e b r a s

& U£2(e)=y

essential,

an e x a m p l e

the

U£1(e)=x

can be represented

binary

and take

~e~UE,

c a n be r e p r e s e n t e d

in categories

To g i v e a role

: 1.

equivalence

fices,

if£

= U~ ( y )

A / ( ~1 • ¢2 ) will be, as a rule,

represented

as

~I:

260

R64. T =~*~

Lemma : L e t with /~

U

preserve

(I)

(~1,~2)

a quotient.

= difKer ~

, let

Then

= coequ(U£1,U~ 2) ,

U~,

(2) Proof: Since

~ = coequ(£1,~ 2) o U preserves limits, we have U~ 1

UP

up to isomorphism,

the pullback

~ UA

U?

UA and hence,

limits,

a m o n o m o r p h i s m and ~

~ UB

UP = ~ ( x , y ) e U A x U A

: UT(x)

= U~(y)~

=

= [ ( x , y ) e UAxUA : U~(x) = U~(y)~ and U~i(Xl,X 2) = x i . Thus, (I) holds. Now, let o~.&1 = oc-E 2 °Then U~-U& 1 = U~--U~ 2 and hence there is an £ with Uoc = foU~ , Since ~ is a quotientj f = U ~ for a satisfying

~

=~-J+

2.5_.._~_. C o r o l l a r y -

: In a concrete

each difference

a coequalizer

A (~pU)-congruence (i.e.,

£1 ~ £2)

still 2o7o with

: ~ore

a stronger

It

is

(if

on

losely

is)

but

(a)

(vi)

if

kernel~ D

it

is

implies

(tel,t£2):E--~B

non-

: a critical

congruence

whenever applicable

In particular,

is

B/(LEI,L£ 2) .

A/(~l,62)~

speaking

on a n o b -

an e q u i v a l e n c e

stronger

(t&I,L6R):E--~B

B is

is

which

a critical

still

a congru-

.

irreduc, ibillty and c o n g r u e n c e s satisfying

on an o b j e c t

There exists =~i-p)

difference

Ucoequ(t~l~L£ 2) = Ucoequ(~l,~2)

Lemma : I n a c a t e g o r y

two s t a t e m e n t s

and

quotients,

couple

on n o s t r i c t l y that

if£

§3.Subdirect 3.1.

the

quotient

worth realizing

(~i,~2)

the

o£ i t s

t o be c r i t i c a l

one h a s

structure. A

with

said

c :A ~ B

a congruence,

Remarks

(i)

and i £

a stronger-structured

is a congruence congruence.

ing

is

satisfying

a congruence,

coincide

is

whenever for

ence

category

is

- the coequalizers

2.6.

ject

kernel

- each quotient

trivial

yields

(2)follows.

and

a system ~ffi ~

A

are

(i)

and

equivalent

(/~i:A--~Bi)

(vi)

the

follow-

:

such that

b u t n o n e o£ t h e / ~ i

is

(V i ~ = a mono-

261

(b}

morphism. There exists

a system

A

i~intersection

such that

o£ n o n - t r i v i a l is

congruences

trivial.

Pro~£ : Given the system {~i ) consider the system £ e r e n c e k e r n e l s o£ t h e ~ i . On t h e o t h e r hand~ g i v e n t h e congruences, consider their coequalizers. a m a t t e r o£ an e a s y c o u n t i n g . 3.2, Lemma : I n a c a t e g o r y s a t i s £ y i n g two s t a t e m e n t s are equivalent £or A with Ca)

For every a ~:A~B

on

Checking

the

o£ t h e d i f s y s t e m o£

properties

is

( i ) and ( v i ) t h e £ o l l o w i n g £inite UA :

~-'A--~A* with cardUA'<~cardUA there ~01o t. = ~ and a ~ 'I B - - ~ A a s u c h t h a t .

is

(b) T h e r e i s no c r i t i c a l c o n g r u e n c e on a . P_r_oo_£ : I . L e t ( a ) h o l d and l e t (£1,£2):E--~A be a n o n - t r i v i a l congruence. Consider ~ = c o e q u ( E l , £ 2 ) - A - - - ~ A , We h a v e cardUA'~ cardUA and h e n c e t h e r e i s a L:A~B and a ~ , ' : B - - ~ A * s u c h t h a t Let ~-c-£ 1 = ~o't'~2 £ o r some T : B - - ~ C . Then t h e r e i s a ~ : A ~ - ~ -~C s u c h t h a t ~$r= ~ c o S i n c e , h o w e v e r , ~ = ~ ' c and t i s an e p i m o r p h i s m , we o b t a i n ~'= ~ . Thus, ~'= coequitel,t~ 2) . Consequentl y , ( L E 1 ) t ~ 2} i s a c o n g r u e n c e on B (we h a v e US-'= U~ and U ( t e i ) = U~ i ) and we h a v e B / ( t ~ l , t ~ 2 ) = A*= A / ( ~ I , £ 2 ) so that i~1)£2 ) is not critical. I I . L e t (b) h o l d and l e t ~ - ' A - - ~ A " be s u c h t h a t cardUA~cardUA. Consider a decomposition ~ =~-ff with ~ a quotient and ]t~ a m o n o m o r p h i s m . Then ~r = c o e q u ( E l , £ 2 ) k e r n e l o£ ~r ( r e c a l l 2 . 5 ) . The c o n g r u e n c e tical, there is a B and B / ( ~ I ) ¢ £ B "=~B/(t~-I,tE 2) 3.3.

(£1,£~) L'A~B

where

(£1,£2}

is

the di££erence

is non-trivial. S i n c e i t c a n n o t be c r i such that (t£1st~ 2) i s a c o n g r u e n c e on

2 ) ffi A / ( £ 1 , £ 2) . T h u s , f o r 2r' = c o e q u ( L £ l , t ~ 2 } : we h a v e ~'°ot= ~ ) h e n c e , ¢~ = / ~ ' a ' = ~ - ' ) - c , [1

Theorem : Let the concrete

conditions (i)-(vi). ducible i££

Then a £ i n i t e

category object

A

(~jU) is

satis£y

subdirectly

irre-

either

A i s m a x i m a l and t h e i n t e r s e c t i o n o£ a s y s t e m non-trivial congruences is non-trivial)

or

b is non-maximal meet-irreducible critical c o n g r u e n c e on A o

~£9~

: By ~ ;

Thm 3 . 3 ~

, A

is

subdirectly

and t h e r e irreducible

the

o£

is

no

if£

262

e i t h e r i t i s m a x i m a l and e a c h s y s t e ~ (~i;A-~Bi) such that (Vi ~i ~ = ~i~ ~ ~ =~) contains a ~onomorphic element, or it is non-maximal meet-irreducible and tile s t a t e m e n t (a) £rom 3 , 2 h o l d s t r u e . Thus~ t h e s t a t e m e n t o£ t h e t h e o r e m £ o l l o w s £rom 3 , 1 and 342o

|{ e £ e r e n c e s

[ [2] [3] [42

C5]

~6] ~7]

1

]

G,Birkhoff, Lattice Theory (A~ Colloquium Publications Vo1.25, P r o v i d e n c e , R I , 1967) N.Bourbaki, Th6orie des Ensembles, Ch.IV Structures (Hermann, P a r i s , 1953) ~.liu~ek, S-categories, CoJ~ent.~ath.Univ.Carolinae 5 (1964), 37-46 A o P u l t r , On p r o d u c t i v e c l a s s e s o f g r a p h s d e t e r m i n e d by p r o h i biting given subgraphs, Coll.~ath.Soc.Janos Bolyai,18, Combinatorics (Keszthely 1976), 805-820 A , P u l t r and J . V i n a r e k , P r o d u c t i v e c l a s s e s and s u b d i r e c t i r r e d u cibility, in particular £ o r g r a p h s , D i s c r o M a t h . 2 0 (1977)~ 159-176 W . T h o l e n , B i r k h o £ £ ' s Theorem £ o r C a t e g o r i e s , S e m i n a r b e r i c h t e Nr,8, Fernuniversit~t Ha~en ( 1 9 8 1 ) , 1 5 3 - 1 5 9 J o V i n a r e k ~ R e m a r k s on s u b d i r e c t r e p r e s e n t a t i o n s in categories~ Comment.~ath.Univ.Carolinae 19 ( 1 9 7 8 ) , 6 3 - 7 0

A L G E B R A I C C A T E G O R I E S OF T O P O L O G I C A L SPACES

Guenther Richter

Abstract It is well known that every epireflective, full s u b c a t e g o r y of the c a t e g o r y Comp 2 of compact Hausdorff spaces is algebraic in the sense of H E R R L I C H [6,§32]. Conversely, every algebraic, epireflective, full subcategory of the c a t e g o r y of all Hausdorff spaces is c o n t a i n e d in Comp2. This g e n e r a l i z e s a result of H E R R L I C H and S T R E C K E R [5] and yields a complete new p r o o f for it. The lattice of such algebraic categories is v e r y large. For a r b i t r a r y full subcategories ~ of topological (not n e c e s s a r y Hausdorff) spaces the following holds: If C is algebraic, closed-hereditary, and c o n t a i n s the ordinal spaces [0,6] for every limit ordinal B then each space in C is c o m p a c t (not n e c e s s a r y Hausdorff). AMS Subj. Class.

(1980): P r i m a r y 18B30,

18CLO, 54D30; Secondary 18A40

O. Introduction

In [5] H E R R L I C H and S T R E C K E R p r o v e d the following nice algebraic c h a r a c t e r i z a t i o n of the c a t e g o r y ComP2 of c o m p a c t Hausdorff spaces: 0. I T h e o r e m If C is a full, isomorphism-closed,

e p i r e f l e c t i v e and nontrivial s u b c a t e g o r y of the

c a t e g o r y ToP2 of all H a u s d o r f f spaces then the f o l l o w i n g c o n d i t i o n s are equivalent: (i) C is v a r i e t a l LINTON

(i.e. the forgetful functor U : C - - ~ Set is v a r i e t a l in the sense of

[8])

(2) C = CompQ "Nontrivial" m e a n s that there is a space in C c o n t a i n i n g at least two points. most d i f f i c u l t p a r t of the p r o o f "(i) ~

The

(2)" is to show that every space in ~ is

compact: First one o b t a i n s that every space X in C is normal b e c a u s e nonempty, d i s j o i n b closed subsets A , B c X become d i s t i n c t points in the q u o t i e n t space X/R w h e r e R :=

(A× A) u ( B × B )

u { (x,x) ~ x ~ X }

is a closed e q u i v a l e n c e r e l a t i o n on X. By assump-

2S4

tion

(i), the space X/R belongs to C, e s p e c i a l l y it is Hausdorff. A and B have dis-

joint n e i g h b o r h o o d s in X/R, hence in X.

Since C is productive, result of N O B L E

[ 9],

each power X

I

of a C - o b j e c t X is normal. This implies, by a

that each C--object is compact.

Essential in this proof is that C is closed under the f o r m a t i o n of c e r t a i n quotients X/R w h i c h are, in addition,

Hausdorff spaces. This enables one to apply the result of

NOBLE. Nevertheless, there is a g e n e r a l i z a t i o n of O.i for w h i c h quotients, axioms, and the result of NOBLE are not needed.

separation

Instead of v a r i e t a l categories, we

consider the weaker concept of a l g e b r a i c c a t e g o r i e s in the sense of H E R R L I C H

[6,§3~:

0.2 D e f i n i t i o n A c a t e g o r y ~ is called algebraic w i t h respect to a functor U : C - - g S e t p r o v i d e d that has coequalizers,

U has a left adjoint and p r e s e r v e s and r e f l e c t s regular epimor-

phisms. Each algebraic c a t e g o r y

(C,U) is u n i q u e l y

(regular epi, m o n o ) - f a c t o r i z a b l e and U

p r e s e r v e s and reflects these factorizations.

Especially,

U reflects isomorphisms.

Moreover, U is faithful and C is c o m p l e t e and cocomplete. Note, that the existence of coequalizers

in C is a

closedness under the formation of c o e q u a l i z e r s Top2 in the proof above. Up to equivalence, variety,

much weaker condition than the

(quotients)

in a bigger category like

an algebraic c a t e g o r y C is a quasi-

i.e. a full s u b c a t e g o r y of universal algebras of a certain t y p e ~

(not

n e c e s s a r y w i t h rank) w h i c h is closed under the formation of subalgebras and p r o d u c t s in the c a t e g o r y Alg-~q

of all a l g e b r a s of type ~& and all ~ - h o m o m o r p h i s m s b e t w e e n

them. On the other hand, a varietal category ~ c o r r e s p o n d s to a v a r i e t y w h i c h is, in addition, closed under the formation of quotients

[2,11].

i. Results

In the sequel, C always denotes a full,

is, m o r p h i s m - c l o s e d subcategory of the cate-

gory Top of all t o p o l o g i c a l spaces and all continuous maps and U : C - - ~ S e t d e n o t e s the forgetful functor.

If

(C,U) is algebraic then every b i j e c t i v e m a p in C is a homeo-

morphism. Let B be an ordinal and [0,6]

:=

{a I a <_5, a o r d i n a l }

the usual ordinal spaces together w i t h the order

, [O,~[ := {a topology.

I a < ~, a ordinal}

265

I.I Lemma If every b i j e c t i v e map in C is a homeomorphism, logy on

[0,6] ~ C

Proof:

~

([O,B[,T)

~ C

The map h: [ O , 6 [ - ~ [O,B] d e f i n e d by h(O) = B, h(n) = n - i for n ( Eq , n ~ I,

and h(a) = ~, otherwise,

is continuous and b i j e c t i v e but not a homeomorphism,

hi {O}1 = {6 } is not a~ open subset in [O,~] in

6 is a limit ordinal, and T a topo-

[O,B[ w h i c h is finer than or equal to the order topology, then

because

(for a limit ordinal 6) but { O }

is open

([O,B[, T).

1.2 P r o p o s i t i o n If C is closed-hereditary,

(surjective,injective)-factorizable,

and has finite pro-

ducts which are p r e s e r v e d by U, and if C contains the two element d i s c r e t e space D2

= {O,I} then ~ is closed under the formation of finite eoproduets

Proof: By assumption,

the t o p o l o g y of finite p r o d u c t s in ~ is finer than the usual

p r o d u c t topology, because the natural p r o j e c t i o n s are still

continuous.

then in the case of XIT C{O,I} the two injections of X, x~-~(x,O), tinuous and, therefore, ~ h e t o p o l o g y on X'D-C~,I} on

(sums) in Top.

X x {O,i} = X 0 X

If X is in C_,

x~-~(x,l),

are con-

is coarser than the sum t o p o l o g y

w h i c h coincides w i t h the p r o d u c t t o p o l o g y in Top. This means

x o x = x × {o,:} : x ~

{o,:} (~_

Now, consider an element x

of X and the c o n s t a n t m a p c:X--~X, x ~ - ~ x . This defines o o a m a p id X 0 c in C w h i c h admits a (surjective, i n j e c t i v e ) - f a c t o r i z a t i o n

idxOc XOX

) XSX

x~{x

o}

The usual coproduct t o p o l o g y is the only topology on X 0 {Xo}

such that s and i be-

come continuous. Hence, the sum X 0 { x ° } b e l o n g s to C . If we have two n o n e m p t y spaces X,Y

( C, x --

o

( X, Yo

( Y' then S := (X x{y~)O ( ~ x

is a d i s j o i n t union of two closed subsets in P := (X 0 {Xo}) ~ c ( Y these

subsets are already closed in the coarser o r d i n a r y p r o d u c t topology.

assumption,

Y)

O {yo}), because Thus, by

S together w i t h the induced t o p o l o g y b e l o n g s to C and, moreover,

it is

266

the t o p o l o g i c a l sum of its subspaces X x { Yo } and {x ° } x Y w h i c h are isomorphic to X and Y, resp. 1.3 Lemma If ~

is c l o s e d - h e r e d i t a r y and

Proof: The closure ~[I]

morphism s:FI-'~,~[I]

~:I--~UFI

of ~[I]

~[I]

in FI belongs to C. Therefore,

such that i !

U - u n i v e r s a l then

is dense in FI.

there is a unique C--

the diagram ) ~(i)

u~cl]

;

,1

US

UFI commutes.

If j:~[I] --$FI denotes the injection we have U ( j s ) ~ = U j U s ~ = Uj~' = ~ = U(id F I ) ~

This implies

js = id FI, hence

j is surjective.

1.4 T h e o r e m If C c o n t a i n s the ordinal spaces

[O,B] for all limit o r d i n a l s B, is c l o s e d - h e r e d i -

tary, and algebraic w i t h respect to the forgetful functor U : C - - ~ S e t then every space in C is compact.

Proof: For a topological space X the following conditions are equivalent: (I) X is compact

(not n e c e s s a r y Hausdorff)

(2) Every d e c r e a s i n g family

(Aa)a~ [o,~[ of n o n e m p t y closed subsets A a ~ X, ~ a

limit ordinal, has a n o n e m p t y intersection, because,

it is well known

[i] that

(i) is e q u i v a l e n t to

(3) E v e r y w e l l - o r d e r e d d e c r e a s i n g family

(Ai)i~ I of closed n o n e m p t y subsets of X

has a n o n e m p t y intersection. Now, every w e l l - o r d e r e d set

(I,~) is o r d e r i s o m o r p h i c to some

[O,B[. Obviously,

it

suffices to c o n s i d e r limit o r d i n a l s B. Otherwise there is a g r e a t e s t element in [O,B[, hence in (I,~), and a smallest member in the family

Therefore,

it

is enough

to

(Ai)i E I

show that every d e c r e a s i n g family

(Aa)a ~

[O,B[ of

267

nonempty closed subsets of a space X ~ C has a nonempty limit ordinal.

gram, where ~ : [ O , B [ - - ~ U F E O , B [ property of ~

intersection,

where B is a

Now, take an x a ( A a for each a < B and consider the following is U-universal

dia-

and f, g are defined by the universal

: xa *

4

a

ux ~

~

a

;

[O,BE

1~ UEO,B]

UF CO,~ E

We will prove that the intersection (This implies

~ ~ f[ Na f - I [ A a ] ] c-

of the family

Na fCf-iEAa]]

c

(f

-I

[Aa] )a ~ [O,B[ is nonempty.

aq Aa') To do this, we show

a) g is surjective b) g-l(B) c f-l[A ] a Now, by the commutativity

of

then the middle space of the

(2),

for all a < B .

[O,B[ c_ g[FEO,BE] ~_ EO,B].

If EO,B[ = gCF[O,B[],

(regular epi, m o n o ) - f a c t o r i z a t i o n

of g would give us a

topology T on £O,B[ which is finer than or equal to the order topology (CO,B[,T)

belongs to C. This contradicts i.i. Hence g[FEO,BC]=[O,B]

For each a < B the set [O,a]

~ [a+l,B]

[O,a] = [o,a+l [ is clopen in [O,~]. Therefore,

is a topological

By 1.2, we have FEO,B[ = F[O,a] to C), and diagram

such that

and g i s surjective [O,B] =

sum. The left adjoint F of U preserves O FEa+I,B[

coproducts.

