Completions of Categories

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann,...

Author: Joachim Lambek

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Forschungsinstitut for Mathematik, ETH, Z0rich 9Adviser: K. Chandrasekharan

24 Joachim Lambek McGill University, Montreal Forschungsinstitut for Mathematik, ETH, Z(Jrich

Completions of Categories Seminar lectures given 1966 in ZOrich

1966

Springer-Verlag. Berlin. Heidelberg. New York

All rights, especiallythat of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomec.hanlcal means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. O by Springer-Verlag Ber]/n 9Heldelber8 1966 Library of Congress Catalog Card Numbex 66-29802. PHnted in Germmy. T/de No. 7M4~

Acknowledgement

These notes contain at the Mathematical 1966,

an embryonic

course

at McGill

The author

Research

is indebted

stimulation

Institute

lectures

of the E.T.H.

given

in F e b r u a r y

in a graduate

1965. to McGill

to the National

for a Senior Research its hospitality,

of seminar

version having been presented

in spring

leave of absence,

their

an account

Fellowship,

for a generous

Research Council to the E.T.H.

and to Bill Lawvere and criticism.

University

of Canada

in ZUrich

and F r i e d r i c h

Ulmer

for for

Contents

Introduction

. . . . . . . . . . . . . . . . . . .

2

Terminology

. . . . . . . . . . . . . . . . . . .

6

G e n e r a t i n g and sup-dense

subcategories

Limit p r e s e r v i n g f u n c t o r s A sup-complete

sup-dense,

The c o m p l e t i o n w h e n The r e l a t i o n s h i p completeness Theorems w i t h o u t

~

. . . . . . . . . . . . sup-preserving

is not small

between d i f f e r e n t

10 17

e x t e n s i o n 24

.......

27

forms of

. . . . . . . . . . . . . . . . .

35

properness

42

conditions

Completions

of groups

Completions

of c a t e g o r i e s

of algebras

Completions

of c a t e g o r i e s

of m o d u l e s

References

......

......

. . . . . . . . . . . . . . ......

51 55

.......

58

. . . . . . . . . . . . . . .

69

-

O. Introduction. and inverse

limits of

The derivative preserving"

-

We shall call the generalized direct Kan

"supremum"

"sup-complete",

and "infimum"

"sup-dense",

respectively.

and "sup-

then also have fairly obvious meanings,

be made precise their duals

terms

2

which will

in the text. One can distinguish these terms from

"inf-complete"

from "right" or "property"

etc., without being able to tell "left" from "co-property".

Can every small category

A

be embedded as a (full)

dense subcategory into a sup-complete

category

also noted by others,

to be the category of all

functors from

-A~ ,

is yes: Take ~'

the opposite category of

A'

~,

?

sup-

to

The answer,

Ens,

the

category of sets. U n f o r t u n a t e l y the embedding does not in general preserve However,

consider

functors from

A._ O

instead the category

~'' of all inf-preserving

to Ens. The embedding of

is sup-dense and sup-preserving. it is an open problem whether

Moreover

A

~''

into

it is also sup-complete.

is sup- and inf-complete with a sup-dense,

of

A__~ ~'''.

To wit,

A.A'' which are subobjects

It is an open problem whether category

A''''

~''

is inf-complete;

Luckily there does exist at least one category

embedding

sups.

let in

A'''

which

sup-preserving

A_A''' consist of all objects ~''

of products of objects in

A~

there exists a sup- and inf-complete

with a sup- and inf-dense embedding

A ~ A''''

in analogy to the Dedekind completion of an ordered set.

-

3 -

Now let us drop the assumption still define all

A--

functors

this means

and

T :

~ Ens

in

D,

is still

sup-dense

A~

A--'

A'' A

is complete

is still

to

A in

A__' and

sup-preserving.