(C is closed hereditary, hence D 2 belongs

(2) splits as follows: [O,B[ = [O,a] 0 Ea+l,B[

UF [

O

) U[o,a]

O uEa+l,B]

= UEO,B]

~

This shows g

-i

(g) C_ g

For each y ~ [~+i,~[ we have f-i [A ] ~

This implies f-l[A a] ~

-I

[a+I,B] =

f~(y)

g,,-1

[a+l,~] = F[a+1,g[

= xy ( Ay and, therefore,

f-I [A ] ~ f-I (xy) ¥

~[[a+l'B[]

~ ~(y)

and, by 1.3 and

(+)

(+) .

268

f

-I

[A]

= f

-]

[A ] _D ~;[[~+I,B[7 =

Thus, we have b) which completes

F[~+I,B] ~ g

-I

(B) .

the proof of 1.4.

1.5 Corollary If C ~ T o p 2 is closed-hereditary

and contains the ordinal

ordinals B then the following conditions (i)

spaces

[O,B] for all limit

are equivalent:

(C,U) is algebraic

(2) ~ is an epireflective Proof:

subcategory

of ComP2

"(i) =) (2)": ~ fulfills the assumptions

the inclusion C =-~ComP2 is algebraic ComP2 and,

in addition,

epireflective,

"(2) =; (i)": Every epireflective therefore,

of 1.4, hence ~ ~ ComP2. Moreover,

[6,§323.

Thus, ~ is a reflective

because

subcategory of

it is closed-hereditary.

subcategory of ComP2 is regular-epi-reflective

and,

algebraic.

1.6 Corollary If ~ ~ ToP2 is epireflective (I)

(~,U) is algebraic

(2) C ~ C o m P 2 Proof: tire

are equivalent:

(varietal)

(C = ComP2 or ~ is trivial)

"(i) =)(2)": ~ ~ T o P 2 [4]. Especially,

all Stone spaces, ~ ComP2.

then the following conditions

is epireflective

and producI for nontrivial ~ all closed subspaces of the powers D 2 , i.e.

are contained

Moreover,

iff ~ is c l o s e d - h e r e d i t a r y

in ~, hence all ordinal

s:D 2

)

[O,i],

s(f)

from the Cantor space D9IN to the unit interval if C is varietal[5].

spaces

Every compact Hausdorff

space of a certain power

:=

0o ~ f(i) i=O 2 i+l

[O,i] which,

' therefore,

space X is homeomorphic

belongs

to

to a closed sub-

[O,13 I, hence it belongs to C, too.

A careful analysis of the proofs above yields the following

topological

of 1.6 which is comparable with DE GROOT's famous topological ComP2

[O,B]. Thus, by 1.4,

there is a well known quotient map

formulation

characterization

of

[12]:

1.7 Corollary ComP2 is the greatest full subcategory ~ of ToP2 which satisfies ditions:

the following

con-

269

(i)

C is p r o d u c t i v e

(ii)

C is c l o s e d - h e r e d i t a r y

(iii) C is (surjective, (iv)

injective)-factorizable

Every bijective map in C is a h o m e o m o r p h i s m

.

1.8 Remarks (i) The powers X

I

of every regid compact Hausdorff space X w h i c h c o n t a i n s at least

two points define a varietal s u b c a t e g o r y ~ of Comp2 w h i c h is not closed-hereditary

E6, p. 292, 38E].

(2) The full s u b c a t e g o r y C ~To__p_pof d i s c r e t e does not contain all the ordinal spaces (3) Every s u b c a t e g o r y C ~ T o p

(indiscrete)

spaces is varietal b u t it

[O,B].

w h i c h fulfills the a s s u m p t i o n s of 1.4 contains neither

nontrivial indiscrete nor infinite discrete spaces, b e c a u s e there are nonhomeomorphic continous b i j e c t i o n s b e t w e e n them and c e r t a i n ordinal spaces

[O,B]

(with

the same c a r d i n a l i t y and a limit ordinal B). (4) If, in 1.4, C is not only c l o s e d - h e r e d i t a r y b u t even c o m p a c t - h e r e d i t a r y then every space in C is T I.

2. The lattice of algebraic s u b c a t e g o r i e s of ComP2

The full, epireflective, plete

(large)

lattice

[.

and i s o m o r p h i s m - c l o s e d s u b c a t e g o r i e s ~ of COmP2 form a comComP2 i s

the

greatest

element

of Land

the

full

subcategory

C of all spaces w h i c h c o n t a i n at m o s t one p o i n t is the smallest one. Moreover, we --o have 2.1 P r o p o s i t i o n (i) There is a greatest p r o p e r s u b c a t e g o r y C max ~ C o m p 2 in L . (2) There is a smallest s u b c a t e g o r y C_min ~

Co in L .

(3) There are a r b i t r a r y large d i s c r e t e subsets in i . (4) There are arbitrary large w e l l - o r d e r e d subsets in i .

Proof: (I) The class

{X ~ ComP2 [ f:[O,l] --)X continuous ~ f

closed-hereditary,

hence

w h i c h is not in C -~nax

cogenerates

constan~

is p r o d u c t i v e and

a member C o f L. Now, e v e r y X ~ Comp 2 --max c o n t a i n s a nontrivial image of the unit interval [O,i].

270

Such a space is a cogenerator

in ComP2

(2) The category C min of Stone spaces, to homeomorphisms,

is algebraic

isomorphism-closed,

I D 2 , up

i.e. closed subspaces of the powers

and contained

in every nontrivial, algebraic,

full subcategory of Comps.

(3) The dual, ComP2°P , of ComP2 is almost algebraic there are arbitrary

large families {Xj

in the sense of

EIO]. Especially

I J ( J } of compact Hausdorff

spaces

with at least two points and C(Xj'Xk)

=

{~d

X~

u

{f:Xj--~Xj

~:Xj---gX k The homeomorphic algebraic

implies

Xk ~ C . .

I of X. define an 3 ] ( J }of L is discrete,

of the powers

o f Com.___.2. .~p The s u b s e t { % 1

for a certain I P i : X j ----l, Xj y i e l d s

j = k

I f const. } , j ~ k

images of closed subspaces

subcategory %

b e c a u s e --~C'" ~ _ j

I f const. } ,

Therefore,

there

J

X

i s a c l o s e d embedding

U:Xk~.-.~X j

s e t I and t h e c o m p o s i t i o n o f u w i t h one o f t h e p r o j e e -

tions

a noneonstant

(4) i s a p u r e l a t t i c e - t h e o r e t i c a l

c o n t i n u o u s map ×k----~X. 3

consequence of

(3).

2.2 Remark A famous example of a member of C which is not in C is the so called pseudoarc --max ~in P [3]. An example of a continuum which cogenerates a "very small" member of [ is

given in

[7].

Moreover, for every family

(~i)i

E I of epireflective

nua in ComP2 there is a continuum which cogenerates

hulls

of conti-

a lower bound of the C, in [. --i

References

I.

P. Alexandroff,

P. Urysohn,

Zur Theorie der topologischen

R~ume,

Math.

Ann.

92

(1924) 258 - 266 2.

B. Banaschewski, of Math.

3.

H. Herrlich,

2 (1976)

Subcategories

defined by implications,

Houston J.

149 - 171

R.H. Bing, A homogenous

indecomposable

plane continuum,

Duke Math.

J. 15 (1948)

729 - 741 4.

H. Herrlich, in Math.

78

Topologische (1968)

Reflexionen

und Coreflexionen,

Springer Lecture Notes

271

5.

H. Herrlich,

G.E.

Strecker,

a n d its A p p l i c a t i o n s 6.

H. Herrlich,

7.

M. Hu~ek,

8.

II, Math.

F.E.J°

Linton,

Theory,

Ch. F. Mills,

Some aspects 1965

Products

Category

Tracts

I15

Allyn and Bacon,

Some v e r y

(1979)

General

Topology

Boston

(1973)

small continua, T o p o l o g i c a l

struc-

147 - 151

of e q u a t i o n a l

(1966)

= Compactness,

283 - 287

categories,

Proc.

Conf.

Categorical

II, Trans.

Amer.

Math.

Soc.

84 - 94

with closed projections

160

169 - 183

iO. A. Pultr, Groups,

Centre

La Jolla

N. Noble, (1971)

Strecker,

J. van Mill,

tures

Algebra, 9.

G.E.

Algebra n Topology

i (1971)

V. Trnkov&,

Semigroups

ii. G. Richter,

Combinatorial

and Categories,

Kategorielle

Algebra,

algebraic

and t o p o l o g i c a l

Noth-Holland Studien

Math.

zur A l g e b r a

Lib.

Representations

22

(1980)

und ihre A n w e n d u n g e n

3

(1979) 12. E. Wattel, Tracts

21

The c o m p a c t n e s s (1968)

Guenther Richter U n i v e r s i t y of B i e l e f e l d F a c u l t y of M a t h e m a t i c s U n i v e r s i t ~ t s s t r a B e 25 D-48OO Bielefeld

1

operator

in set theory and topology,

Math.

Centre

of

EXTENSIONS OF A THEOREM OV P.GABRIEL

Tiberiu Spircu

The theorem of Gabriel about the renresentation type of a oramh now well-known.

This theorem,

(see [6]) is

first appeared in 1972, is the source for many results

in graph representation theory_. Several extensions have been made, esmecially by Dlab and Ringel

(see [J],[4],[5]), Loupias

([ii]),

Zavadsky and Shkabara

([14]). We

cbtain a common extension of these results. i. Preliminaries Let F,G be two division rings, such that F c G and

dim G F = dim F G . Let W

be a G-vector space and let U be an additive subcmoun contained in W F-subspace). Following Diab and Ringel

(eventually an

([4]), denote U the largest G-subsnace of W

contained in U, and ~ the smallest G-subspace of W containing U. The case [ l,y }

dim ~

= dim F G = 2

will be of major interest in this namer. Let

be a basis of F G (obviously it is also a basis of ~ ) .

canonical isomorphism of F-vector spaces G

~ F

given by

Lenma

G

~- F 2

and by

Denote by r

j

the

the F-linear man

r(1) : 0, r(y) = 1 . We need the followina:

(see [4], lemmm 2.4) . Let

F c G

be division rings such that

dim C~ =

= dim F G : 2. Let U be an F-subspaee of a finitely dimensional C~-vector space W, such that

U = w

v I ..... v n

is a basis of V, then there exists a basis of W

and

U = 0 . Let V be a C~subspace of W, such that

...,z m , contained in U, such that Our definition of a Gabriel definitions I. Let

v i = x i + yiy

Uf%V = 0 . If

x I ..... xn,Y 1 ..... yn,Zl,

for 1 ~< i ~< n .

species will be somewhat different from other

(see [4],[53,[7]); this definition is presented in three staqes. (G,F) be a (finite)

graph, G bein~ the set of vertioes and F the set

of arrows; each arrow y has a source s(v) and a sink a(y). Denote by G the vector space over GF(2) - the field with two elex~nts - having mapping

8:G

~G

the property that ned by d-cyclic

GUF

as a basis. The

given by 6(x) = 0 for x~G and 8(y) = s(y)+a(y) 68 = 0 . Denote by

rk(G,F)

rk(G,F) = dimGF(2 ) (Ker 8 ) - card(G) if it has the following fore :

for y~F has

the cyclic rank of the ~raph, defi. We say that a subqramh of

(G,F) is

273

y{ :1,- . ~ .

X{

~-

X I

X ~ o

n ~

(1)

xl .

Yl

x I

with

n

~

...

~

x"

m

m,n ~ 1 . The maximum number of d-cyclic subqraphs, independent over C/F(2), is called

the d-cyclic rank of (G,F); obviously, it is less than the cyclic rank (in Ker 6 may occur cyclic subgraphs which are not d-cyclic). We say that a graph is perfect if it has the d--cyclic rank equal to the cyclic rank. II. By [3], a valuation over

(G,F) is obtained if every vertex x has a

height p(x) # 0 and every arrow y has two capacities q(y),q(y) such that the natural ntm~0ers p(x), xcG, are c ~ r i m e , •q(y)

. A valuation

cne has

p , q

and for every arrow y, p(a(y))a(y) = P(s(Y))-

is called perfect if for every d-cyclic subQTaph

q (yn)...q (yo) = q (ym)...q (yo) . Obviously, the trivial valuation

(i)

(having

all capacities i) is perfect. III. Let F be a division rinq. Also by F3], an F-modulation on a valued graph (G,F,p,q) is: a) a set [Fx]xsG of division rings, such that

d ~ F F x = o(x) ;

b) a set [M~y~F of abelian groqms, such that every My has a structure of special Fs (y)-Fa(y)-bimodule, and

d~a(y

) My = q(y)

. (An ~u-Fv-bim°dule M is called spe-

cial if the two duals HornF (M,Fu) and HornF (M,Fv) are ison~rphic as Fv-Fu-bimodules. ) u v An F-modulation is called admissible if, for every cyclic subqraoh can choose in a coherent manner an isomorphism

i

between the F

-F xo

My; x{My i ... eFx, M, and Yn

Myo ~ x ~

(i), we

-bimodules x1

My ..... i

Fx,, n m Definition i. Let F be a division rinq. A Gabriel F - s ~ c i e s

consists of a

perfect graph with a perfect valuation, and an admissible F-modulation on it. Let (G,F) be a graph. We say that we suppress the vertex xsG if we eliminate x from G and all arrows which are adjacent to x from F. We say that we reduce the arrow yEF if we identify s(y) with a(y) and we eliminate y from F. For a valued graph

(G,F,p,q) , the reduction of an arrow y ir~91ies modifica-

tions in heights and in capacities : the new height of s (y) is max (D (s (y)) ,p (s (y)) ) ; the capacities of all arrows which are adjacent to s (V) or to a (V) must be modified coherently. For example, if in (G,F) we ha\~ the subgraph x -and

if

~(z)q(y)

p(s(y))~< p(a(y)) , instead of

z

S(y)

y - ' ~

a(y)

, then the new capacities of the arrow z are c(z)q(y),

q(z),q(z)

.

274

Given an admissible modulation

[Fx] , {fly} on the valued gramh

(G,F,D,g),

after reducing the arrow y, this modulation must be modified coherently;

for e-

xar~ple, in the case above, the new b-module correspondin a to the arrow z is FmmF a (y) (My'Fa (y)) (~ Mz Fs (y) These operations do not alter the eo~ality between the cyclic rank and the d-cyclic rank. Thus we give the following: Definition 2. Let S ans S' be two Gabriel F-species. We say that S' is subordinate to S if S' is obtained from S after a finite number of suppressions of an extremal vertex and/or reductions of an arrow, followed by coherent modifications in valuation and in modulation. Let S be a Gabriel F-species, having as a support the perfect valued graph (G,F,p,q). let

r = rk(G,F)

be his cyclic rank. Choose a basis of Ker 6 , contai-

ning all vertices x and the d-cyclic subqraphs FI,... ,F r of the form (i), and let Xos (resp. Xls) be the initial also

t

r

qs = q (Yns)'" "q (Yos)

(resp. final) vertex of F s , for 1 ~< s <, r. Denote

for this subgramh F s

Definition 3. The Tits form of the species S (see [1],[8]) is the following quadratic form in variables Xx, xcG : T s(X) = x ~ e p(x)X x - ~ a ( y ) n ( a ( y ) ) X

, ,X , ,

The Brenner form of the species S (as sketched in [2]) is the followinq quadratic form : Bs(X) = Ts(X ) + s~=rl qsP(Xls)Xx0sXXls Proposition i. a) If the species S' is obtained from S by suppressing the vertex x, then TS, resp. BS,

is obtained from T S res D. B S by identifying X x = 0 .

b) If S' is obtained frc~ S by reducing an arrow y, then TS, resm. BS, is obtained from T S resp. B S by identifying

Xa(y ) = g(y)Xs(y ) .

The proof is obvious. We say that a Gabriel F-species S is excellent if its Brenner quadratic form B S has the following property: (E)

B S(X) > 0 Corollar~.

for any vector

X >i0 , X # 0 .

If S is an excellent Gabriel ~-species and S' is a subordinate of

S, then S' is an excellent Gabriel F-species. For

exanple, the Brenner quadratic form of the species 0 5 :

i~4~5 (where ====~ denotes an arrow with capacities 2,1 and both capacities i) is the following

:

~

denotes an arrow with

275

2

2

2x21 + 2 x 2 + 2x~ + x 4 + x 5 - 2xlx 2

--

2x2x3

2x3x5- 2xlx 4 - x 4 x 5 + 2xlx5 .

- -

Writing this quadratic form as a s~n of squares :

¼(2X1

2X2 + X5)2 + I(2XI

2X 4 + X5)2 + ~(X 2

2X3 + X5)2 + ix2

, it is obvi-

ous that 0 5 is excellent. 2. Representations of F-species Let F be a division ring and let S be a Cxabriel F-species. A representation

(V) of the species consists of :

a) a vector space Vx, such that

dimF

V x = dx < ~

, for every xeG ;

X

b) an Fa (y)-linear map

%

: Vs(y ) ~ M ~ F s (y)Y

Va(y ) , for every arrow yeF .

A morphism (a) from the representation (V) to the representation (W) is simply a family [ ~ x~G ' ~x being an Fx-linear mad V x ycF,

~ a ( y ) % = Wy(~s(y)~l) Denote

~(S)

~ Wx, such that for every

"

the abelian category of all representations of the Gabriel

F-species S . Remember that, given a species S, for every d-cyclic subgraDh of the form (I) there exists an isomorphism

i : M y ~ . . . ~ M y . ~~ ~

My0.,,~...~MVm "

Definition 4. A representation (V) is called commutative if for every d-cyclic subgraph of the type (I) we have : Vyn,O ( % n _ l ~ l ) ~

... o (v~vu' ~ l ~ ' ' ' ~ l )

= W..O(VmYm-l"~ l ) a

.... (%~i~...~I)6

o(l ®i) , where

i

is the iscrnorphism for this subqraph, chosen in the definition of S .

Denote by

~

(S)

the full subcategory of

~

(S)

whose objects are the

commutative representations. Then, obviously, R_~ (S) is an abelian cateqory and every object (having a dimension) d e ~ s e s

as a finite direct sum of indecc~po-

sable objects. This decomposition is unique. 3. Commutative representations of the species 0 5 The F-species 0 5 is based on the following valued graph (the heights are indicated above each vertex) : If F is contained in another division ring G, such that

dim F G = dim C~F = 2

2

2/2 1

2

----~3~.. 1 ~ 5

and G is a special

G-F-bimodule, then the admissible modulation of the species is the following : A cc~mutativ~ representation (V) of this species is ognpletely described by the following ccmanutative diagram of F-vector spaces :

G

G ,G

276

D

~here ~

is the F - v e c t o r

space structure o b t a i n e d

ture o f X b y restriction of scalars; If E = 0, then thermore,

from the G - v e c t o r

the maps a F = a and b F = b are C~linear.