Ens

Theorem.

is known

equivalent

hand,

which

appears

(This result

While

A'

from some

such

y ~ T(D),

x = T(f) (y).

is sup-complete,

All proper

completeness"

inf-preserving

on

Adjoint

implies

is also a symmetric

theorem

of

functors

A.A are equivalent

A__ is

A_~

Ens,

to the corresponding

category.

to be slightly more

Adjoint

general

Functor

conditions

Theorem

than any in the literature.

to show the sup-completeness

We also give new sufficient

Functor

completeness

of certain

a form of the Special

imply sup-completeness.

to a general

There

to the

also announced

form of sup-completeness

-A ~ , the opposite

is required

relate

inf-completeness

This fact,

representation

to the r e p r e s e n t a b i l i t y

We obtain

A.

and the embedding

On the one hand,

that a m o d i f i e d

on

of

A'',

completeness.

and that both conditions conditions

comes

D

to Isbell,

such that

to be equivalent

On the other

asserts

According

subcategory

sense:

a kind of sup-completeness. which

one restricts

i.e.,

"representation

representation

by Benabou,

One m a y

are representable.

forms of completeness?

implies

be small.

provided

A,

f : A ~ D, in

A.

"proper".

a small

in a different

How does this older

to be

x ~ T(A),

via some'map

functors

as before,

that there exists

that every element D

A--'

that

of

~'''

for inf-completeness

to

above.)

-

4 -

To illustrate completions of small categories, sider the example in which

A

is a group

is the category of all permutational In another application,

we let

an equationally defined category operations.

If

A

many generators, of

~

.

A__'' = A__' of

G~

be a subcategory of

of algebras with finitary

contains a free algebra with sufficiently then

A__I' is equivalent to a

A__* consists of all algebras

preserves infs.

Then

representations A

~

G.

we first con-

When

~

subcategory

C such that

A__ w

[ -, C] : _A~ ~ Ens

is the category of all R-modules,

this

result was first obtained by Ulmer. Finally,

when

C

is the category of all

fairly generous conditions on C

such that every

Aj

A__* consists of all

nonzero submodule of

nonzero factor module in

R-modules,

C

under

R-modules

has a

A__. Prior to showing this, we make a

general study of certain pairs of classes of

R-modules,

as

exemplified by the following pairs of classes of Abelian groups: torsion,

torsion-free;

divisible,

reduced.

It will be assumed that the reader is familiar with what is common to the standard expositions of Category Theory [MacLane,

Freyd, Mitchell],

concepts:

category,

and epimorphism, pushout,

functor,

natural transformation,

monomorphism

subobject and quotient object, pullback and

Yoneda's Lemma

categories,

in particular with the foll~ving

[see MacLane,

adjoint functors,

p.54],

representable

equival~nce of functors.

-

Subcategory

5

-

will always mean full subcategory, e m b e d d i n g will

always m e a n a full and faithful

functor.

Some other well-known

concepts will be redefined in Section I, to allow for some idiosyncracy in terminology. I have attempted to make these notes readable, risk of including some so-called in the literature,

For proofs

the reader is sent to the recent book

b y Mitchell whenever possible. however,

"folk-theorems".

at the

For some important results

the papers by Isbell m u s t be consulted.

-

I_o T e r m i n o l o g y . A,

B

It is u n d e r s t o o d

of a c a t e g o r y

of m a p s

a : A ~ B.

of sets,

called

if the c l a s s

"A",

a.

[A, B]

for e v e r y p a i r of o b j e c t s a set

is i t s e l f

Hom

(A, B) = [A, B]

the o b j e c t

of the c a t e g o r y

this m a y be taken (1962)].

of o b j e c t s we do not

to be any of the u n i v e r s e s

The c a t e g o r y

is a set,

i.e.,

"category",

we

shall

consider

"object"

and

categories

A

A

is c a l l e d

an o b j e c t

find it c o n v e n i e n t

for

Functors

that

is g i v e n

"a"

Frequently maps

there

[see G a b r i e l

Regrettably, "A",

A_

Ens;

of G r o t h e n d i e c k small

6 -

to use

"map"

of Ens.

the

consistently.

with objects

w i l l be d e n o t e d b y c a p i t a l

styles

a

and

R o m a n or G r e e k

letters. A diagram

is the same thing

this t e r m i n o l o g y particular, dia@~am

is used,

I is c a l l e d

w i t h e a c h o b j e c t A of

A I : I ~_A,

defined

i

An U p p e r b o u n d of an o b j e c t (Of course,

A

or s u p r e m u m exists

diagram-

and m a p

~

of

(A, u) F

i

of

of

index category.

we m a y a s s o c i a t e

In

the c o n s t a n t

of I.