(V) is in fact a representation

if A = 0, then

(V) is a representation

of the valued graph C 4 ; fur-

of the v a l u e d qraph F 4 . These

trivial cases are e l u c i d a t e d by Dlab and Ringel in [4]: there exists indeconposable

co~utative

Proposition

representations

deccmposable

commutative

Then the F-species

representation

F

F2

G

w h e r e i and r are the canonical maps

are, r e s p e c t i v e l y , (i,i,i,2,1)

(from section j (i) =

rV

i), ~ is the p r o j e c t i o n on

(l,y). The dimensions o f these re-

:

, (1,1,1,2,2)

, (1,1,2,2,2)

and

W e sketch the proof, w h i c h contains several stages; deccr~oose

(V) as a direct stun (V')(~(V")

Ix)sable and c o ~ u t a t i v e . -

-

-

-

Ker (d) (] Ker (ba) : 0 ; = 0 ;

Ker (d) (] Ker (a) = 0 ;

- Ker(a) ~ -

-

-

-

Ker(d)

Ker(ba) C

Ker(d)

a(Ker(d))

= 0 ;

Ker(ba)

;

= 0 ; Ira(b) ;

- Ker(c) C

I_m(ba)

-

;

Ker (ba) r~ Ker (d) = 0 ;

- Ker(c) C

Ker(c)

= 0 ;

(1,2,2,2,2)

.

at each sta~e w e try to

, (V) b e i n g supposed nontrivial,

indeccm-

In this m a m m e r we obtain the followin 9 conclusions

Ker (d) ~3 Ker (a) = 0 ;

Ker(d)

F2

r~

the second co _r~ponent and j is given by

(i,i,i,i,i)

(that is E # 0 and A # 0) i n d e x -

F2

G

presentations

= 2

is of one of the followin~ five types:

F

G

dim F G = dim ~

05 has exactly 41 types of in-

representations.

Proof. W e e s t a b l i s h that any nontrivial sable c ~ t a t i v e

36 types of

w h i c h are trivial.

2. Let F c G be division rinqs such that

and G is a special G-F-bimodule.

space struc-

;

:

277

- Ker(b)

= 0 ;

-

Ker(e) ~

-

E

=

Im(d)

~(e)

;

.

A t this stage, denote b y K the intersection i) First, Denote

X = ba(Ker(d));

in Ker(c).

Then

then, obviously,

Ker(c)

= X•Y

and

XC

= a ( ~ )

, S" = bl(c"),

qhen

= E'

, e(D")

where A' = Ker(d')

Supposing that i-i)

(V) =

A' = K e r ( - ~

= E"

and Ker(d")

Ker(c).

andwe

.

Im(ba)

and D = Ira(d).

Cheese a cc~plement Y of X Denote C' = X and let C"

. Denote E' = c(C'), E" = c(C"), B' =

, A" = al(B"), decompose

D' = d(A'),

D" = d(A")

(V) as a direct s u m

(V')~(V"),

= 0 .

(V) is indecomposable,

(V')

Ker(c) ~

X = ba(~----~).

he a complement o f C' in C, c o n t a i n i n g Y

e(D')

Ker(c)~Im(ba)

suppose K # 0 ; then it follows

, that is, ~

that, in this case, the r e p r e s e n t a t i o n G

I /

w e n e e d to consider two possibilities.

= A and ~

= C . It follows e a s i l y

(V) has the following tvoe

:

1 ---------- G

"~'~

G /

F

-'>-r --.. F 1-2)

(V) =

(V") , that is Ker(d)

let C" be a c ~ l e m e n t

= 0. In this case, denote C' = ~

of C' in C. Denote A' =

(ba)-l(c ') , B' = a(A'), D' = d(A')

E' = c(C') a n d a n a l o g o u s l y A", . . . .

It follows that

s~n

and

(VI)~(VI) f

"

,

with

C'

=

~

Ker(c")

A g a i n w e n e e d to consider two possibilities 1-2-1) r~nsion

(V) =

(V{)

, that is

Ker(c)

and

=

(V) is d e , r e p o s e d as a direct 0 .

:

= C. In this case

(V) has the type o f di-

(1,1,1,2,1).

1-2-2)

(V) =

isc~orphisms.

(V~') , that is Ker(c)

The representation

2) Suppose n o w

= 0. In this case a,b,c,d and e are all

is of the type o f dimension

K = 0 ; then, obviously,

Ker(d)

= 0 and, because Ker(e)

contained in Ira(d), it follows inmediately that also Ker(e) In the next stages w e obtain -

Im(ba)

-

C

=

~

Ker--6~7~7

- dR

(1,1,1,2,2). is

= 0 and D = Im(d).

:

; ;

C = 2dim G Im(ba)

;

- dimG A = 1 . There are two possibilities, Im(ba).

debendind on the coincidence

It follows n o w easily that these two p o s s i b i l i t i e s

of dimensions

(1,1,2,2,2)

4. Extensions

and

(1,2,2,2,2).

The p r o p o s i t i o n

o f ImCo) w i t h

c o r r e s p o n d to the types is proved.

of a t h e o r e m o f Gabriel

The result of Gabriel about the representation

type o f a graph is well-known:

278

Theorem 1 - .positive form (Gabriel,[6]). Let (G,F) be a graph and F a division ring. The category

Re_F(G,F)

is of finite representation type (that is, it

has only a finite number of types of i n d e ~ s a b l e if and only if

i) A n (n >i i) List 1

objects), independently on F,

every connected subgraph of (G,F) is contained in the list i : 1--2--...--n 21 ~ 3 - - . . . - - n

2) D n (n >i 4)

; ;

3) E n (6~n,<8)

i--2--4--...--n 3/ (with arbitrary orientation for each edge). Theorem 1 - negative form. Let (G,F) be a graph and F a field. The category Be_F(G,F) is of infinite representation type (independently of F) if and only if the graph (G,F) has as subordinate a graph from the list i' below :

List i'

;

° -/- -°- o~/ O

I) A1

o~

2) A4

4) Z 6

o - - o - - o C . ...-o-_O_~ O °

6) E8

O--°---°--O~

;

O ~

o o--o o--o o

5) E7

O

3) D5

;

O o J~ ~ O ~ o / O /~

o --o

; ;

o

(again, arbitrary orientation for each edge). This theorem, first appeared in 1972,

is the source for many results in

graph representation theory. The beatiful proof oiven by Bernstein, Gel'fand and Ponemarjow ([i]) permitted one extension of this theorem, oiven by Dlab and Rin9el ([4],[5]). In our notations, their theorem, in positive and negative form, is the following: Theorem 2 - positive form (Dlab-Rinqel,[5]). Let F be a division rinm and let S be a Gabriel F-species. The category ~ p (S) is of finite representation type (independently on F and on the modulation) if and only if every connected valued subTraph of (G,F,p,q) is contained in the list 1 or in the list 2 :

List 2

i) B n (n >i 2)

1

(1,2)

2 --...--n

;

2) C n (n ~ 3)

1

(2,1)

2 --...--n

;

3) F 4

1--2

(1,2)

3--4

;

4) G 2

1

(1,3)__ 2

(with arbitrary orientations for each edge, a single edqe having both capacities ~ity). Theorem 2 - ne.gative form. Let S be a Gabriel F-snecies based on the valued graph (G,F,p,q). The category Re~(S) is of infinite reDresentation type (indemenc~ntly on F and on the modulation) if and only if S has as subordinate a species from the list i' or from the list 2' below : --O-. o.'=-f..o_l:~o

;

3)

o

(x,y)

4)

o--o

i)

List 2'

(u,v)

o

Oo--" ....O

2) o

6)

o

O

with uv = 2 ;

with uv = xy = 2 ;

(u,v) o--o---o

with uv = 3 ;

(u,v)

(u,v)

with uv = 2 ; o

with u v ~

5) 4

o

(u,v_____~) o---o

279

(with arbitrary orientation for each edge, a simmle edge having both capacities i, a dotted edge having arbitrary capacities). The next extension of the theorem of Gabriel was given, in negative form, by Loupias

([ii]) and, in positive form, by Zavadsky and Shkabara

([14]). In our

notations, their theorem, in positive and negative form, is the following : Theorem 3 - negative form (Loupias,[ll]). Let (G,£) be a qramh and F a division ring. The category RCF(G,F) is of infinite representation type (that is, it has an infinite number of indeconposable cc~nutative representations) if and only if the graph (G,F) has as subordinate a graph from the list i' (except A4 ) or from the list 3' : O--O

I) ~4

3) R{

°"~°~"°

,-~ o ,,L..--

oo- o -

o1°"--_%°

~J

o

,

8)

/o

I0) R 6

o--o--o--o

;

5) R~ = dual of R 2 ;

o ~ o__ o/'°~".o~o__o ; 7) R 4 o ~'o/ o/ ~'o--o--o~o---o ; 9) R~ =

6) R 3

~1) i~

;

; ~k.o 7 = dual of R 1 (obtained by reversing the orientation of arrows)

4) P~

List 3'

2) R 1 o/" ~ ' o - - o - - o - - o - - o

7 ° ~'o~o ~o J

o~0~0---o~o

;

dual of R 5 ;

o

/ " o#~ o

\ o/

f~. o

; 12) R 7

o--o--o

= dual of R6 ;

O--O \

~o o/

(with arbitrary orientation for each edge). Theorem 3 - positive form (Zavadsky-Shkabara,[14]). Let (G,£) be a qraph and F a division ring. The category _~_F(G,F) is of finite representation tyne (and independently of F) if and only if every connected suboranh of (G,F) is contained in the list 1 or in the list 3. List 3 is too big to be presented here; it can be found at the end of Zavadsky-Shkabara's preprint [14]. This list contains graphs havinq the cyclic rank at most 4 . The theorems of Dlab-Ringel and of LeuDias-Zavadsky-Shkabara admit a conmon extension, which is also an extension of the theorem of Gabriel: Theorem 4 - negative form. Let S be a C~nbriel F-species, F being an infinite field. The category R c F (S) is of infinite representation type

(independently

cn F and on the modulation) J f and only if S has as subordinate a sbecies from the lists 1',2' or 3'. Theorem 4 - positive form. R cF (S) is of finite representation type (independently on F and on the modulation) if and only if every connected valued subgraoh of S is contained in the lists i, 2, 3 or 4 :

280

I/st 4

i) 0 4

o o ~~ . ~ o ~~ ' ~ o

;

2) 0 5

;

3) 0 5' -- dual of 0 5

(any double arrow has capacities 2, ]). Pr_oof. An easy computation shows that the species listed in lists 1' , 2' and 3' are not excellent. Let (V) be a representation of the species S, with ously,

dim (V) : (dx)xsG. Obvi-

(V) correspond to an element from the set

R' = ]-] H ~ ysr

(%

- a (y)

(Y)

My , v a (V)) " (y)

R' is a vector space over F, havinm the ~ s i o n m = Zq(Y)P(a(Y)) ysF

d s(y)d a(y) .

Denote by R the subset of R' containing all elements which correspond to commutative representations. R is an algebraic variety, having cc~oonents of dimension less than

r D - s~=l qsP(Xls) d d = X0S XIs

Consider the group

S_~' = ~ Au~ (Vx) xe G x

is an infinite field, then S is

and the quotient

S = S'/F { . If

an alqebraic gro~o having the dimension over F :

~ . p(x) d 2 x xsG

i

The group S operates on the set R' ; an easy commutation shows that S omerates also on R ; R deco~9oses in orbits, and each orbit contains families

(W) cor-

responding to isomorphic con~nutative representations. Suppose the category ~qF (S) has only a finite number of types of indecomposable objects. Then, qiven a dimension vector number of indecomposable c o m u t a t i v e

(dx)x~C, there exists only a finite

reuresentations havina dimension d = (dx) .

Under the action of S, R deconposes in a finite number of orbits. Every such orbit has, over F, the dimension less than the dimension of S; so, the dimension of R is less than the dimension of S, that is B S(d) >/ 1

if

d # 0 ,

BS being the Brenner quadratic form of the species S. Pence S is an excellent species. Now, if S is a trivial excellent species, then eve~, connected subgraDh must appear in the lists 1 and 3. If S is not trivial and has the cyclic rank 0, then every connected subgraph must anDear in the list 2 ; if the cyclic rank is not 0 , then from the corollary of proposition i it follows that S appears in list 4. Conversely, from Dlab-Ringel and Loupias-Zavadsky-Shkabara theorems [11],[14]), every species from the lists 1,2 and 3 is of finite

(see [5],

(commutative) re-

presentation type. For the sbecies 0 5 we showed this in the proposition 2, and the species 0 4 is a subordinate of 0 5 .

The theorem is oroved.

281

The theorem of Dlab and Ringel ([5]) describes more co~mletely the species with finite representation type; namely, for such a smecies, each indeccr~sable representation correspond to a positive root of the Tits quadratic form. The Tits ouadratic form of a Gabriel species S is positive definite if and only if the smecies has finite representation type. In the proof of theorem 4 we used the Brenner quadratic form in the same manner Dlab and Ringel used the Tits form. The Brenner form for a s~ecies with commutative representation type is no loger positive definite, it has only the oromerty (E). But still remains true that each positive root of the (Brenner) quadratic form determines an indeccmposable commutative representation! This fact is not fully explained and the problem of describing (axiomatically) the dimensions of such representations remains open. RET~RENCES [i] Bernstein, I.N., Gel'fand, I.M., Ponomarjow, V.A., Funktory Kokstera i teorema Gabriel'a, Uspehi Mat.Nauk, 28(1973), 19-33 [ 2] Brenner, S., Quivers with co~utativity conditions and some Dhencrnenology of forms, Representations of Algebras-Ottawa 1974, Springer Lecture Notes in Math. , 488 (1975) [ 3] Dlab, V., Representations of valued graphs, Universit4 de Montr4al (1980) [4] Dlab, V., Ringel, C.M., On algebras of finite representation type, J.Alqebra, 33 (1975), 306-394 [5] Dlab, V., Ringel, C.M., Indeconposable representations of graphs and al~ebras, ~emoirs Amer.~th.Soc. 173 (1976) [6] Gabriel, P.,Unzerlegbare Darstellungen I, Manuscripta Math., 6 (1972), 71-103 [7] Gabriel, P., Indeccmloosable representations II, Syr~=osia Math., Ist.Naz.Alta Mat., ii(1973), 81-104 [8] Gabriel, P., Repr4sentations i n d 4 ~ s a b l e s ,

S4minaire Bourbaki 1973/74

[9] Gel'fand, I.M., Poncr~arjow, V.A., Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector soace, Coll.Math.Soc.Janos Bolyai V, Tih~ny-Hungary (1970), 163-237 [i0] ioupias, M., Repr4sentations ind4conzx)sables des ensembles ordonn4s finis, S4minaire d'alg'ebre non-conmutative Orsay 1974/75 [ii] Loupias, M.

Indeco~oosable representations of finite ordered sets, Representa-

tions of Algebras-Ottawa 1974, Springer Lecture Notes in Math., 488 (1975) [12] Ovsienko, S.A., 0 sistemah kornei dl'a proizvol'nyh c[rafov, Matri~nye zada~i, Inst.Mat.AN UkSSR, Kiew (1977) [13] Zavadsky, A.G., Nazarova, L.A., Ciasti~'no qnoriado~ennve mo~estva ru~no~o tipa, Matri~ye zada~i, Inst.Mat.AN UkSSR, Kiew (1977) [14] Zavadsky, A.G., Shkabara, A.S., Konmutativnye kol~any i matri~nye algebry kone~nogo tipa, Preprint IM-76-3, Inst.Mat.AN UkSSR, Kiew (1976)

Characterization

of b i c a t e g o r i e s

Ross

of stacks

Street

Introduction Although gories, modify

the paper

it was p o i n t e d the work

present

paper

[11] was w r i t t e n

in the setting

out in the i n t r o d u c t i o n

in order

to make

is to make

of that paper how to

it b i c a t e g o r i c a l .

these m o d i f i c a t i o n s

of 2-cate-

The purpose

precise

of the

and to give

an

application. The main

theorem

(= champs

stacks

in French)

and size c o n d i t i o n s Giraud's

stack.

result

The e x i s t e n c e

construction

Exactness obvious

properties

generalization associated

Exactness

nite b i c a t e g o r i c a l cotensoring

with

ated stack needed

formula.

limits

finite

stack

L

L

is immediate:

categories

objects,

used

for the

it p r e s e r v e s

all fi-

bipullbacks,

L

not

in [I]

in the sense of of

were

the obvious

[9]).

(recall

and bi-

The associthat two are

case).

The a p p l i c a t i o n the formula

on fibrations.

construction

of the functor

of

of

category-valued

The formula we give uses

(bitermina]

version

for the associ-

construction

is given by three a p p l i c a t i o n s

in the sheaf

exactness,

stack on a c a t e g o r i c a l

the a s s o c i a t e d

of the a s s o c i a t e d

of

[I].

is given

of the a s s o c i a t e d

to b i c a t e g o r i e s

sheaf.

of sheaves

and a s t r i c t i f i c a t i o n

from G i r a u d ' s

colimit,

a bicategorical

a formula

[5] using

of b i c a t e g o r i e s

of limit,

of c a t e g o r i e s

to this

site was proved by Giraud sheaf

in terms

on the b i c a t e g o r i e s :

characterization

On the way ated

is a c h a r a c t e r i z a t i o n

we w i s h

to p r e s e n t

for the a s s o c i a t e d

st~ck.

is really

We give

an a p p l i c a t i o n

an easy proof

of

of the

v

torsors and Cech cocyles. C o m b i n i n g this w i t h a

relationship

between

very general

theorem giving

c lass i f y ven

objects

family

mation

about

§I.

local

the c l a s s i f y i n g

structure

of m a t h e m a t i c a l

tor bundles, SO

locally

structures),

structures

locally

finite

property

isomophic

for example,

in a topos,

Azumaya

(they

of a gi-

we are able to deduce

in mathematics;

objects

of torsors

to some m e m b e r

infor-

about vec-

algebras,

and

on.

Regular

and exact b i c a t e q o r i e s .

The n o t i o n transformation Hom(A,B)

of bicategory,

and m o d i f i c a t i o n

for the b i c a t e g o r y

and m o d i f i c a t i o n s

homomorphism

of b i c a t e g o r i e s ,

are those of B ~ n a b o u

of h o m o m o r ph i s m s ,

from the b i c a t e g o r y

A

strong

strong

[3]. We w r i t e transformations

to the b i c a t e g o r y

~.

283

The notion

of

For homomorphisms is an o b j e c t

limit

for b i c a t e g o r i e s

F : A ~ Cat,

{F,S}

of

K

is t a k e n

from Street

the F - i n d e x e d bilimit

S : A ~ K,

satisfying

an e q u i v a l e n c e

[9].

of

S

of homomor-

phisms: K(-,{F,S}) As

special

bicotensor

in

K

K

K(K,m) is f u l l y

is a b i c a t e g o r y

arrows

K (e,X)

sketch takes

Given

can define using T

what

an a r r o w

2 E2

d2

do

the

K

"essentially

squares

in

theory

of categories There

into

is a n o b -

categories.

we can

form

2-cells

the

following

are bicomma

isomorphisms

di-

object

are bipullbacks.

A f

. A

of arrows

of b i c a t e g o r i e s

to b i l i m i t s .

weak K

containing

containing

2 dl ~ E1 - -

for a f a m i l y bipullback.

for t h e

cones

f : A~B

squares

~ A

for

is a b i p u l l b a c k

is a h o m o m o r p h i s m

theory)

do dl

(short diagram

it m e a n s

of weak functor b e t w e e n

the

do

e.s.o.

following

a many-legged

in

(= Gabriel

and

2 E1

An arrow

L K ( A Y)

the d i s t i n g u i s h e d

in w h i c h

diagrams

bilimits.

functor

!K(e,Y)

e.s.o,

K

notion

finite

the

K (B,Y)

K(A,m)

one

to be

from the

the

~

A weak category

agrams

biproduct,

m : X-Y.

I

generally,

vious

with

when

is c a l l e d when

K (B,m)

K(A,X)

which

bipullback,

K.

e : A--B

on objects")

f.f.

B

f.f.

for all

K(B,X)

into

objects,

: K(K,X)~K(K,Y)

An arrow

More

biterminal

is c a l l e d

faithful

surjective for all

Hom(A,Cat) (F,K(-,S)).

product.

Suppose m : X~Y

=

cases we have

f

EI

. B

do

If

. A

]

~

A

f

I f

,B

, B f

We

obtain

a weak

functor (a)

f.f.;

two weak

categories

j : E I -- E 2

EI = E2 = A o o

t

E2 : E2 2 -- E 2I ~ E 02 a n d

with Jo

the

following

is an i d e n t i t y ,

E] :EI ] I = E O,

properties: I 2 Jl : El " El

and

is

and

284

(b) bration

the

span

in

the

(c)

E1

This

leads

(d o ,E 2 1 , d I)

sense is

of

Street

a weak us

equivalence

j : EI - E2

satisfying

has

a congruence

associated

g :A - X

and

I El

for

JI

, E2 I

(a) ,

to

A

relation

with

(b),

is

a bidiscrete

fi-

on

on

(c) . S o

A.

A

to

each

be

a weak

arrow

func-

f :A - B

it.

a congruence

a 2-cell

A

a congruence

to d e f i n e

tor

A quotient

from

[ 9];

j :EI ~

X : g d o -~ gd I s u c h

E2

consists

of

an

arrow

that:

dl ~ A

do

g

A

' X

is

invertible,

and

g E~

d2

2 '" E 1

dl "

dO

A

~

E1

dl

~

A

g

g

~

X

and

which

K

has

~

A

g

g

X

;

-- X

I

is b i u n i v e r s a l

If

' A

do

I

-- X

g

• EI

=

g A

E

with

finite

these

colimits

properties.

then

every

congruence

has

a quo-

tient. An

arrow

a congruence form

E

a quotient

K

Call all

-f.f.

each

is

K

f

is

has

a quotient

a 2-cell

map

~ : qd ° ~ qd I

map

is

e.s.o.

the

following

when

there

exists

such

that

Q,q,~

(Compare properties

[11;(1.17)].) hold:

exist; isomorphic

to

a composite

me

where

m

is

e.s.o.;

exact

a quotient

of

an

e.s.o,

bicategory,

is

e.s.o.

every

e.s.o,

is a q u o t i e n t

[11;(1.22)].)

associated

bicategory

called

when

bilimits

bipullback

Call

and

regular

(Compare

congruence

is

quotient

In a r e g u l a r

map.

the

E.

arrow e

each

Theorem.

ence

A,

finite

and u

on for

Every

ProPosition.

--

q : A~Q

in

when

it

with

some

an

is

exact

Hom(c°P,cat)

is

regular arrow.

and It

bicategory. exact.

each

follows For

congruence

is

that

congru-

all

every

bicategories

the

C,

285

§2.

Bitoposes.

A topology C,

a set

CovU

tisfying

the

TO.

of

C,

the

all

S

~ C (-,V)

R

~ C (-,U)

the

top

T 2. if the

R' ~

R ' V -- C(V,U) in C o v V

for

such

gories

is

is

the

Write consisting

R -

as

C

with

For RS S

when

in

C

for

is

and

of

Hom(C°P,cat)

sa-

arrows

U

; all

and U

in

u : V-

as

equivalent together

R -- C(-,U) the

image

in TI to an

with

CovU

f.f.

arrow

with

the

top

arrow

in

CovU

of b i c a t e g o r i e s

in

is

of

.

A stack

a topology.

F : C°P - Cat

, an e q u i v a l e n c e

= Hom(C°P,cat)

of

cate-

K

homomorphisms

the

for

(R,F).

on a b i c a t e g o r y

full

the

is t h e are

largest

all

sub-bicategory

bisite

is c a l l e d

underlying

K- Stack

there

CovU

CovU u : V~

a bipullback

topology

a bisite

R

is

C(-,U)

stacks

small

equivalence:

U

V ;

each

representable

the

in

object

follows:

A bicategory site

Cov

(C(-,U),F)

Stack

of

in

CovU

a homomorphism

each

canonical

for w h i c h

to each

C(-,U)

is

in

is a b i c a t e g o r y

Hom(C°P,cat)

gy

for

C(-,U)

induced

The

in

exists

for

R ~

C(-,U)

C(-,U)

that

a bisite

that,

is

R -

A bisite

such

of

there

, then

assigns

a bipullback

arrow

property

arrows

R -- C(-,U)

exists

C

conditions:

identity

there

in w h i c h

with

f.f.

following

T I. f o r in

on a b i c a t e g o r y

topolo-

stacks.

of

Hom(C°P,cat)

C.

a bitopos

bicategory

when

such

there

that

exists

there

is

a bia bi-

C.

C,

exists

regard

CovU

as

an o r d e r e d

set

by

taking

a diagram:

~S

c(-,u)

If

C

define

is

small

for

a homomorphism (LP)U

where

then,

R

= colim R

runs

over

A homomorphism

each

homomorphism

LP : C °P - Cat Hom(COP,cat) the

directed

P : C ° p - Cat,

we

can

by:

(R,p) set

of b i c a t e g o r i e s

( C o v U ) °p which

preserves

finitary

in-

286

dexed

bilimits Since

bilimits,

will

be c a l l e d

filtered

L

left exact.

colimits

is a left

exact

in

Cat

commute

homomorphism

with

from

finitary

indexed

Hom(c°P,cat)

to it-

self. Theorem.

For any

small

bisite

C, the

left biadjoint

of the inclu-

sion Stack C is o b t a i n e d P ~ F

~ Hom(c°P,cat)

by applying

is f a i t h f u l

stack

of

P. If

L

and

P ~ F

LP

is the a s s o c i a t e d

§3.

Characterization

rating

when,

is fully

in the

and

faithful P.

L2p and

left exact.

If

is the a s s o c i a t e d F

(Compare

of a b i c a t e g o r y

object

set,

A bicategory tegories

of

and is hence

then

is a stack

then

[11; (3.8) ].)

D

theorem.

for e a c h

sources

times

is a stack

stack

A set of o b j e c t s

with

three

F

K

U

is

of

is c a l l e d

C, the

e.s.o,

set of a r r o w s

gene-

into

U

e.s.o.

is c a l l e d

the Y o n e d a

C

lex-total

when

it has

small

homca-

embedding

Y : K ~ Hom(K°P,cat) has

a left-exact

left b i a d j o i n t .

Bicoproducts served

in a b i c a t e g o r y

by b i p u l l b a c k s .

bicoproduct

have

When

any

a bi-initial

are u n i v e r s a l

two d i s t i n c t

bicomma

when

they

coprojections

are preinto

a

object

then

the b i c o p r o d u c t

is

is no g r e a t e r

than

the

of

disjoint. A set w h o s e the

set of

Theorem.

small

cardinality

sets

is c a l l e d

The f o l l o w i n g

homcategories

are

K

is a bitopos;

(~)

K

is

jects

of

K

with

(iii) every e.s.o, (~)

such

canonical

and there for all

stack

small

on

K

with

small

set

M

of ob--

exists such

exists

a moderate

X

in

K, there

K

is r e p r e s e n t a b l e

exists

an

e.s.o.

a small that

and

K

has an

set of objects;

bicategory

and has an e.s.o,

dexed bilimits

on a b i c a t e g o r y

M;

is an exact

there

conditions

that,

in

generating K

bicoproducts (v)

lex-total

M

moderate.

equivalent:

(i)

M ~ X

cardinality

which

has disjoint

generating

canonical

K ~ Stack C .

small

bisite

G

(Compare

universal

small

set of objects; with

finitary

[11;(4.11)].)

in-

287

~.

Application Let

to torsors.

E

denote

a finitely

complete

E/U

and such that each of the c a t e g o r i e s K

denote

the

2-category

Regard into

U

E

cells,

we obtain

Regard

E

A

in

the usual

in

K

of

of

U,V

n~gular epimorphism only

which

K

by taking

as c o n t a i n e d

in

Let

epimorphisms

identity

are stacks

objects

2-

for

in

F

of

E

by taking

as each

E(-,A).

is called admissible when,

F

with

closed.

stacks in this section.

to the r e p r e s e n t a b l e X

with

F

coequalizers

F =Hom(E°P,cat).

regular

as a b i c a t e g o r y

Regard

E

E. Let

single

The objects

with

is c a r t e s i a n

in

generating

E

as c o n t a i n e d

, y : V-X

E, there

is a b i c o ~ a

for all object

x/y

E. Define

ling back

S E F

along

F(Y,PX) For

A

in

and

S

on arrows

is given by pul-

X

in

F

there

exists

PX

in

with

the

F

satisfying:

= F(X °p × Y,S)

K, we can i d e n t i f y

spans

spans

S U = E/U

by

them.

For each

the

and

simply be called

An o b j e c t

in

U

a bisite.

categories.

category

x : U-X

of

E. R e g a r d i n g

this b i s i t e will

discrete

of c a t e g o r i e s

as a site by taking

as covers

topology on

category

A ~ E ~ U

for w h i c h

the

in

K

(PA)U

from

following

U

to

A

full

subcategory

consisting

of

of those

is a pullback.

dI

Pl

Po AI

In st a n d a r d

dI ~ Ao topos

terminology,

For any a d m i s s i b l e YX : X - PX. For y A U : E(U,A) A ~ A/a-

-

along

(PA) U

A

colim(E,f) q

X

in

F, there

K, the y o n e d a that

is a yoneda arrow

arrow

functor w h i c h

YA

takes

has c o m p o n e n t a : U-A

to the span

is a d m i s s i b l e of

f : A-- X

and

E E (PA)U.

is the p o i n t w i s e

The E-indexed colileft e x t e n s i o n

of

fp

as below: E

P~

Pl A Here

in

~ E A°p×U.

U.

Suppose

mit

A

(PA)U

U

[colim(E,f) ~

X

pointwiseness

. means

that the

left

(Kan)

extension

property

is

288

stable

under

pullback

Call

X

and

f : A- X

all

B

in

with K

f : A~ X of

when AV,

an a r r o w when

in

[8],

K.

U. colim(E,f)

In p a r t i c u l a r ,

locally

is

exists

PB

isomorphic

a regular

an i s o m o r p h i s m

e

into

it a d m i t s

for a l l

E

is c o c o m p l e t e

for

[I~).

z E XU

there

and

V

A

(see

An object

a

along

cocomplete

to a value

epimorphism

of

e : V~U,

an o b j e c t

(Xe)z ~ fv a-

~ U

el

Iz

A------~X Let

LOCx(f)U

Since

be the

the p u l l b a c k

an object

subcategory

of a r e g u l a r

LOCx(f)

For

full

of

F

A 6 K, an o b j e c t

to a v a l u e

of

epic

which

Y A : A-- PA

of

XU

consisting

is a r e g u l a r

epic,

is a s u b h o m o m o r p h i s m

E ~ (PA)U

is c a l l e d

which

of

this

of

is l o c a l l y

such

z.

defines

X. isomorphic

an A - t o r s o r .

dI A/a

A

~ V

, E

~ U

P Put

q

T o r A = L O C p A ( y A) .

An o b j e c t

Proposition.

mits for

all

B

Theorem

ble

colimits

al'i

in

X

indexed

K.

Each

by

F

is a s t a c k

torsors.

by torsors.

x : W~X

in

F

with

morphism

as a c o m p o s i t e

of an a r r o w

identity

on o b j e c t s

an a r r o w

nents

are

fully

if and

In p a r t i c u l a r ,

only PB

if it adis a s t a c k

[]

on c l a s s i f i c a t i o n

stack.

of

and

faithful.

The

Suppose W

in

X6 E

F

is an a d m i s s i -

factors

up

to iso-

W - X[x]

in

K

which

is the

i : X[x] - X

in

F

whose

compo-

functor

E -- c o l i m ( E , i )

provides

an

equivalence: TorX[x]

~LOCx(X)

Theorem

relating

K

U

and

of

E

torsors

there

(TorA) U = colim V-U e

where

e : V~U

[]

runs

and ~ech

cocycles.

For

each

into

U

object

A

is an e q u i v a l e n c e K(eru(e),A)

over

the

r e g u l a r ~ epics

and

eru(e)

of

289

denotes

the category

Proof.

Bunge

A.

The

given

[4] h a s

colimit in §2.

So o n l y

ciated

stack.

one

× N

~

+

The lue of dable)

of

the

last

objects

Z are

of

This

we

$I

locally

in

The

is a t o p o s the

filtered

proof

that

allow

that

the

cohomology

spaces over

situation

X 6 F

over

~ ; that

Take mensional

vector

R

is,

as PA

the

a

asso-

N.

denote

In the

the o b j e c t

locally

isomorphic

to a va-

( =Kuratowski-finite

deci-

is the

usual

category

Efi n

of

give:

in

E

o n the o b j e c t

[6] a n d

colimit

on the

locally

involved

right

finite

R

above

objects

category-valued

of

E.

is a n e s s e n t i a l l y

are basically

is a topos. in

E

form

cohomological

coefficients

and

coefficients).

take

E

regular

to b e a n i c e epics

internalization

XU

topos

is the

to

of to-

local homeomorphisms.

of t h e

category

category

theory

of m o d u l e s

of v e c t o r

E/U

in

to b e the

family

R n , n E ~ , of f i n i t e

di-

spaces. Z

bundles

over

X[Euc]

is t h e E.

object

× U~U.

Euc : ~ ~X

vector

in

the

to be the

Objects

category

of §4,

Restrict

Take

the ring

numbers

two t h e o r e m s

with

bundles.

spaces.

are

category

the

ideas

§6.

pological

to o b t a i n

F i n C SN = E/N

finite

E

last

abelian-group-valued

In the

LA

with

[]

natural

which

not merely

Vector

is n e e d e d

let

the

the c h a o t i c

another

showing

with

of

S[Fin]

Efi n

notion,

(provided

for

= c o l i m K ( R c , E f i n) R--~I

denotes

gives

a topos

L

formula Y A : A-- PA

of

~ Loc(Fin)

Tor(Efin)1

algebraic

of

the

arrow

as r e q u i r e d .

section,

objects.

T o r ( E f i n)

Since

faithful

stack

E.

category

Rc

is the a s s o c i a t e d

E/N.

of

objects

cardinal-finite

where

= LA

Tor A

is p r e c i s e l y

to b e a t o p o s

Fin : N~ S

The

that

application TorA

--~ N suc

N

shown

e.

in a topos.

E

terminology N

d e t e r m i n e d by the kernel pair of

is a f u l l y

So

Finiteness Take

E

of the T h e o r e m

There

stack.

~5.

in

of

XU

locally

isomorphic

to a v a l u e

of

Euc

are

U. category

Mat(~)

of matrices

over

R

as a

290

The

two theorems

of §4 give equivalences:

Tor

(Mat(A))

Loc(Euc)

Tor

(Mat(~))U

=

= colim

=

(vector b ~ d l e s )

K(eru(e),Mat(~

))

V~U where

e

runs

over

surjective

Thus we o b t a i n bundles over

over

U

surjective

compact

and the c o l i m i t

inherited

S @ T ~ S' @ T necessary ving

the general

K-theory valued

deduce

implies

linear

therefore

appearing

cohomology.

These

category

group

brings

in books

with

properties

U. of v e c t o r e

runs

is a finite

are t h e r e f o r e

. The r e s u l t

is a p r e c i s e

construction for v e c t o r bundles

for the c o n s t r u c t i o n

equivalence

as

U. Now M a t ( A )

of v e c t o r b u n d l e s

of the clutching

w h i c h we can i m m e d i a t e l y

into

additive

idempotents.

by the c a t e g o r y

into

the c a t e g o r y

of K ( e r u ( e ) , M a t ( ~ ) )

closed m o n o i d a l

and s p l i t t i n g

formulation

between

local h o m e o m o r p h i s m s

symmetric

prod u c t s

local h o m e o m o r p h i s m s

an e q u i v a l e n c e

from

the property: S Z S' of K-theory. GL(n,~

together such as

)

The usual

colimit

invol-

is also a consequence.

much

[2],[7]

The

of the i n t r o d u c t o r y as an aspect of c a t e g o r y -

291

References I

2

M. Artin,

A. Grothendieck

~ta~e des schemas, Lecture Notes in Berlin, 1972).

269

(Springer,

K-Theory,

M.F. Atiyah,

47

(Springer,

1967)

Lecture Notes

in

1-77.

P.T. Johnstone,

7

M. Karoubi,

Topos Theory,

Street,

Wissenschaften,

(1980)

R.H.

Fibrations

Street,

R.H.

R.H.

Street,

Algebra

24

21

in bicategories, XXI

(1981)

der

Transactions

Amer.

(1980)

Cahiers

2Opp.

de topologie et

111-160. J. Pure and

307-338.

Two dimensional (1982)

1978)

(Springer Berlin, 1978).

Conspectus of variable categories,

Appl. Algebra

1971).

271-318.

diff~rentielle

Street,

Berlin,

Grundlehren

Band 226,

Cosmoi of internal categories,

Math. Soc 258

g~om~trie

(Springer,

(Academic Press,

K-Theory: An introduction,

Mathematischen R,H.

401-436.

Cohomologie non ab~lienne,

6

11

to bicategories,

Berlin,

Stack completions and Morita equivalence for categories in a topos, Cahiers de topologie et g~om~trie diff~ren-

J. Giraud,

10

Lecture Notes Ser.No.7

M. Bunge,

5

9

Math.

1967).

Introduction

J. B~nabou,

tielle XX-4(1979)

8

Th~orie des

topos et cohomologie

Math. 4

editors,

Math.

(Benjamin-Cummings, 3

and T.L. Verdier,

sheaf theory, J. Pure and Appl.

On h o m - f u n c t o r s

and

tensor

products

Walter

It is e a s y spaces with

to see t h a t the

with

the

But

topologies

many

the

a subset

AE

of

the

In this

paper we

HB

and

first

defined HB

discuss

their

by Fischer its

paper

left

(O)

Preliminaries K

be

a fixed

the c a t e g o r y

and a b s o r b i n g

for all

(See A d a s c h ,

a generalization Un

Let

E

power

are and

and e a c h

W(A,U)

tensor

subsumes

valuated

vector

vector

space

subsets Ernst,

the k n o t s

and

U

spaces

[I].

convex

of the vector

Iyahen

[6].