F

F : I -~ A_A c o n s i s t s

transformation sufficient

a : A -~ A'

u

w i l l be c a l l e d

such that

situation

: F-~ A I.

to s p e c i f y

if for e v e r y u p p e r b o u n d

I. This

If

(t) = 1 A,

(A, u) of a d i a g r a m

it w o u l d h a v e b e e n

a unique map

all o b j e c t s

AI

and a n a t u r a l

The u p p e r b o u n d

~

the

F : I ~ A.

by

A I (i) = A,

for e a c h o b j e c t

as a f u n c t o r

u

alone.)

a least u p p e r b o u n d

(A', u')

of

F

there

au (i) = u' (i) for

is i l l u s t r a t e d

by a commutative

-

A

.....

7

-

~- . . . . .

u(i)

~ A'

' (i)

r(i) We w r i t e Actually,

sup

F =

the o b j e c t

We shall use

(A, u), A

I A. Let

and o n l y if [F, A~] A', w h e r e A'

~ [A, A']

one m a y a l s o F, w r i t t e n

The

is the same

The or

A

over

of

inf

suggested

by various

by Amitsur.

a directed

free g r o u p s Other I

cokernel a, b

are c l a s s i c a l special

~i

F (i)

as s p e c i a l

cases.

isomorphism

in

of sup

cok

lower b o u n d : A I -~ F.

"co-limit",

direct

as a s p e c i a l limit~

k i n d of sup, every

but only torsion-

of free ones.

are th~ sum

"discrete" (a, b)

kar

limit

For example,

groups,

limits F

u

limit",

the c l a s s i c a l

direct

the k e r n e l '

if

The p r e s e n t ' t e r m i n o l o g y

[i~iF(i),

category,

of a p a i r

and the pull.back. D u a l l y ,

~I

F = A

A._ ~

"direct

free A b e l i a n

(or " c o - e q u a l i z e r " )

sup

(A, u), w h e r e

direct

is j u s t a set or small

.- A ~ A',

product

cases

isomorphism.

the g r e a t e s t

set I m a y be v i e w e d

is a sup of

case

F =

While

Abelian

F = A.

A.

authors.

is a c l a s s i c a l

group

. Then

as the sup in

not e v e r y sup

sup

Kan:

is a n a t u r a l

s u p r e m u m has b e e n c a l l e d

"right root"

was

of

introduce

or i n f i m u m of inf in

result

loosely

o n l y up to

F 9 I -~ A

is a n y o b j e c t

Dually,

sometimes

is u n i q u e

the f o l l o w i n g

PROPOSITION

and

(a '

in

the

of m a p s

inf

F

b)

and the p u s h o u t '

has

the

-

We shall every

call

diagram

following

with

if and o n l y

if e v e r y

sup-

into a more category

and

general

of all

A__ is small.

functors

PROPOSITION and

A

I

is small,

Proof

context.

The m a p s C.

If

C

then

~,

~]

diagram

F - I ~

objects

A

Let

F' (A) = (F(A),

sup

of

[A_A, C]. A

F (i) (A) ~ F(A). functor v

and

: F ~ GI

Put

One e a s i l y

: F ~ FI

is a n o t h e r

: F(A)

natural

is to say,

~ G(A)

such

fu(i)

= v(i) .

functors put

be

that

from

this

[_AA, ~]

assumed

that

(inf-complete) (inf-complete).

and c o n s i d e r

any

and w r i t e

for all

: I-~ C

is a functor.

u(i) (A) = u'(A) (i)

that

F

9 A-~ C

transformation.

transformation,

there

for the

= F(i)(A),

that

to Ens

transformations.

F' (A)

verifies

A

Moreover,

sup-complete

F' (A)(i)

I. Then

(kernel).

last o b s e r v a t i o n

, it b e i n g

small

and v e r i f y Then

~

and e v e r y

inf-complete.

are n a t u r a l

a natural

transformation.

and

is also

u'(A)),

v' (A) (i) = v(i) (A)

a natural f(A)

u

of

by

(inf-complete)

(product)

is s u p - c o m p l e t e

Define

and i

sup-

__A to

I

The

discovered

a cokernel

We shall w r i t e

[~, ~]

Let

has

We shall

from

of

(sketched).

has a sum

of all

inf-complete.

(inf).

is s u p - c o m p l e t e

same o b j e c t s

the c a t e g o r y

has a sup

if

and Maranda:

A category

the

(inf-complete)

to h a v e b e e n

that Ens is b o t h

_AA is small,

is b o t h

seems

set of o b j e c t s

of m a p s b e t w e e n It is clear

suP-complete

and Hilton,

I B~

-

index c a t e g o r y

result

Eckmann

PROPOSITION

if

small

fundamental

Grottendieck,

pair

a category

8

exists

is a

Suppose

where

v' (A)

:

G : ~

~.

: F' (A) ~ G(A)

a unique

map

f(A)u ~ (A) (i) = v" (A) (i) , that

Finally,

one v e r i f i e s

that

f 9 F -~ G

is

-

is a natural transformation.

9 -

It follows that sup F = (F, u).

This proof is illuminated by the isomorphism of categories

[ A , [ I , C]] = [IXA, C] ~ [AXI, C] ~- [I, [A__.,C]]. We have presented it here, because the construction of the sup in [A, C]

will be of importance

later.

-

~.

Generating

With

any

9nd s u p - d e n s e

functor

category

~

T

: A ~ Ens

we

shall

assume

A__ to d i s j o i n t

sets.

This

b u t even

if

construct we m i g h t

s u b c a t e @ 0 r i e ~.

and a d i a g r a m

simplicity,

T

put

we

that

which

[[A, -],

and

A ~ A'

hence

T'(A)

does

not m e e t

the

in the usual sets

T(A),

where

if also

~ =

such

set,

the e l e m e n t The c o n d i t i o n

x'~

distinct

that

T'(A').

of

objects

of

satisfied;

it. For example,

[~, Ens],

implies

a

it is e a s y to

does have

then

T' ~ T,

[A, -] ~ [A', -],

(The r e a d e r

will

it is a s s u m e d

that

Let

A

are

of

T(A')

recall

that

x' = T(a) (x). a'),

we put

is d e f i n e d

of the d i s j o i n t

triples

are o b j e c t s

x r T(A),

~ =

of

X T,

We w r i t e ~'~ =

FT(~)

and

a)

a : A ~ A'

~ : x -~ x';

(x, x''

by stipulating

and

(x, x',

I

that

a'a)

9

r T (x)

is the a such

a) {O} be a typical

according x ~ T(A)

to L a w v e r e the m a p

satisfied

b y the c o m m u t a t i v e

is the u n i o n

XT

(x', x'',

: ~-~

(x, x'

Remark 9 element

of

such that

~' = FT

A

and

XT

The d i a g r a m

that

of o b j e c t s

x 9 T(A)

is the

the sake

is f r e q u e n t l y

of a c a t e g o r y

A in _AA. The maps

is a m a p of and,

definition

canonically

[A, B] are all disjoint.)

The class sets

sends

T] in

Lemma,

that

associate

this property,

by Yoneda's that

T

assumption

T' = T

T' (A) =

shall

P T : X T ~ A__. For

does not have

a functor

I O -

by

diagram:

~

one-element (1964)).