[9].

let

that

In s o m e

TVS

denote

(Un I n 6 ~ )

Un+ I + Un+ I c U n

sense

a string

is

neighbourhood.)

string spaces,

a neighbourhood

of u l t r a b a r -

is a s e q u e n c e such

to

category.

of

thesis

K.

been

restricted

sense

and

A E-

having

closed

over

E

U nc E

Keim

are

field

of

If the h o m -

the n o t i o n s

in t h e

E

of t h e h o m - f u n c -

spaces.

product

space

HB(E,L)

latter

monoidal

p a r t of the a u t h o r ' s

be topological E

U 6 U

W(A,U) Then

L

set of

convex

spaces

of an a b s o l u t e l y

called

properties

locally

space

space

the

topolo-

vector

on the e l e m e n t s

products,

a symmetric

non-discrete

in a t o p o l o g i c a l

n.

for

adjoint

of t o p o l o g i c a l

of b a l a n c e d

The

[5]

space

of

where, r o u g h l y

topological

categorical

space

the t o p o l o g i e s

TVS,

the v e c t o r

and

closed

function

function

on

convergence

tensor

one gets

is a s h o r t e n e d

These

vector

monoidal

namely

HB

every

and

quasi-ultrabarrelled

Let

A strin~

the

of a B - b a r r e l l e d

resp.

to

set,

related

spaces,

spaces,

subsets.

of u n i f o r m

spaces

convergence

interesting

hom-functors assigns

of t o p o l o g i c a l

of p o i n t w i s e

is a s y m m e t r i c

vector

its p o w e r

and

B-barrelled The notion

B

topology

tors

functor

product

on c e r t a i n

internal

functor

carries

TVS

category

are a lot of m o r e

convergence

spoken,

This

tensor

there

vector

Sydow

hom-functor

for t o p o l o g i c a l

gies yield

relled

internal

inductive

category.

uniform

the

of t o p o l o g i c a l

base

(Un). A in

a subset L.

For

of the each

AE A

let

:= {f I f ~ T V S ( E , L ) , :: { W ( A , U ) I

f(h) c u } .

A 6 A, U 6 U}

is a n e i g h b o u r h o o d

base

of a

293

linear

topology

gy of u n i f o r m

on

TVS(E,L),

convergence

called

o n the

the

A E A,

A-topology, if

A

or the t o p o l o -

fulfills

the

follow-

ing conditions: (i)

Each

AE A

(ii) A , B 6 A

is b o u n d e d .

imply

It is c o n v e n i e n t A

is c a l l e d

(BI)

Every

(B2)

UA = E.

A U B 6 A.

to r e q u i r e

a bornology A6 A

some more

on

E

conditions

if the

following

implies

A U B,

C C A.

(B4)

implies

A + B,

~ A ~ A.

For

Now we

(I)

~ 6 K

a bornology

TVS(E,L)

A

on

together

extend

A:

hold:

is b o u n d e d .

(B3) A , B E A, C C A A , B 6 A,

for

E

with

this

we denote

the

by

HA(E,L)

the v e c t o r

space

A-topology.

construction

t o get

internal

hom-functors

on

TVS.

Definition

A bornological vector

space

A mapping

topological

together

f:

vector

with

(E,A) ~

space

a bornology

(L,B)

(E,A)

is a t o p o l o g i c a l

A. N o t a t i o n :

is c a l l e d

bounded

if

A(E,A)

fAE B

:= A.

f o r all

A6A. The

category

ded

continuous

(2) The

linear

mappings

topological is d e n o t e d

forgetful For

:= { A C E

functor

each IA

V:

source

BorTVS

(E,

is b o u n d e d

~ TVS

(fi: E ~

and

spaces

and boun-

BorTVS.

the

source

(3)

Definition

A functor identity

B:

((E,A),

TVS

functor

~

BorTVS

on

TVS.

(fi)i)

is t o p o l o g i c a l .

(Li,Ai))i),

f.AE A 1

Then

!4)

by

vector

Proposition

Proof: A

of all b o r n o l o g i c a l

for all

define i C I}.

1

is

V-initial.

is c a l l e d

a B-functor

if

VB

is the

Examples

The B-functors (I)

Bf(E)

(2)

B

Bf,

B

, Bc,

Bt

and

Bb

=

(E,

{ACE

IA

finite}).

(E) =

(E,

{ACE

IA

bounded

Bf

and

are d e f i n e d is

left

by

adjoint

c a r d A < e})

to

for an

V. infinite

294

cardinal

e.

(3)

Bc(E)

=

(E,

{ACE

(4)

Bt(E)

=

(E,

{ACE

to

V o. Vo(E,A)

with

(5)

the

finest bounded.

Bb(E)

(E,

In the

(5)

Definition

Let

B:

tor

TV S

IA

bounded}).

we n e e d

in

be

E

are c l o s e d

(2) T h e logy

closed

on

E,

E,

Bb

such

E

of

that

is r i g h t

and

Hausdorff).

is r i g h t

space

be

adjoint

E

together

all

AE A

adjoint

of a B - b a r r e l l e d

to

are

V.

space.

a topological

vec-

(4) E

is c a l l e d

full

(6)

SB:

TVS

~

vector

TVS

subcategory

in

E

topolo~

if all

of

if

generate

of

is d e f i n e d

space

B-barrelled

(B-bornivorous)

its

A E A(BE)).

strings

the B - s t r o n g

(underlying

=

closed

each

B-bornivorous

called

funetor

a linear

topo-

E. by

E, B - s t r o n g

topology

of

E).

is d e n o t e d

by

BBar.

E = SBE.

of the B - b a r r e l l e d

spaces

Proposition each B-functor

flective Proof:

in

(7)

Remark B

tains

be all

the

B

SB

on

E 6 0 b TVS

a B-functor

such

zero-sequences

the B - b a r r e l l e d

spaces

A circled

sequences

of B - b a r r e l l e d

[all

U

that

spaces

is b i c o r e -

spaces

as n e c e s s a r y .

for e v e r y rapidly

are e x a c t l y

absorbs

rapidly

as o f t e n

[or all

the q u a s i - u l t r a b a r r e l l e d Proof:

category

TVS.

Apply

Let

such

Bt

vector

on

a B-functor,

is c a l l e d

(3) T h e

(8)

the n o t i o n

(absorb

SB(E)

For

bounded}).

without

space.

knots

For

totally

topology

~ BorTVS

(1) A s t r i n g

The

]A

(compact

linear

{ACE

following

compact})

is the u n d e r l y i n g

totally =

IA

TVS

a circled

A(BE)

con-

sequences].

the B b - b a r r e l l e d

of I y a h e n

decreasing

E 60b

decreasing

spaces,

Then

i.e.

[6].

A,

sequences]

if

U in

absorbs

all

zero

A.

Proposition each that

B-functor for all

HB(E,L)

there

is an i n t e r n a l

E , L E Ob TVS

= H A ( B E ) (E,L).

hom-funktOr

H B on

TVS,

295

Proof: and

Let

B(Ei)

u: E I -- E o, v: L o ~ L I =

(Ei,Ai)

for

HB(U'V) : H A o ( E ° ' L ° ) is o b v i o u s l y

linear,

i =0,1.

be c o n t i n u o u s

linear m a p p i n g s ,

Then

~ HAl (EI,L I ,

and c o n t i n u o u s ,

f -- vfu

s i nce

HB(U,V) (W(uA I, -vI UI)) C W ( A I , U I) holds

for e a c h

AI E A

(9)

Proposition

Let

B:

space.

TVS

Then

(I) T h e r e

~

and

U1

a nelghbourhood

be a B - f u n c t o r

BorTVS

and

E

in

LI.

be a t o p o l o g i c a l

vector

the f o l l o w i n g hold:

is a n a t u r a l

isomorphism

ITV S ~ HB(K,-)

I:

defined

by

(2) T h e r e

tL: L - H B ( K , L ) , is a n a t u r a l

x ~

(a ~ ax),

L 6 0 b TVS.

transformation

~: K -- HB(-,-) defined

by

~L: K ~ H B ( L , L ) ,

(3) The p a r t i a l (initial

hom-functor

(4) T h e c o n t r a v a r i a n t

Proof: (3) Let source because

(I) a n d

assume

partial

(HB(E,L),

by

that

of

HB(E,-)

HB(E,-)

((fi)'L)' B

b e an i n i t i a l

is c o n t i n u o u s

Functor (cp.

carries

those

preserves

this colimit.

(L,A), B ( L i) = W(fiAi,U)

T h e n the

TVS.

is initial, for

A C E, U i C L i"

and h e n c e has a left

fi: Li ~ L, be a c o l i m i t

B(L)

A i e Ai

in

holds

Theorem).

Let be

=

source

In

(15) we c o n s t r u c t

the

(17)).

HB(fi,E) : HB(L,E ) ~ HB(Li,E)

and

sources

B, to limits.

source

E

HB(-,E)

-I = H B ( E , f i ) (W(A,Ui))

(Special A d j o i n t

that

hom-functor

initial

left adjoint.

(HB(E,fi) : H B ( E , L ) ~ H B ( E , L i ) ) I )

-I fi(Ui))

W(A,

preserves and has

(2) are obvious.

(L, (fi: L ~ Li) I)

left a d j o i n t (4) Let

sense),

t h a t are p r e s e r v e d

We c o n c l u d e adjoint

HB(E,-)

in the t o p o l o g i c a l

colimits,

~ ~ el L, L 6 0 b TVS.

(Li,Ai).

of

We h a v e

D: I ~ T V S to s h o w

and

that the

is initial. For e a c h n e i g h b o u r h o o d

-I = H B ( f i , E ) (W(Ai,U))

holds.

Since

U

in

296

is a colimit,

((Bfi) ,BL) ZfiAi,

A i 6 A i, h e n c e

each

A f A

the p r o o f

is a s u b s e t

is c o m p l e t e

as

of a f i n i t e

sum

W ( Z f i A i, ZU) D

D N W(fiAi,U ) N o w we d i s c u s s

the

symmetry

s: H B ( E , H B ( L , X ) )

-- H B ( L , H B ( E , X ) ) f-

For e v e r y

continuous

s(f) (i) : E -- X X

there

(e-

linear

s(f)

because UE

is c o n t i n u o u s

neighbourhood

UX

in

X

s(f) (U L) C W ( A , U x )

Consequently

s(f)

equicontinuous.

there

latter

X

and

be

topological

B-barrelled

such

for e v e r y

iff

condition

Let

L

E

and e v e r y

1 6 L

,

any n e i g h b o u r h o o d that

UX

in

f(UE) c W ( { I } , U x ) -

A E A(BE)

is a n e i g h b o u r h o o d

is c o n t i n u o u s

The

for

and e v e r y

UL

in

L

such

.

(Banach-Steinhaus)

L

in

iff

(10.) T h e o r e m

Then

f(e) (i))).

f : E--HB(L,X )

is c o n t i n u o u s ,

is a n e i g h b e u r h o o d

Moreover

that

(i ~

,

implies

vector that

for e v e r y

AC

A(BE)

holds,

if

L

is B - b a r r e l l e d .

spaces

and

every

B

bounded

f(A)

is

a B-functor.

subset

of

HB(L,X)

is e q u i c o n t i n u o u s . Proof:

Let

A C HB(L,X)

logical

string

in

hence

topological,

(11)

Proposition

Let

B:

TVS

I) For

~

be b o u n d e d

X.

Then string

all B - b a r r e l l e d

aB(E,L,X) : HB(E,

(2) For

is c o n t i n u o u s Proof: The

in

E,

TVS

-- H B ( H B ( E , Y ) , and

L

HB(L,X))

all B - b a r r e l l e d

HB(X,E)

-I (~f U n) f6A in L.

linear,

and

is o b v i o u s

.

(U n)

be a c l o s e d

topo-

B-bornivorous,

Then:

all

X E Ob TVS

the

symmetry

-- H B ( L , HB(E,X))

and yields E

all

HB(X,Y)), and y i e l d s

(I) By B a n a c h - S t e i n h a u s

rest

and

let

is a c l o s e d

be a B - f u n c t o r .

BorTVS

is an i s o m o r p h i s m

and

(10)

a natural

isomorphism.

X,Y 6 0 b TVS

the

composition

h - HB(h,Y) a natural OB(E,L,X)

transformation. is w e l l - d e f i n e d .

law

297

(2) Let in

AC HB(E,y )

Yo T h e n

A(UE) c Uy.

there

be b o u n d e d ,

is a n e i g h b o u r h o o d

Since

is c o m p l e t e .

There

special

H B. U s i n g ducts

these

that

left

(12)

Definition

Let

Eo,E I ,E 2

a bilinear (I) b

mapping

U2

mappings,

mappings

adjoint

and

a neighbourhood

E2

let

U.

and

we w i l l

in

b(Ao,UI) C U 2

E.

such

and

those

b: E °

pro-

x E I

-- E 2

be

be B - f u n c t o r s .

BoATVS

provided

tensor

H B.

that

i = O,1

for

any n e i g h -

, there

exists

b ( U o , A 1 ) c U2.

, b(Ao,A1)

C A(B 2 E2)

B =B o= B I = B 2

we

if for

every

A i 6 A(B i Ei) ,

holds• say

B-hypocontinuous,

B-bounded.

Remark all B - f u n c t o r s

mapping

listed

in

(4) any

B-hypocontinuous

bilinear

is B - b o u n d e d .

(14)

Proposition

Let

Bo,B I

spaces • For

be B - f u n c t o r s every

bilinear

and

let

mapping

Eo,EI,E 2 b:

be t o p o l o g i c a l

E° x E I ~ E 2

the

vector

following

equivalent:

(I) b

is

(Bo,B1)-hypocontinuous.

(2) bo:

E O -- H B I ( E I , E 2 ) , x ° -- b ( X o , - ) ,

and

any

for

(3) b 1 : E I ~ H B and

to the h o m - f u n c t o r

that

i = O,1

are

h 6W(Ax,UE)

1

(Bo,B1,__BB2)-bounded

(13)

~

that

for all

define

A i 6 A(B i Ei),

is c a l l e d

For

TVS

a neighbourhood

such

spaces, let

Bo,BI,B2:

(2) b

In case

Uy E

belonging

vector

any

1

in holds

(Bo,B1)-hypocontinuous

in

and

to the h o m - f u n c t o r s

be t o p o l o g i c a l

is c a l l e d

bourhood

bilinear

bilinear

are

UE

HB(h,y) (A) c W ( A x , U y )

the p r o o f

are

A x E A(X)

for any

A ° 6 A(B ° Eo)

b°

and

bI

from

continuous,

is e q u i c o n t i n u o u s .

(Eo,E2), x I ~ b ( - , X l ) , is w e l l - d e f i n e d , o A 1 6 A(B I El) b1(A1) is e q u i c o n t i n u o u s .

(4) T h e m a p p i n g s continuous.

bo(Ao)

is w e l l - d e f i n e d ,

(2) and

continuous,

(3) are w e l l - d e f i n e d

and

298

~15)

Proposition

Let

Bo,B I

be B - f u n c t o r s

Then there

exists

continuous

bilinear

the f o l l o w i n g For e v e r y into

and

Eo,E I

a topological mapping

space

~(Eo,EI)

from

L

vector

and a

E° × E I

spaces.

(Bo,B1)-hypo-

into

L

with

property:

(Bo,B1)-hypocontinuous

E 2 , E 2 E Ob T V S , t h e r e

f: L ~ E 2

be t o p o l o g i c a l

vector

such that

bilinear

is a u n i q u e

fT(Eo,EI)

mapping

b

continuous

from

E° × E I

linear m a p p i n g

= b.

(Eo,E I ) Eo ×E I

~ L

E2

Proof: of

Let

L

be the t e n s o r

Eo,E I , e q u i p p e d w i t h the

canonical

bilinear

(16)

Definition

The

L

of

E°

or by

and

EI

Eo ®B El

For fixed

finest

proposition

in case

linear is

topology,

vector

spaces

such that the

(Bo,B1)-equicontinuous.

is c a l l e d ( B o , B 1 ) - t e n s o r T ( B o , B I ) (Eo,EI),

by

product

Eo B o B I El

B = B o = B I.

all t h e s e ~

of the u n d e r l y i n g

~ ( E o , E I)

and d e n o t e d by

Bo,B I

:= T(Bf,Bf)

(17)

mapping

in the a b o v e

T ( B o ,BI) : T V S × T V S ®f

product

TVS,

is u s u a l l y

tensor

called called

products

yield

(Bo,B1)-tensor the i n d u c t i v e

a bifunctor product

tensor

in

TVS.

product.

Proposition

For e a c h B - f u n c t o r the h o m - f u n c t o r Proof:

Look

the t e n s o r

product

T(Bf,B)

is left a d j o i n t

H B-

at the

following ,

i

L

HB (E,X)

diagrams E

L×E

and apply • (L,E)

(14). ~ L B~B f

HB(I ,g) X

to

299

In general

the tensor product

is restricted

to B - b a r r e l l e d

theorem and prop.

T(Bf,B)

is not symmetric,

spaces,it

but if it

is: The Banach-Steinhaus

(14) imply the

(18) Lemma Let

B

be a B-functor,

b: E o x E I -- E 2

topological

Eo,EI,E2

a bilinear mapping.

If

Eo

vector

and

EI

spaces and are B-barrelled,

the following are equivalent: (I) b

is

(Bf,B)-hypocontinuous.

(2) b

is

(B,Bf)-hypocontinuous.

(3) b

is

(B,B)-hypocontinuous.

(4) b

is

(Bb,Bb)-hypocontinuous.

(19) Corollary For e_ny B-functor × BBar

T(B,B)/BBar

the following holds: T(Bf,B)/BBar

× BBar

= T(B,Bf)/BBar

x BBar

=

=

T(Bb,Bb)/BBar

The following p r o p o s i t i o n the t e n s o r - p r o d u c t

®B

× BBar.

implies

that for all B-functors

in

(4)

of two B - b a r r e l l e d

spaces

is B - b a r r e l l e d

Eo,E I

and let the canonical bi-

again. (20) Proposition Let

B

be a B-functor,

linear m a p p i n g

B-barrelled

E ° x E I -- E ° ~B El

be B-bounded.

Then

Eo ®B El

is

string in

Eo ®B El

and

B-barrelled. Proof:

Let

(Un)

A ° C A(B Eo). Then

(U~)

gical.

be a closed B-bornivorous

Define

U In := {x I C El I Ao ® x l c Un}.

is a closed B-bornivorous

The analogously

quently

(Un)

defined

(U~)

string in

El, hence topolo-

is topological,

too.

Conse-

is topological.

(21) Proposition Let

B

modifies

be a B-functor,

R

the coreflector

only the topology.

Let

E,L

for

BBar

be B - b a r r e l l e d

~

TVS,

and

Then the following hold: (I) The canonical mapping ~: HB(E ®B L, Z) ~ HB(E,

HB(L,Z)),

f ~

(x ~

(y ~ f ( x ® y ) ) )

which Z 60b TVS.

300

is a v e c t o r (2) ~

space

isomorphism.

is c o n t i n u o u s ,

(3) ~-I: Proof:

R HB(E,

if

is B - b o u n d e d .

E × L ~ E ®B L

is c o n t i n u o u s .

HB(L,Z.)) ~ HB(E ®B L,Z)

(I) is a c o n s e q u e n c e

(2) ~ ( W ( A E ® A L , U z ) )

of

= W(AE,

(17) and

W(AL,Uz) )

(19). holds

for

AECE,

ALCL,

U Z c Z. (3) The c a n o n i c a l m a p p i n g E ® B L ~ H B ( R H B ( E , H B ( L , Z ) ) ,Z) -I hence ~ is c o n t i n u o u s , too.

is c o n t i -

nuous,

(22) T h e o r e m Let

B

be a B - f u n c t o r ,

mapping

is B - b o u n d e d .

which modifies hom-functor

such that any

Let

R

the t o p o l o g y only.