We m a y

: {O} -~ T(A)

(x, x',

a)

in

set

associate

such may

(or the one-

that

with

R(O)

= x.

then be r e p r e s e n t e d

-

T(m

1 1 -

T(A')

-

T(a) We

shall u l t i m a t e l y

inf of _A

FT . T h i s m u s t be m a d e

b u t o n l y of

being

assumed

a canonical We

shall

[_AA, Ens] that

HoF T

or, b e t t e r is small.

case

FT

as a d i a g r a m HO : A ~

result H~

Let

F

/%

T~

the d u a l

:

~ AA .

XT~

says t h a t

A in

G

T h i s m a y a l s o be e x p r e s s e d G'

: B ~

[A_A O, Ens]

In the s p e c i a l calls

A

is d e f i n e d

a ~eneratin@ dually.

T

A~ ~

:

B

section,

FT Ens,

in

to v i e w this embedding

the a b o v e

is the sup

and c o n s i d e r

of the

a functor

if for a n y p a i T

in B there e x i s t s

by aaying

G

T

: G(A) ~ B

defined by

case when

concerning

the c a n o n i c a l

~enerates

b

= [A, -]

in the p r e s e n t

convenient

form:

b I, b 2 : B ~ B'

A_A and a m a p

H(A)

= [ -, A]. T h e n

takes

of d i s t i n c t m a p s object

H~

it

there does e x i s t

where

a functor

be a small c a t e g o r y

: A ~ BB . One

result

It is o f t e n

where

[A_A, Ens] ~

of

However,

consider

is the

is in fact the inf of the

.- X T o ~ __A- C o n s i d e r

[_AA O, Ens],

mentioned diagram

p o

T

a weaker

: X T ~ ~A

still,

T

is not an o b j e c t

T

Ens] O

[~, Ens] O.

w e shall

as

sense,

Fortunately,

H : A_A~ ~

shall be c o n t e n t w i t h

which

in some

precise,

in [~, Ens] O,

: XT ~

Sometimes

G

~

embedding

see that

diagram we

s h o w that,

such

that

an b i b ~ b2b.

t h a t the functor

G'(B)

= [G -,

B]

is faithful.

is the i n c l u s i o n

of

A

subcategor7

of B__t. The t e r m

in

B , one

"cogenerate"

-

PROPOSITION G

generates

2.1.

__B if e v e r y

o f a sup o f s o m e holds

when

First,

form.

Suppose

p

B.

blpu(i)

generated

B_ ~

Conversely,

functor

X

X

the

= x G(a).

for

1 .- G

l(x) o F

Now p

: B* ~

bl,

let

in

Bx

such

B.

Then object

The converse

F(~)

X,

B'

[A,

sake

= a. ~

B,

and hence

b I = b2.

B

of objects of

= [G(a),B](x) x

: G(A)

[G(A),B]

~

B

are

~ [G(A),B], verified

it is e a s i l y

that

transformation.

(B*,u),

= b2P

that

x

the

G

the m a p s

x'

sets

ifrom

category

class

that

A.

is a s s o c i a t e d

and

any object and

then

pu (x) = x. blP

A_j

in

a l s o ithat

small

The

the

B, A

Thusl

and

the

such

that

For

is a n a t u r a l =

A

is the

~

there

B

such

a)

a

It f o l l o w s

B,

in

the

a n d an e p i m o r p h i s m

: G(A)

X ~ A. A

has

bI = b2.

hence

B],

(x, x',

: G(F(x))

b I.

in

-~ _B,

B

then we have

generates B

of

(B*, u),

in

F = FT ~

F(x)

that

i

G

: ZA

of all

sup(GoF)

b2 : B ~

all x

that

B]

B',

= b2P , hence

simplicity's

= x

-~

B

blP

~ =

and

object

for all

any object

Moreover,

all disjoint), put

: A ~

F : I ~ A.

(GoF) =

= b2b

the diagram

triples

(we a s s u m e ,

we

With

is the u n i o n

are

G

is a q u o t i e n t

where

for all

that

T = [G -,

and

sup

blb

assume

is s u p - c o m p l e t e o

of

of

b 1, b 2 : B ~

= b2Pu(i)

of sup

X = ~o

small,

that every

with

Suppose

definition

the

Go~,

assume

F : I ~ A,

Then

be

object

diagram

diagram - B* ~

A

_BB is s u p - c o m p l e t e .

Proof. indicated

Let

12-

there

Moreover would

Thus

yield B

exists p

a unique

is epi, blx

= b2x

is a q u o t i e n t

map

since for object

=

-

of sup

(GoF),

Again

and

consider

Following

Isbell

i.e.,

adequate

B_~

sup-dense

we call

H~

A ~

:

embedding

following

has

B

A

T

Let

B_B.

if

G

We shall

is

is left call

G

When

G

is the

inclusion

or s u p - d e n s e by Isbell,

of

_BB,

_BB,

Right

adequate

For example,

is right

the

adequate.

The

b y Ulmer:

is small, then

: A__~ ~ Ens

o

in

the c a n o n i c a l

dually.

[_AA, Ens]

A

subcategory

is left adequate.

H : _AA ~

of

B

is sup-complete,

G

is sup-dense.

and

In p a r t i c u l a r ,

is the sup of r e p r e s e n t a b l e

functors

this r e s u l t

holds

and will

even w i t h o u t be p r o v e d

the a s s u m p t i o n

in g r e a t e r

generality

5.

Proof.

In v i e w of P r o p o s i t i o n

F 9 I -~_AA,

: B* -~ B

Thus,

Ens]

is the sup of some d i a g r a m

are d e f i n e d

is s u p - c o m p l e t e

in S e c t i o n

p

[A__~

_BB

in A.

Actually,

we have

:

for

B

[A o , Ens]

A

A__ is small.

in

is left adequate,

functor

B

If

G'

full.

also b e e n o b s e r v e d

COROLLARY.

that

and

where

left a d e q u a t e

generates

object

functors

canonical

[-, A],

faithful G

G

functor

As had b e e n n o t e d

and i n f - d e n s e

every

we call

_AA a left a d e q u a t e

embedding

: A~

G : A ~__B

F : I ~ A.

respectively..

G

a functor

then

if e v e r y

G ~ F, w h e r e

is complete.

associated

an embedding, for

the p r o o f

(1960),

if the c a n o n i c a l l y

1 3 -

such t(A)

t(A) (x) = u(x).

sup (GoF)

that pu(x)

: [G(A),

=

2.1

(B*, u) ,

, or rather

its proof,

and an e p i m o r p h i s m

= x.

B] -~ [G(A),

It is e a s i l y

B *]

verified

be d e f i n e d that this

by

is n a t u r a l

in

A,

-

hence

we h a v e Now

the

functor

the m a p p i n g

hence

Therefore that

a natural

there

in

x

Therefore sup F =

X

B],

[G -,

= B ~ B*

(see the p r o o f

(B, I),

is a s u p - d e n s e dually

and so

pbp = p, h e n c e

As we have

the

where seen,

[G -,

to be

B*].

full,

is onto.

B*]]

[ G -,

such that

b]

= t,

embedding

functor

[A_AO, Ens] H~

the does

rationals

not

in g e n e r a l

due

of integers

the same d i r e c t e d

in the c a t e g o r y

functor

of

direct

We saw that a n y it a functor

An e a s y c o m p u t a t i o n

: _BB~

shows

G

Thus

is complete.

H O 9 A.A~ [A_A O, Ens] category,

it not While

hence

to call

for the many

The A b e l i a n

with

embedding

fact

examples

9 and I0, we m e n t i o n group

of

isomorphic

denominator

has d i r e c t

limit

n),

zero

groups. : A - ~ _BB has c a n o n i c a l l y

[AAO, Ens],

that

is epi.

limit of s u b g r o u p s

set of g r o u p s

functor

G'

(1966)-

(all f r a c t i o n s

of free A b e l i a n

sups.

in S e c t i o n s

to U l m e r

of sup.

be t e m p t e d

Aj were

preserve

Thus

is an i n f - d e n s e

One w o u l d

systematically

example

p

and the p r o o f

into a s u p - c o m p l e t e

category.