R HB/BBar

B-hypocontinuous

be the c o r e f l e c t o r Then

product

is a s y m m e t r i c m o n o i d a l

closed

Proof:

(21) and the f o l l o w i n g

(9),

(11),

(20),

bilinear

BBar

~

TVS,

t o g e t h e r w i t h the

BBar

and the t e n s o r

× BBar

for

®B/BBa~

× BBar

category. lemma.

(_~23) L e m m a Under

the a s s u m p t i o n

of the p r e c e d i n g

R HB(E,L) = R HB(E, holds

R HB(E,L)

transfinite (24)

and e v e r y

~ R HB(E,SBL)

induction

L 6 0 b TVS . (see

is c o n t i n u o u s

R H B(E,L)

B

be a B - f u n c t o r

be the c o r e f l e c t o r 19)

E

~ R HB(E,RL)

(5)), h e n c e by

is c o n t i n u o u s ,

too.

Remark

(I) Let R

RL)

for e v e r y B - b a r r e l l e d

Proof:

theorem

for

i m p l y the s u r p r i s i n g R HB/BBar

such that BBa~

theorem

~ TVS.

T h e n the a b o v e

relled,

subcategories

holds

of

for u l t r a b o r n o l o g i c a l

namely

TVS,

quasi-ultrabarrelled

In the same way one g e t s the c a t e g o r y

resp.

symmetric

of l o c a l l y c o n v e x

cally p-convex

spaces,

(7)) , let

t h e o r e m and

× BBar.

too. T h u s we get at l e a s t t h r e e n o n - t r i v i a l closed

(see

= BbBar

fact that

× BBar = R HBb/BBar

2) The a b o v e

BBar

0 < p s I.

spaces (Iyahen ~6]),

symmetric

the c a t e g o r i e s

monoidal

of u l t r a b a r -

ultrabornogical

spaces.

monoidal

subcategories

closed

s p a c e s and of the c a t e g o r y

of lo-

of

301

References [I]

Adasch, N.,; Ernst,B.;

Keim,D.: Topological vector spaces,

Lecture Notes in Mathematics, New York [2]

Berlin Heidelberg

Duske, J.: Analogie zwischen k-R~umen und bornologischen Vektorr~umen,

[3]

Springer,

(1978).

thesis, Universiti~t Kiel

(1967).

Duske, J.: Adjungierte Funktoren in der Kategorie der p-bor nologischen R~ume, Manuscripta Math. 4 (1971), 169-177.

[4]

Eilenberg,

S.; Kelly, G.M.: Closed categories,

Conference on Categorical Algebra, Berlin Heidelberg New York [5]

Fischer,

Iyahen,

S.O.: On certain classes of linear topological spaces, Soc. 18

Ligon, T.: Galois-Theorie Universit~t MHnchen

[8]

(1966), 421-562.

(1963), 242-258.

Proc. London Math. [7]

Schipper, W.J.De.:

(1968), 285-307. in monoidalen Kategorien,

thesis,

(1978). Symmetric closed categories, Mathem.

Tracts 64, Mathematisches Centrum, Amsterdam [9]

Proc. of the

1965, Springer,

H.R.: Uber eine Klasse topologischer Tensorprodukte,

Math. Ann. 150 [6]

La Jolla,

Centre

(1975).

Sydow, W.: Uber die Kategorie der topologischen Vektorr~ume, thesis, Fernuniversit~t Hagen

(1980).

Walter Sydow, Fernuniversit~t,

FB Mathematik und Informatik, Postfach

D 5800 Hagen, West Germany

940

UNNATURAL ISOMORPHISMS OF PRODUCTS IN A CATEGORY V~rs Trnkov~ Prsha I. Introduction. Though n s t u r s l i t y is the essence of the category t h e o r y , c a t e g o r i c a l m e t h o d s c a n be u s e f u l s l s o i n some p r o b l e m s , w h i c h seem t o be f a r f r o m b e i n g n a t u r a l . I n t h e p r e s e n t p a p e r , we i n v e s t i gate "unnsturel isomorphisms of products", i.e. the situstions when products are isomorphic without any natural resson for being isomorp h i c . This field of problems has rather sn old origin. In 1933, S.

Ulam put the problem (see[21~) whether there a r e two non-homeomorphic topological spaces X, Y with homeomorphic squares (this was solved positively in[6]). The implication X~X~Y~Y ~ X~Y is called the unique square root propert7 snd it has been investigated not only in topology, but also for vsrious algebraic snd relational structures, see e.g.[ll].By[9],[lOJ, the unique square root property is vslid in Shy cstegory, mhich hss only a finite set of morphisms between any psir of its objects. On the other hand, there exists e.g. a, countable poset (= partially ordered set) which has 2 o non-isomorphic square roots, by[5].The Tsrski cube property is implied by the unique equsre root property because if X is isomorphic to its cube ~ = X ~ X x X but not to its squsre X 2 = X x X , then X and ~ are non-isomorphic objects with isomorphic squsres. Hence the Tsrski cube property is valid in any category which hss only s finite set of morphisms between any psir of its objects. The Tsrski cube property is not fulfilled e.g. in the category of Boolean slgebrss ([7], [8]), Abelisn groups [4] or topologicsl spaces [12] . The cancellation ( X x Z ~ Y x Z ~ X~--Y), the Csntor-Bernstein property ((X~X~-Zy) snd some other problems concerning iscmorphisms of products hsve been investigsted by many suthors in s lot of pspers for more than fourty years. All these problems sre specisl cases of the investigation of productive representations of semigroups. Let us present the following Definition. Let ~ be s category with finite products, let (~,+) be a commutative semi@roup. A collection iXs ! s ~ S ~ oz" objects of ~ is cslled s productive representation of (S,+) in ~ if (i) Xs is not isomorphic to X s, for s,s'6 S, s~s';

303

(ii)

Xs.s ~

XaxX s

for all s,a'6 S.

Which commutative semi~roupa have productive representations in which categories? This field of problems has been investigated in the ,Seminar from General Mathematical Structures" held in Prague. A brief survey of the obtained results is presented in[14]; a more detailed description with some later results is given in[17] for relational structures and in[18~ for topologi,:al structures. (More general questions - isomorphisms of infinite products, isomorphisms of products and coproducta - are investigated In[14],[2~.) The present paper is also a contribution to this field or problems. We describe some general methods in a categorial language and apply them to functor categories. II. The basic method i. Let o~ be an infinite cardinal.. We say that a category ~C is e6distributive if it has products and coproducts of all collections of the cardinelity _~o~ and fJnl~e products commute with coproducts of collections of the cardinslity

--~__~o , i.e.

(;l~i Xi) ~ (~t~ Yj) -~(~,~)~1i~~ Xix Yj whenever card I ~ o c and card J ~ ~ . 2. Let o, be sn infinite cardinal, let ~

be an oc-distributive

ca-

tegory, let ~ = 4 X ~ I~ eoc~ be a collection of objects of .~ (as usual, ordinals are the sets of all smaller ordinals and the cardinals are initial ordinals, so ~ is also a set of the cardinelity o~ end ranges over it). Denote by oj the set of all finite cardinals. For any map f: oc ~ co put

Xf = ~ X

f(~),

where X f(~) is e product of f ( ~ ) copies of the object X/~ (if f(~) = = O, t h'" e n X f(~~ ; is a terminal obJe:t) and for any A ~ co°c with card A = o¢ put X A = ~e~L%~A(Xf)IB where (Xf)~-- Xf for all /~ e OCo Definition. We say that the collection ~ is oc-productively independent if, for every f ~ co~c and every A c co°c with card A ~ oc XA~-~XflIY for some Y ~ 3. Theorem.

f6A.

Let oo be an infinite cardinal, let ~

be an oC-distri-

butive category which contains an o6-productively independent collection of objects. Then any commutative semigroup S with card S ~ oo has

304

a productive representation Proof. s) The s e t 60 i s

in ~. s commutative semigroup with respect

to

the usual addition + . The set co ~" of all maps of o~ into 6o is also a commutative semigroup, by the rule

(f ,.g)(~)

= f(~)

+ g(~).

Finally the set exp co °~ of all subsets of co~" ture of a commutetlve semigroup, by the rule

admits also the struc-

A ÷ B = ~ f + gl f ~ A, g ~ B } .

All the sets A c co ~c with card A = o~. form s subsemiMroup of the above commutative semiEroup exp ~ , denote it by ~ c " b) What we really prove is that the semigroup ~o¢ has a productive representation in .q~ . Let q~= ~X~ I (~ c-oC } be an o~-p~oductively independent collection of objects of JC , let Xf end X A be as in 2. Clearly, X f + g ~ X f ; < X g snd, since .~ is o0-distributive,

"~ h i

(X-× X )

--~

LL

_(X~)..~ --'X, .~

for every A,B ~ ~ . If f ~ A \ B , than Xf is a s,,m~snd of X A (i.e. X A ~ X f l i Y) but it is not e s,,mmsnd Of X B because ~ is o~-productively independent. We conclude that ~ X A ~A e ~ } is s productive rep r e s e n t a t i o n of 5~c in '~C . c) The proof o£ our theorem is finished by the fact, proved in [14~, t h a t e v e r y c o m m u t a t i v e s e m i g r o u p S w i t h c a r d $ ~ o~ i n t o ~oc"

can be embedded

4 . Remarks. A w e a k e r and i n c o m p l e t e v e r s i o n o~~ t h e above t h e o r e m appears already in[13~ and, as s method for constructions of productive representations, is used in 8 lot of papers, see the quotations below. A similar version for infinite products is in[14~, for sumproductive representations in[2]. Let us mention some results obtained by the described method. A) The categories of unary universal algebras are o~-dietributive for every infinite cardinal ~ . By[1], the category AI@(1) of unary algebras with one unary operation contains an c~-productively independent collection of objects for every infinite oc , hence every commutative semigroup has a productive representation in Alg(1). Some further results concerning the productive representations in the categories of unary algebras, partial el~ebras and their subcstegories can be found in[l]~ ,[.13~ . B) The categories of relational structures and their eubcateMories (like posers, graphs, tolerance spaces) are examined in[17], where

305

also a survey of the previous results with mar~ quotations is given. C) The categories of continuous structures are investigated in [2 , 14 , 15 , 16 , 18 , 19 , 20 , 22 3 For example, every commutative semigroup can be represented by products of metrizable topological or uniform or proximity spaces, by[15] . 5. In the present paper, we apply the described method on some functot categories. If a commutative semi~roup S has e productive representation in a category ~ , then it has s productive representation k also in S~ for every smell (non-empty) category k~ obviously. Hence we concentrate our attention to the categories Set K . Nevertheless, the problem, for which small categories k every (countable) commutative semigroup can be productively represented in Set k, is still far from being clarified. The case that k is s monoid on one generator is investigated in[lJ.Here, we present two theorems with k being a poser (= partially ordered set, considered as a thin category). 6. Theorem. The following properties of a poser k are equivalent. (0) The semigroup ~co o£ all countable subsets of co co has a productive representation in Set k. (i) Every countable commutative semigroup has a productive representation in Set k. (2) ~et k does not fulfil the Tarski cube property. (3) At least one component 0£ k is not 8 finite target. (Let us recall that a target is a poser of the form B u {c J, where b < < c for all b ~ B and every two distinct elements of B are incomparable. ) Proof. (0) ---~(i): Every countable commutative semigroup can be embedded in ~ > , by [14~. (1) ~ (2) i s easy. (2) ~ ( 3 ) : a) F i r s t , l e t us suppose t h a t k i s a f i n i t e tarset B ~c~, denote by mb:b ---> c the morphism of k, b ~ B . We show that Set k fulfils the Tarskl cube property. Let F:k~-->Set be an object, let ~'~:F--~ be a natural equivalence. If x~_F(c) (or x ~ F 2 ( c ) ) denote by ~.(b,x) (or ~(b,x)) the cardinality of the preimage of x in F(m b) (or F2(mb) , respectively). Since F~---~ 3, one can construct, for every element x ~ F ( x ) , an element ~ F ( o ) such that 0
of cardinals put

V~=~X~F(~) W~=~X~(C)

Ic/~(b,x) = ~ b f o r a l l b~B~, ] ~(b,x) = ~ b for all b ~ B ~ .

306

We prove V~-- card W~ f o r e v e r y c o l l e c t i o n ~ • I f ~ i s non-empty, choose x ~ V ~ ; t h e n ~ ( z , ~ ) I z ~ V ~ ~ W~ , hence card V.~.-~ c a r d W~. I f W ~ i s non-empty, choose ( x , y ) ~ Y ~ ; t h e n ~ - l ( x , zl,Z2}~(Zl,Z2)~ e1[~

c V ~ , hence card W ~ card V ~ . This implies W ~ ~ . b) Mow, let us suppose that every component of k is s finite target. If F:k--~ Set is an object of Set k with F ~ , then F/h

N~F

/ 2 f o r e v e r y component h o f k, by a ) , hence F,~F 2.

-- /h (3) ~ ( 0 ) :

Clearly, Set k is c~-distributive. First, we shall construct co-productively independent collections of objects in the following three special cases. a) k I is an infinite target B ,~c~: we may suppose B = co . For every n e co d e n o t e by Xn:k 1 > Set t h e £ u n c t o r such t h a t Xn(n) = ~ 0 , i ~ , Xn(C) = ~0~ = Xn(b) for all b e co\~'n~. Xn(m b) is a constant map for every b e co obviously. We show that ~ = = %X !n e co } is c~-oroductively independent. If f: ~ - - - >
to a set which has precisely 2f(n) points; hen.) tX ~. and X A = ~ A ~ f~n is

ce f can be recovered from Xf. If A c ~ c °

as in 11.2, then every X ~ X A ( C ) determines a subfunctor of X A which is of the form Xf and f can be recognized from it. Thus the set A can be recognized from XA, so ~ is co-productively independent. b) k 2 consists of three objects a, b, c and a < b , a < c : denote by mb:~---~ b and mc:a ---~ c its morphism. ~e may suppose that c is not smaller than b, so either b < c or b and c are incomparable; if b < c , denote by ~:b---~ c the morphism. Choose an increasing sequence ~Pn 1 !n ~ co~ of primes such that po E 2. For every n 6 co denote by Xn: :k2----> Set the functor such that Xn(a) = ~O,l,...,pn} , Xn(b) = {0,I~, Xn(C) = {0~, Xn(m b) sends 0 to 0 and every point of ~l,...,pn ~ to i, Xn(m c) (and Xn(~) if b < c ) is constant. We show that ~ = ~Xn~ n ~ c ~ is cD-~roductively independent. If f: ~ > co is a map and Af = ~ c a ~ n , then f can be recognized from Xf because, for every n ~ co , f(n) is the number of ell x such that its preimage in Xf(m b) consists of Pn points. (Indeed, any such point x has necessarily all the coordinates equal to 0 except one which is equal to i; and this coordinate corresponds to a copy of X n in the product Xf.) If A c ~ end X A is as in 11.2, we decompose the functo~ X A on subfunctors corresponding to the points Y e X A ( C ) and recognize every l e A from these

eXf(b)

subfunctors.

307 c) Now, we f i n i s h t h e p r o o f o f t h e t h e o r e m . I f a p o s e t k c o n t a i n s k 1 o r k 2 , t h e n e v e r y f u n c t o r Xn, d e s c r i b e d i n e) o r b) c a n be e x t e n d e d t o k s u c h t h a t we o b t a i n a n ~ u - p r o d u c t i v e l y independent collection of o b j e c t s i n S e t k . I f k c o n t a i n s n e i t h e r k 1 n o r k 2 ( w i t h b < c o r b and c b e i n ~ i n c o m p a r a b l e ) , t h e n e v e r y i t s component i s n e c e s s a r i l y s finite target. 7. Theorem. Let k be e poset such that there exists an object in it, in which three distinct arrows iniciate. Then every commutative semi~roup has 8 productive representation in Set k. Proof. Since Set k is cO-distributive, it is sufficient to construct, for every infinite c a r d i n a l o~ , s n c ~ - p r o d u c t i v e l y independent coll e c t i o n o f o b j e c t s o f S e t k. L e t t h e t h r e e d i s t i n c t a r r o w s o f k be rob:S---> b, m c : a > c , m d : a - - - > d . Then t h e f u l l s u b c a t e g o r y g e n e r a t e d b y ~ a , b , c , d } h a s one o f t h e f o l l o w i n g f o r m s .

h of k

Hence we may suppose that there is no arrow from b to c and from b to d and there is no arrow from c to d. Let ~ { 3 I ~ e c6 ~ be s collection o f d i s t i n c t cardinsls. We d e f i n e a f u n c t o r Xt3 : h > Set by

X~ (a) =~O,l]u('r~×~}),

X~(,b) =~0}, X~(c) = ~O,l}, X~(d)

= ~0,I,2~,

X/~ (m O) sends O to 0 and all the other points to I, X ~ ( m d) sends 0 to O, 1 to 1 and (x,2) to 2 for every x e ~ a n d , i f t h e r e i s an a r r o w d > c i n h , i t s X ~ - i m a g e s e n d s 0 t o 0 and both I and 2 to i (for the arrows ending in b, their X ~ - i m a g e is evident). We prove that ~ X ~ I ~ eoc~ is sn oc-productively independent collection of objects of Set h . First, we show that every f e co~

can

be recognized from Xf = /~TTocxf(~). For every z ~ X f ( c ) denote M z =

=

(Xf(ma))'l(z) and put L = ~ z e X f ( c ) I card (Xf(ms))(M.) = 2~. One can verify that, for every z eXf(¢) = ~ o c ( ~ O , l } f ( ~ ) ~ , z~L

iff all its coordinates except one are equal to O.

Hence f ( ~ ) is the number of all z e L with card M z = P//3 ÷ I. Now, let A c co ~ , card A = c¢ , X A be as in 11.2. We show that A can be recognized from X A. Indeed, consider all subfunctors of X A determined by the points of XA(b). Each of them is of the form Xf end £ can be

308

recognized from it, hence A can be recognized from X A. Finally, e v e r y f u n c t o r X~ :h ~ S e t can be e x t e n d e d t o a f u n c t o r k ~Set such t h a t t h e c o l l e c t i o n o f t h e e x t e n d e d f u n c t o r s i s an o ~ - p r ~ u c t i v e l y independent

collection

of objects

i n Set k.

III. Pr_oductive representation with a ~iven subob~ect I. Let o¢ be an infinite cardinal, let ~C be an ct_-distributive category. Let ~ = ~ X / ~ i ~ e ~ b e a collection of its objects, let Xf and X A be as in II.2. We say that an object Y of ~C is ~ - s o f t if, for every A,B c co ~c with card A = ~ = card B, x s iff

x,

x B.

2. Theorem. Let oc be an infinite cardinal, ~C be an c~-distributive category, let ce = { X ~ i ~ e oc~ be an ~.-productively independent collection of its objects such that S ~ ( T , X ~ ) ~ ~ for every ~ ~ oc ,where T is a terminal object of JC . Let ~L be 8 class of monomorphism~ closed with respect to the composition and the forming products and containing all coproduct-inJections and all morphisms inioisting in the terminal object T. Let ~cc be as in 11.2. Then, for every ~ soft object Y of ~C , there exists a productive representation ~Ys ~s ~ ~oc ~ of ~ such that Y is an ~t-subobJect of every Ys" Proof. For every A ~ So~ put rA = A ~

Y k × XA"

Then Y is an ~t-subobJect of every YA a~d ~YA IA ~ ~oc ~ is a productive representation of ~ c • Indeed, YA K ¥B ~ YA+B (the reasoning is similar to 11.3); since Y is c~-soft, X A is isomorphic to X B whenever YA'~--YB; but then A = B because c6 is c~-productively independent. 3. In a lot of concrete cases mentioned in 11.4, the co-productively independent collection ~ is constructed such that every object of the category in question can be embedded in a q - s o f t object. This is fulfilled also for the co-independent collections of objects in Set kl , setk2, constructed in the proof of Theorem 11.6. For example, if T :k1 ~ Set is given, we embed it in a functor Y:kl---> Set such that the preimage in Y(m b) of any x e Y(c) is infinite (for every b eB). One can verify easily that X A can be recognized from ~ = = A ~ c ~ y k ~ x A if we consider only subfunctors of ~ , determined by the points x 6 ~ (c) which have finite preimages in the ~ (m b)'sAn analogous reasoning works for k 2 as well. Hence we can enrich

309

Theorem I I . 6 by t h e f o l l o w i n g a s s e r t i o n s , e q u i v a l e n t t o t h e o t h e r s . (4) E v e r y f u n c t o r ~ : k - - , S e t can be embedded i n t o s f u n c t o r i s o m o r p h i c t o i t s cube but not t o i t s s q u a r e . (5) E v e r y c o u n t a b l e commutative s e m i group has a p r o d u c t i v e r e p r e s e n t a t i o n i n S e t k by f u n c t o r s , c o n t a i n i n g a g i v e n f u n c t o r ~ :k---> -- - ~ S e t . A l s o , i f k i s a p o a e t w i t h an o b j e c t a i n which t h r e e d i s t i n c t arrows i n i o l a t e and eeoc= ~X~ I ~ r=~_~ i s t h e ~ - p r o d u c t i v e l y i n d e p e n d e n t c o l l e c t i o n o~ o b j e c t s o f Se t k, c o n s t r u c t e d i n I I . 7 , t h e n e v e r y o b j e c t o f S e t k can be embedded i n a c e ~ - s o f t o b j e c t , hence e v e r y commutative s e m i g r o u p has a p r o d u c t i v e r e p r e s e n t a t i o n i n S et k by f u n c t o r s c o n t a i n i n g 8 g i v e n f u n c t o r ~" :k 4 S e t . IV. How l a r g e a r e t h e r e p r e s e n t i n g

objects?

1. I f we i n v e s t i g a t e p r o d u c t i v e r e p r e s e n t a t i o n s i n a c o n c r e t e c a t e g o r y , t h e r e i s a n a t u r a l q u e s t i o n : how l a r g e a r e t h e u n d e r l y i n g s e t s o f t h e r e p r e s e n t i n g o b j e c t s . I n many c o n c r e t e c a t e g o r i e s , t h e f o l l o w i n g e a s y m o d i f i c a t i o n o f t h e b a s i c method p e r m i t s t o d i m i n i s h t h e c e r d i n a l i t y o f t h e u n d e r l y i n g s e t s : one c o n s t r u c t s t h e c o l l e c t i o n c~ = ~ X ~ ~ ~ e o c ~ such t h a t any X~ c o n t a i n s e d i s t i n g u i s h e d p o i n t , say ~ . , . a n d , f o r any f ~ ~ , Xf i s not t h e whole p r o d u c t ~I'~e~cx f ~ J as in II.2, but only its subobJect consisting of ell those points, which differs from

~f = B e~Yoc ~f(~) in at most finitely

many coordinates. The collection ~ and the distinguished points ~/~ , ~ oc , have to be constructed such that any set A c c~°c with card A --~ oc can be recognized from the object X A being a coproduct o f t h e s e new X f ' s . 2. If this modification is combined with the application of Theorem Xll.2, one can obtain, for example, the following assertions: every countable commutative semigroup has s representation by products of a) b) c)

countable topological spaces, containing a ~iven countable space (see[20]), countable posers, ~raphs, tolerance spaces, containing a given countable poset, graph, tolerance space (see [17~), countable unary algebras, containing a given countable unary al~ebra.

S. Let us show an application of this idea on the category (Set~) k, where Setco denotes the category of all countable sets (and k is a poser)° We prove the following

310

_Proposition. The assertion (6) below is equivalent to (1)...(5) in 11.6 and 111.3. (6) Every countable commutative semi~Toup has a productive rePresentation in (Set~) k by functors, containir~ a given functor ~J{: :k ~ S e t ~ . Proof. If • poser k contains k I (or k 2), define X n as in I1.6. Let c~n be its suhfunctor, sending any object of k I (or k 2) to a onepoint set, namely O-n(p) = tO} for every object p of k I (or k 2) and define Xf to be the subfunctor of ~ = ~n n such that, for every object p of k I (or k2, respectively), X~(p) consists of those x e f(n)( p) "at most in finitely coordie ~ ( p ) , which differ fr om ~ O ~ n nates. Then X f x X is still isomorphic to Xf.g end f still can be recognized from Xf ~by the same reasoning as in II.G°). The rest of the Proof is the same ee in II.~. end III.3.

4. Remark. L e t us m e nt i on one t r i c k more, which p e r m i t s t o o b t a i n the following assertions: every countable graph (poset, tolerance space, unary algebra, topological space) can be embedded into a countable graph (poser, tolerance space, unary algebra, topological space) which has 2 o nonisomorphic square roots [17], [18]) and also the assertion if a poser k contains k I or k 2 from 11.6, then every fUnctor ~" :k--,Seto~ is a subfunctor o£ some X:k---> Set~o , which has 2 xo non equivalent square roots. This follows immediately from the above results and the Proposition. The semigroup 5~o~ of all countable subsets of co ~ contains a subset T such that card T = 2 ~° and s + s = s" + e" for every s,s'e T° Proof. Let S be a semlgroup with a countable set of generators, say {shin 6 60 ~ and defining equations s n + s n = s n + an. for ell n,n'~ c~ • By[14~, there exists e disjoint homomorphism h:~ ...... ~ 5 ~ i.e. h(s)~h(s') -- ~ whenever s=@s ". Pu~ T : {~_~eAh(Sn)IA ~ , A4=~,

t h e n T has t h e r e q u i r e d p r o p e r t i e s . References

1. 2°

J . Ad~mek, V. Koubek. On • r e p r e s e n t a t i o n o f eemi~roupa by p r o d u e t s o f a l g e b r a s and r e l a t i o n s , C o l l . Math. 3 8 ( 1 9 7 7 ) , 7 - 2 5 . J° A d ~ e k , V. Koubek, R e p r e s e n t a t i o n o f o r d e r e d commutative s e m i g r o u p s , C o l l . Math. Soc. J e n o s B o l y s i 20, A l g e b r a i c t h e o r y o f s e m i ~ r o u p e , Szeged 1976, 15- 31.

311

3. 4.

5. 6. 7. 8o 9o i0. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

J. Ad~mek, V. Koubek, V. Trnkov~, Su.~ of Boolean spaces represent every group, Pacific J. Math., 61(1975), i-7. A.L. Corner, On a conjecture of Pierce concerning direct decomposition of Abelian groups, Proc. of Coll. on Abelisn groups, Tihany, 1963, 43-48. Sur l'~quation n = pour types d'ordA.£. C.R. DavisAcsd. (Morel),, des re, Sci. Paris 235(19521 ~, 924-~926. R.H. Fox, On a problem of S. Ulam concerning Cartesian products, Fund. Math., 34(1947), 278-287. W. Hanf, On some fundamental problems conceding isomorphisms of Boolean algebras, Math. Scand. 5~1957), 205-217. J. Ketonen. The structure of countable Boolean algebras, Annsls of Math., I08(1978), 41-89. L. Lov~sz. Direct product in locally finite categories, Acts Sci. Math., 3311972), 319-322. A. Pultr, Isomorphism types of objects in categories determined by numbers of morphieme, Acts Sci. Math., 35(1973), 155-160. A. Tsrski, Cardinal algebras; with an appendix by B. Jonsson and A. Tsrski, Cardinal products of isomorphism types, New York, 194~ V. Trnkov~, X n is homeomorphic to X m iff n..4n, where ~ is a congruence on natural numbers, Fund. Math. 80(19~3), 51-56. V. Trnkov~, Representation of semigroups by products in s category, J. Algebra, 34(1975), 191-204. V. Trnkov~, Isomorphism- of products and representation of commutstive semigroup, Coil. Math. Soc. Janos Bolysi 20, Al~ehraic theory of semigroups, Szeged 1976, 657-683. V. TrnkovA, Productive representations oi" semi~Toups by pairs of structures, Comment. Math. Univ. Carolinae 18(1977), 383-391. V. Trnkov~, Cstegorial sspects ere useful for topology, Lecture N. in Math. 609, Springer-Verlag 1977, 211-225. V. Trnkov~, Cardinal multipllcstion of relational structures, Coll. Math. Soc. Jsnos Bolyai 25, Algebraic methods in Graph theory, Szeged 1978, 763-792. V. Trnkov~, Homeomorphisms of products of spaces (in Russian), Uspechi Math. Nauk 34(1979;, vyp. 6(210), 124-138. V. Trnkov~, Homeomorphisms of powers of metric spaces, Comment. Math. Univ. Carolinse 21(1980), 41-53. V. TrnkovA, Homeomorphisms of products of countable topological spaces, to appear. S. Ulam, Problem, Fund. Math. 20(1933), 285. J. Vin~rek, Representation of countsble commutstive semigroups by products of weakly homogeneous spaces, Comment. Math. Univ. Carolinae 21(1980), 219-229.

CATEGORIES

O F KITS,

COLOURED

GRAPHS,

AND GAS~S

by Antoni

O.

Introduction The main

category

games

between

games,

there them.

an a b s t r a c t

game which

combination

of t w o

automaton Section

and

different one may these

shall

consider

deal here with

between

prove

by Section

graph.

e.g.,

confine ways

categories

ourselves

to

of d e f i n i n g

be way

games,

notion

of

case of the

shown

in

at l e a s t

16

and consequently

of abstract in g a m e

some g e n e r a l

games,

of an output-state It w i l l

abstract

and

a category

the g e n e r a l

in a n a t u r a l

to b e u s e f u l

4 where

of g a m e s ,

as a p a r t i c u l a r

the n o t i o n

to d e f i n e

16 d i f f e r e n t may

natural

of a coloured

of morphisms

categories

if w e

various

notions:

of applying

noncooperative

However,

may be considered

simpler

it is p o s s i b l e

types

illustrated

We

sorts

of g a m e s ,

of c l a s s i c a l

etc.

a method

are many

categories

are a l s o

the n o t i o n

3 that

There

many

a category

of dynamic type,

is to o u t l i n e

theory.

define

games,

of one

morphisms

this p a p e r

in g a m e

one may

of two-person a category

and preliminaries

aim of

theory

accordingly

Wiweger

games.

theory

as

constructions

Some of

it is of p r o d u c t s

and

coproducts of abstract g a n ~ s a r e descri0ed, and b y S e c t i o n 5 w h e r e t h e i n t e r p r e t a t i o n of these constructions i n t h e p a r t i c u l a r c a s e of t h e t w o w e l l - k n o w n t w o - p e r s o n games isgiven.

if

We use

the t e r m i n o l o g y

(At) tC T

is an i n d e x e d

the d i s j o i n t also write identify same

s u m o f the

If

A I + A 2.

TO a v o i d sets w i t h

shall

f :A

, B

A x B

. B denotes

P o w (A) Pow+ set

tacitly

of

o f sets,

At .

Instead

cumbersome their assume

[I]

and

then of

In p a r t i c u l a r , will

StE{I,2}A t

notation

images

[5].

StcTA t

we

in d i s j o i n t

if n e c e s s a r y

that

denote

we

shall

shall

sometimes

sums;

for the

the s e t s

in

are disjoint.

the r e s t r i c t i o n and

sets

considered

reason we

question

and notation family

functor

of

f

will the

denotes and

is a f u n c t i o n

Pow

to

C.

and

C c A,

The canonical

be denoted

by

pr I

then

fI C

projections

and

pr 2

will

denote

A x B

, A

respectively.

set o f all r e a l n u m b e r s . the p o w e r denote

respectively.

set of

A.

the c o v a r i a n t

and

the c o n t r a v a r i a n t

power

313

I.

Output-state

automata,

An output-state

(I)

K =

where

A,X,Y

Y.

are

Every

next-output

A kit

(of s t a t e s , Y

to

output-state 1

A,

inputs, and

automaton

and the

1

and outputs,

respectively),

is a f u n c t i o n

(I)

is a M e a l y

next-state

[4])

function

is a n o u t p u t - s t a t e is the c a n o n i c a l

from

A x X

automaton

function

automaton

projection

any output-state

automaton

(I) w e

= { (x,x') E X x X I V a E A k ( a , x )

It is o b v i o u s

that

A monokit relation

on

~

with

defined

the

as

(I)

onto

such

the

that

first

the

axis,

(2)

M =

where

X



(cf.

graph

is a f u n c t i o n

and

from

the d

from

G =

that

graph.

relation

such

that

on



X. is the i d e n t i t y

is a t r i p l e

D

(of v e r t i c e s into

the

set

(x,x') C p(d)

x

to

and colours

respectively),

P o w ( X × X)

means

that

of all

there

subsets

is an a r r o w

and of

of

x'.

is a 7 - t u p l e

(A,X,Y,~,I,D,p)

UIG =

(A,X,Y,~,I)

The pregame

condition

sets

condition

A pregame

(3)

is a k i t K

}.

(X,D,p),

are

colour

= k(a,x'

is a n e q u i v a l e n c e

[4])

D

X x X;

define

X.

A coloured

such

and games

z o I = p r I. For

p

graphs,

~ = ~ 0 I.

(cf.

next-state

sets from

function

the c o m p o s i t e

i.e.

coloured

is a q u i n t u p l e

(A,X,Y,~,I) ,

is a f u n c t i o n to

kits,

automaton

(3)

is a k i t a n d

is r_egular if

U2G =

for all

(X,D,p)

is a c o l o u r e d

Xl,X~,X2,X ~

(Xl,X ~) C < U I G > & (x2,x ~) E < U I G > & (Xl,X 2) E p(d)

in

X

the

implies

(x{,x~) E p(d) •

UIG

An

a__bstract

is

a monokit.

A pregrame then

~

equivalence

is

a

game

(a ~ame

It (3)

is

is

surjection

classes

for

obvious

short) that

n__on-degenerate and

~-1({a}),

yields a CA.

is

every if

a pregame game

X ~ ~.

a partition

(3)

is If of

such

a regular it

the

is set

that pregame.

the

case,

Y

into

314

Every The

non-degenerate

elements

of

The e l e m e n t s

of

Active lose

players

in t h e

that

y

D

are passive

result

The e l e m e n t s

their

X

means

are

situation

x.

situation

The

players.

condition

(x,x') C p(d)

situation

x

The

over

notion

The

the

F =

where

and

(Ya) a C A

strategies subset

of

A

of

p : D

a.

cases

sense

the p a y o f f Every

and

~acAYa

that various

o f this

k(a,x)

game

and

have

of sets.

that each by

function;

player

d

the

prefers

the

equivalent

to

[6].

notion.

If,

that

A x X

>H(a,x)

noncooperative

and O.Morgenstern

is the

to e a c h

a-th

every

(3) o n e m a y

associate

game.

Y

game

(a g a m e

the e l e m e n t s H(a,x)

as an a b s t r a c t union

of is

game

of the sets

the c o r r e s p o n d i n g

index

x.

the c__anonical p r e s e n t a t i o n

precisely,

the V o r o b ' e v

that

x).

of the e l e m e n t

game has More

X =~acAYa such

(the n u m b e r

is the d i s j o i n t in

coordinate

abstract

f o r m of a V o r o b ' e v

Y y

[3]),

function

situation

(3), w h e r e

on

are

> H ( a , x ' ) ],

(4) c a n be p r e s e n t e d

form

of g a m e s A = D,

a in the

assigns

types

defined

of the p l a y e r

~

as above,

is a n o n - e m p t y

in p a r t i c u l a r ,

is the p a y o f f

game

X

H

(4) is a c l a s s i c a l

H

same meaning

is the s e t o f a l l

.

important

function

Neumann

the Ya

Vorob'ev

Conversely,

game

means

strategies

is e s s e n t i a l l y

it is a s s u m e d

product

[6]

of J . v o n

in the

(a E A ) ,

in t h e

here

. P o w ( X x X)

is a r e a l - v a l u e d

are players,

Ya

a in t h e

is the p r e f e r e n c e

by N.N.Vorob'ev

family

Moreover,

in

the V o r o b ' e v

G = @(F)

o f the

a.

means

x'

V a C A V x , x , C X [ (x,x ') 6 p(a) <

in t h e

of the p l a y e r

y = l(a,x)

is a m o n o k i t choice

gain or

strategies.

is a q u i n t u p l e

the c a r t e s i a n

particular

then

are

the p l a y e r

the p a s s i v e

presented

is an indexed

It is s h o w n

and there

Y

(A,X, (Ya) a c A , D , p ) ,

A,X,D,

while

situation

by

UIG

p

of

of i n t e r e s t s ) . players

is a s t r a t e g y

b y the

that

introduced

A V_orob'ev g a m e

(4)

means

of a c t i o n ) .

passive

condition

chosen

function

of a g a m e

of a game

y The

that

determined

all active

the n o t i o n

actually

interpretation.

(or c o a l i t i o n s while

The elements that

requirement

is u n i q u e l y

following

(or c o a l i t i o n s

players

situations.

is the s t r a t e g y

the

strategies,

of the game.

a = z(y) of

(3) h a s

are a__cctive p l a y e r s

choose

The c o n d i t i o n

game

A

game

with

every

abstract

a,

,

315

H(G)

where

~

=

(A,~(X), (7 -I ( { ~ ) ) a E A , D , P o w + ( l

is the

the b i j e c t i o n Remark. also

and

features

the t h i n g s elements

of

Y

(3) are

D

yA

induced

induced

by

Following

[4] w e m a y

given

the p r e f e r e n c e

regular

imagine

features,

of

2.

Categories Let

K =

of

automata. K'

A first

is any

functions

kind

the e l e m e n t s and

K'

morphism

triple

(f,K,K'),

fA : A

, A',

=

fx : X

d

nad categories

f =

has

does

of

the

his o w n condition thinqs

but

not

of k i t s

for short)

fy : Y

values

be o u t p u t - s t a t e

(fA,fx,fy)

, X' ,

then

on a c t u a l

(A',X',Y',z',I')

the

are u s e r s

regularity

(the u s e r

(a 1 - m o r p h i s m

where

user

not d e p e n d

automata

and

D

are m o t o r - c a r s ,

p(d) ;

of

admissible

the set

each

of o u t p u t - s t a t e (A,X,Y,,z,I)

is

can be

by the r e l a t i o n does

~

of the

that

d e p e n d s o n l y on v a l u e s of f e a t u r e s of t h i n g s d i s t i n g u i s h b e t w e e n two i d e n t i c a l things).

and

programe

of m o t o r - c a r s ) ;

relation

I,

interpretation

things,

X

by

~.

the

the e l e m e n t s

the e l e m e n t s

are b u y e r s

criterion

that

way.

in

e.g.,

to

of a n o n - d e g e n e r a t e

given in

X

~(X)

respectively;

(if,

preference

from

onto

in a n o t h e r

of a k i t

sets A,X, of the

X

The n o t i o n

interpreted

notion

says

function

from

× I) 0 p) ,

from

K

to

is a t r i p l e

~ Y'

such

of

that

the

the d i a g r a m

A×X

. Y

A'xX'.---~-r-.a. Y'

. A

~,.

A'

is c o m m u t a t i v e . A second is any

triple

fA : A'

. A,

kind morphism (f,K,K'), fx : X

(a 2 - m o r p h i s m

where ~ X',

f =

fy : Y'

AxX

, Y

AWxXw.......-..:-~

y'

that

. A

fY

fA

3,1 t

,

from

K

is a t r i p l e

such

Y

A' xX

is c o m m u t a t i v e .

for short)

(fA, fx, fy)

A ~

to of

K' functions

the d i a g r a m

316

For -state

i = 1,2

automata

let

os-Aut, be the c a t e g o r y w h i c h has all o u t p u t 1 and all i - m o r p h i s m s as arrows. The c o m p o s i t i o n

as o b j e c t s

of a r r o w s

is d e f i n e d

composite

of two 2 - m o r p h i s m s

The c a t e g o r y and the c a t e g o r y Some p r o p e r t i e s

3.

Categories Let

M =

in an o b v i o u s w a y and it is easy

os-Aut i

has

the full s u b c a t e g o r y

of c o l o u r e d (X,D,p)

graphs

and

M' =

be a s u b s e t of the set

and categories (X',D',p')

{1,2,3}. where

A

fx : X

. X',

fD : D

(ci, j)

are s a t i s f i e d

(ci, I)

VdEDPOW+(fxX

(Ci, 2)

V d E D P O W + ( f X × fx ) (p(d)) D P ' (fD(d)) ,

(ci, 3)

V d E D P O W _ ( f x × fx ) (p' (fD(d)) c p ( d ) .

: D'

f =

(fx,fD)

j

in

M

to

M'

conditions

fx : X

are s a t i s f i e d

, X',

fD : D' j

(c2, 3)

V d , E D , P O w _ ( f x x fx) (P' (d')) c P(fD (d')).

morphismsof Note graphs

a triple and

ion

while

relational

D = D',

and a

p : D S' D'

systems

every

fD = ID"

in the s e n se of

In fact, where

, Pow(X)

satisfying

and

D

transformations

if we r e g a r d

of c o l o u r e d of t o p o l o g i -

a topological

s p a c e as

is the f a m i l y of the o p e n s u b s e t s

transformation

similar S

fD : D'

of

then e v e r y o p e n t r a n s f o r m a t -

a p a i r of f u n c t i o n s

the c o n d i t i o n s

X',

relational

[2].

(2,{1,3})-morphisms

is the i n c l u s i o n ,

fx : X - -

coincide.

are i d e n t i c a l w i t h s t r o n g h o m o -

m a y be i d e n t i f i e d w i t h

continuous

such

~:

In this case the

(2,~)-morphism

to o p e n and c o n t i n u o u s

(X,D,p),

a p a i r of f u n c t i o n s

where

, D

in

identical with homomorphismsof

(1,{1,2})-morphisms

respectively. S =

S --

fD : D - -

case

(1,{1,2})-morphisms

are a n a l o g o u s

cal s p a c e s

X

the s p e c i a l

(1,~)-morphism

that

M'

fx) (P(fD(d'))) C P' (d'),

(1,{1})-morphismsare while

(f,M,M'),

for each

V d , 6 D , P O w + ( f x × fx) (P(fD(d'))) D P ' (d'),

systems,

to

conditions

is any t r i p l e

(c2, 2)

of a

M

e:

Vd,ED,POw+(fx×

Moreover,

from

and let

is a p a i r of f u n c t i o n s

(c2, I)

notions

graphs

fx) (0(d)) C 0 ' (fD(d)),

is a p a i r of f u n c t i o n s

that the f o l l o w i n g

Consider

of all kits,

of games

be c o l o u r e d

such that the f o l l o w i n g

for each

(2,~)-m_oorphism f r o m

(fx,fD)

Kt i

(1,e)-morphism

(f,M,M'),

f =

that the

Kt. has the full s u b c a t e g o r y mKt. of all m o n o k i t s . 1 1 of the c a t e g o r y Kt I are d e s c r i b e d in [4] and [7].

is any t r i p l e

A

to c h e c k

is a g a i n a 2 - m o r p h i s m .

, S' - D

to

fx : X

(ci, I) and

, X',

(ci,2),

m a y be i d e n t i f i e d w i t h satisfying

the conditions

317

similar

to

For which

has

arrows.

The

with

Pgai, ~

and

functions

(3)

triple

way,

(f,G,G'),

fA : A --~ A'

f =

fx : X -- X'

t

way.

~ Set

forgetful

VI(A,X,Y,z,I)

Set

of

pregame

where

as

in an o b v i o u s be

the

= V2(X,D,p)

can n o w be d e f i n e d

V2

the o b j e c t s

to an a n a l o g o u s

category

(i,~)-morphisms

by

= X.

the p u l l b a c k

~ Kt i

Cgri, ~

from

all

is d e f i n e d

of p r e g a m e s

be the

Cgri, s and

V 2 : Cgri, a

Pgai, ~

In an e x p l i c i t

let

as o b j e c t s

of a r r o w s

~ Set

the o b j e c t

category

{1,2,3} ,

ac

graphs

composition

V I : Kt i

functors

(c2,3).

and

all c o l o u r e d

Let

The

(c2, 1) and i 6 {1,2}

.

Pgai, ~ G'

=

are all p r e g a m e s .

(A',X',Y',~',I',D'

(fA, fx,fY,fD)

fY : Y ~

,

Y'

t

An a r r o w

,p ' )

is a q u a d r u p l e

is any

of

functions

fD : D -- D'

in the c a s e

i=I

fD : D' -- D

in the

i=2,

I

and

fA : A'

such

~

that

automata

A,

fx : X -- X',

fY : Y'

((fA, f x , f y ) , U I G , U I G ' ) and

-- Y'

is an i - m o r p h i s m

((fx,fD) ,U2G,U2G')

is an

case

of o u t p u t - s t a t e

(i,~)-morphism

of c o l o u r e d

graphs. The Since

category

i

three-element

set

ent c a t e g o r i e s

4.

Products Let

(5)

T

(D,X,p)

{1,2,3},

a set and

family

we

have

in some

subcategory

Gai, e

is an a r b i t r a r y thus

obtained

of games.

subset

of the

2 • 2 3 = 16

differ-

categories

(coproduct

in

respectively) t 6 T,

Cgri,e, are

t C T It is o b v i o u s

respectively)

is a p r o d u c t

(coproduct

of @ a m e s

let

of p r e g a m e s .

(U2Gt)t6 T G~,

full e

(At,Xt,Yt,zt,lt,Dt,Pt),

is a p r o d u c t

all

the

of games.

be

Gt =

has

I or 2 and

and coproducts

be an i n d e x e d

and

Pgai, ~

can be e i t h e r

(coproduct

then

G =

of the

games

and

the

family

respectively)

(A,X,Y,~,I,D,p)

family G

of

that

(5) in

happens

if

(A,X,Y,~,I)

(UiGt) t6 T of the

in

Kt i

family

is a p r o d u c t

Pgai, ~.

to be a game,

If, m o r e o v e r , then

G

is

318

a product

(coproduct

This In e a c h

general

case

the

product

are

defined

ion

of a coproduct A)

any

which

family

A =[~tcTAt , the

function

~

function

of

The

: Y

. A

(5)

I : A x X

there

as

the

d =

(dt) tE T

function

p(d) B) in

p

in

=

For

Yt

is d e f i n e d

. P o w ( X × X)

the

can

A = StETAt,

respectively) the

construct-

exists (3),

,

a product

of

(5)

in

where

D = ~tETDt

,

in

Yt'

by

for

assigns

a t in A t a n d

x t in Xt,

to an e l e m e n t

set

family

which

on

below•

by

for

{ ((x t ) t C T , (xl)tET) any

Ga1,{1},

: D

D

Ga.

in A)-D)

injections

is b a s e d

a game

k((at) tET, ( x t ) t E T) = ( k t ( a t , x t ) ) t E T and

in

[71.

is d e f i n e d

, Y

B)

Y = ~tETYt

(zt(Yt))tET

(5)

listed

(coproduct

games

,

family

results

result

in

constructed

the

the

given

X = ~tETXt

~((Yt) t E T ) = the

way.

of k i t s

can be

of

yields

projections

in an o b v i o u s

For

Ga1,{1}'

respectively)

procedure

E X × X I VtE T ( x t,xl)

of

games

be

constructed

X = StcTXt ,

(5)

there as

exists a game

E pt(at) }" a coproduct

of

(5)

(3), w h e r e

y = S t E T Y t + S t , u c T ( A u x Xt) , t#u

D = StETDt, The

function

Section t

in

T,

T,

C) Ga2,{1}'

= At

u ~ t,

PlDt' = Pt

: Y

nlY t = u ~ t,

hi (A t × X t) in

~

O)

For

t any

which

nt the

for and

for

, A for

the

in

in

by

T,

(cf.

~I (Au × X t)

: A × X T,

function

the

. Y

II (Au × X t) p : D

conventions

in

= pr I

u

for

is d e f i n e d

= IAu×X t

, P o w ( X × X)

for

and

by u

and

is d e f i n e d

t by

T.

family c a n be

A = StETAt ,

t

function t

in

is d e f i n e d

of

games

constructed

X =]-[tETXt ,

(5) as

there

exists

a game

Y = StETYt ,

a product

(3), w h e r e

D = S t E T D t,

of

(5)

in

319

the

function

~

the

function

I : A × X

for

u

in

p : D

: Y

T,

, A

au

in

, P o w ( X x X)

is d e f i n e d , Y

Au,

by

(xt)tE T

assigns

~IYt

is d e f i n e d

by

in

= nt

for

t

in

T,

k(au, (xt)t6 T) = k u ( a u , X

X,

to an e l e m e n t

the d

u)

function

in

D

c D

(u 6 T) the set

U

p(d) D) (5)

= { ((xt) tET, (xl)tE T) C X × X I (Xu,X u) 6 P u ( d ) } .

For

in

any

family

Pga2,{1},

of p r e g a m e s

which

can

be

the

set

(5)

there

constructed

exists

a coproduct

as a p r e g a m e

of

(3), w h e r e

X A

is

the

subset

of

(~tETAt)

×~t,uET(Yt

consisting

u)

tgu of

all

elements

a =

VtETVuETVx

u

((at) tCT, (~t,u : Xu

EX

u

~t(~t,u(Xu

,. Y t ) £ , u 6 T , t # u

such

)

that

)) = a t '

L

X = StETXt, X Y

is

the

subset

of

the

set

(~tETYt)

×~t,uET(Yt

u)

consisting

t#u of

all

elements

y =

VtETVuETVx

D =~tETDt the

function

the

=

function

l(a,x)

EX ~ t ( ~ t , u ( X u u

: Xu

, Yt)t,uET,tju

)

such

that

)) = ~ t ( Y t ) '

, ~

z(y)

u

( ( Y t ) t 6 T , (gt,u

: Y

A

((~t(Yt)

is d e f i n e d

tET, (~t,u

I

: A × X

, Y

=

( ( y ~ ) t E T , (~t,u

by

: Xu

' Yt)t,uET,t~u

is d e f i n e d

: Xu

),

by

Yt) t , u C T , t # u ) , V

where in

v

T,

assigns Pt(dt), the set (3)

is

the

t ~v,

index

and

to an

y~

element

pregames

coproduct

need

contains

of g a m e s

coproduct

T

such

that

x E X v,

Yt

= ~ t , v (x)

= kv(av,X),

and

the

d =

the

(disjoint)

union

then,

contrary

to the

(dt) tE T

function

p : D of

for

t

, Pow(XxX) the

sets

t 6T.

If all

Yt

in

of

(5) not

at

are

be

least

games

a game. two

(5)

in

Pga2,{1 }

(5)

in

G a 2 , { 1 }.

However,

elements

if

and

is a game,

and

for

each

X t ~ ~,

cases t

then

consequently

A) - C),

in

T

the

coproduct

is a

the

320

5.

An

example

We C)

shall

described

now

discuss

in S e c t i o n

cooperative

two-person

as V o r o b ' e v

games

I and Si

4. L e t games

us

case consider

chess

and

the

constructions two well-known

draughts

(= c h e c k e r s )

( { 1 , 2 } , S I x $2, ($I,$2) ,{1,2}, (pl,P2)) ,

draughts

=

( { 3 , 4 } , S 3 x $4, ( $ 3 , $ 4 ) , { 3 , 4 } ,

2 are

chess

we

to e a c h

induced

on

players,

by

is

the

opening,

the

while set

understand

the

here

i.e.

board.

payoff

of

and

nonregarded

each

4 are

(p3,P4)). draughts

strategies

a full

to

The

3 and all

of

procedure

players.

the

which

appropriate

preference

player

i.

assigns

the

sequence

relations

Pi

of

By next

consecutive

(i = 1 , 2 , 3 , 4 )

are

functions

H (ch) : {1,2} x (S I x S2 )

defined

B)

(4) :

(i = 1, 2, 3, 4)

IR ,

H (dr) : {3,4} × (S 3 x $4)

. ]R

as wins

I 1/2

H(i, (S ,s' )) =

if

i

The

coproduct

whose

chess

U

draughts

in the

follows

B)

(cf.

the

situation

(s ,s t

loses

presentation

as

in

~draws

0

defined

the

=

positions

game

of

chess

a strategy move

a special

form

in t h e of

in S e c t i o n

category

a Vorob'ev

Ga1,{1 } game

(4)

is

can

the

be

4) :

A = D = {1,2,3,4}, X = X (ch) U X (dr) ,

the

x(Ch)

= { ( s 1 ' s 2 , (3, ( S l , S 2 ) ) , ( 4 , ( s 1 ' s 2 ) ) )

X (dr)

= { ((I, (s3,s 4) ) , (2, (s3,s 4) ) ,s3,s 4) I (s3,s 4) 6 S 3 × S 4},

YI

= $I + {I} x ($3 x $4),

Y2

= S2

+

Y3

= S3 +

Y4

= S4

+ {4} x (S I x S2) ,

function

function

I (s1's 2) E S I x $2}'

H

{3} × (S I × S2),

p : A : A x X

' P o w ( X x X) o

, ~

,

where

is

{2} × (S 3 x $4) ,

induced

by

the

partial

payoff

321

= H(Ch) (a, (Sl,S2)) is u n d e f i n e d

for

for

a 6 {1,2}

a 6 {1,2}

and

and

x = (Sl,S 2 .... ) 6 X (ch) ,

x 6 X (dr),

H(a,z) = H(dr) (a, (s3,s4)) is u n d e f i n e d

for

for

a 6 {3,4}

We see that the c o p r o d u c t four-person players

game.

a e {3,4}

The p l a y e r s

and

chess ~ draughts I and

imagine

playing,

2 intend

of p l a y e r s

If the c h e s s p l a y e r s

choose

players

must choose

(4, (Sl,S2))

respectively,

and a g r e e w i t h s t r a t e g i e s p a y o f f of the d r a u g h t s The p r o d u c t

(cf. C)

sI

and

the

I and 2 (we m a y

s2

atthe board). respectively,

(3, (Sl,S2))

that they stop p l a y i n g

then

and draughts

in this case

the

undefined.

in the c a t e g o r y

in the f o r m of a V o r o b ' e v

in S e c t i o n

looking

the s t r a t e g i e s

remains

chess ~ draughts

whose presentation f o l l o ws

and vice v e r s a

c h o s e n by the chess p l a y e r s ;

players

is a

h a v e o n l y one c o m m o n b o a r d and

strategies

what means

Ga1,{1 }

to p l a y chess w h i l e

are not a b l e to p l a y w i t h o u t

the d r a u g h t s

in

but if the p l a y e r s

then 3 and 4 m u s t wait,

that the two p a i r s

that the p l a y e r s

x = ( .... s3,s 4) 6 X (dr),

x 6 X (ch)

3 and 4 i n t e n d to p l a y d r a u g h t s ,

are a c t u a l l y

and

game

Ga2,{1 }

is the game

(4) can be d e f i n e d

as

4):

A = D = {1,2,3,4}, X = S I × S 2 × S 3 × S 4, Yk = Sk the f u n c t i o n H : A× X

for

k = 1,2,3,4,

p : A

P o w ( X x X)

~ ~ ,

is i n d u c e d by the p a y o f f

I = H (ch) (a, (s 1,s 2))

H(a, (Sl,S2,S3,S4))

Thus the p r o d u c t four-person

function

where

game w h i c h

for

a 6 {I,2},

for

a C {3,4}.

i = H(dr) (a, (s3,s4))

chess ~

draughts

in

can be c o n s i d e r e d

Ga2,{1 }

is a n o n c o o p e r a t i v e

as a " d i s j o i n t

sum" of chess

and d r a u g h t s . U s i n g the c o n s t r u c t i o n s

A) and D) g i v e n

similarly

describe

the p r o d u c t

coproduct

chess U

coproduct

can be e a s i l y p r o v e d ) .

draughts

in

is S e c t i o n

chess ~ d r a u g h t s Ga2,{1 }

in

4 one may

Ga1,{1 }

(the e x i s t e n c e

and the

of this last

322

References [I]

S.MAC LANE, Categories for the w o r k i n g mathematician, -Verlag, New York - Berlin, 1971.

[2]

A.I.MAL'CEV, - Heidelberg,

[3]

J.von NEUMANN and O.MORGENSTERN, Theory of games and economic behavior, P r i n c e t o n U n i v e r s i t y Press, Princeton, 1947.

[4]

Z.SEMADENI, categories,

[5]

Z . S E ~ D E N I and A.WIWEGER, K a t e g o r i e n und Funktoren, Leipzig, 1979.

[6]

N.N.VOROB'EV, The present state of game theory, Uspehi Mat. Nauk 25 (1970) no. 2 (152), 81-140 (Russian), (English t r a n s l a t i o n in Russian M a t h e m a t i c a l Surveys 25, no. 2, 78-136).

[7]

A.WIWEGER, (1976).

A l g e b r a i c systems, 1973.

Springer-Verlag,

Springer-

New York -

On classification, logical e d u c a t i o n a l materials, and automata, Colloq. Math. 31 (1974), 137-153. E i n f H h r u n g in die Theorie der BSB B.G. Teubner V e r l a g s g e s e l l s c h a f t ,

On concrete categories,

D i s s e r t a t i o n e s Math.

Institute of M a t h e m a t i c s of The Polish Academy of Sciences ~ n i a d e c k i c h 8, P . O . B o x 137, 00-950 Warszawa, Poland.

135