2.1).

by definition

H 9 A__~ [A, Ens] O

is a c l a s s i c a l

to the group

= x,

"sup-completion"

be d i s c u s s e d

n o w a simple

bp = I,

also bp = I, since

l(x)

= bx,

of P r o p o s i t i o n

the c a n o n i c a l

into an i n f - c o m p l e t e

with

b

B] ~

is a s s u m e d

= t(A) (x) = [G(A),b](x)

bpu (x) = bx = u (x),

but

[A_O, Ens]

[[G -,

a map

t : [G -,

is to say

for any

will

: B ~

[B, B*] ~

u(x)

that

transformation

G'

exists

14-

where

G'o G "--Ro.

G' (B)(A)

associated

= [G(A),B].

-

In particular, such that

H'o H ~ H ~

[~, Ens] ~ to verify

there

that

H~

H O'

maps

there

o

composite

T* ~ T*

not enough

to establish

and

of all U +.

out,

+

that

"reflexive" T *+ ~

T

In this way,

and

all

[.AAO, Ens]

and

is rather

small.

When

elements,

Isbell

(1964)

three objects, G~

[A, Ens] ~

the regular

and two other trivial

(see Section

the c a t e g o r y

such that

G

functors,

with more

hence

> 4-

A

such U +* m U. of

than two

representation

[~,

has

just

of

is far from complete

,.-~--

<

between

that this intersection

permutational

pointed

this intersection

8 below).

[A~ Ensl

(1960)

a kind of intersection

A__ is a group

exist

this is

T in [A.A, Ens] O

Unfortunately,

showed

There

an equivalence

in [A O, Ens]

one can construct

Ens]

such that their

As Isbell

of all

U

= U +.

between

do induce

subcategories

: [A_A ~

Ens]

H'o

Unfortunately,

of all T* and the c a t e g o r y

the

of

T *+* ~ T*

an equivalence

and

H~

[A__ ~

It is not difficult

= T*, H~

is the identity.

*

: [A, Ens] O ~

is a functor

H O ~ H.

H'(T)

T ~ ~ T *+*

the functors

H'

is the left adjoint

We shall abbreviate canonical

is a functor

Dually

such that

1 5 -

~ns] ~

-

The a b o v e Let

situation

~ : A__ ~ x ~

Ens

admits

small c a t e g o r i e s .

functors

F z A~

[B, Ens] ~

G' o G = H the

give rise

and

left a d j o i n t

F'

and

9 ~ B ~

and

A O , Ens] [..

defined by

= G(B) (A).

G' and

F'

such that that

G'

is

F'. |

< >

[B, Ens

A

]o

B

m

T h a t the o r i g i n a l

~

canonically

It m a y be v e r i f i e d

F

[A O, Ensl

G

to f u n c t o r s

o F = H O.

of

where

This d e t e r m i n e s

F(A) (B) = ~ (A,B)

in turn,

a generalization.

be a g i v e n b i f u n c t o r ,

are g i v e n

These,

1 6 -

m

situation

is s e e n b y c o n s i d e r i n g

is i n d e e d a

the b i f u n c t o r

Hom

special

case of this

: A ~ x A ~ Ens.

-

3~3. L i m i t ~ r e s e r v i n q We say that the

functor

F : A ~ B

preserves

r =

the small c i r c l e

ci)

i

r

u)

. sup

denotes

C

I.

A_~ h e n c e we cannot,

Proposition

IB

to r e p l a c e

of sums and

cokernels.

Hilton)

result

F

if

for all

and If

preserves

PROPOSITION sups

3 A.

: A ~ Ens

COROLLARY. then

ou) c i )

=

does not p r e s u m e

in general,

preservation

Preservation

apply

of sups b y p r e s e r v a t i o n

of

infs

and its c o r o l l a r y

FoU)

is d e f i n e d

(due to E c k m a n n

dually.

and

are w e l l - k n o w n :

PROPOSITION [A, -]

if,

of c o n c r e t e m a p p i n g s z

The d e f i n i t i o n

of

following

o r) = (F(A),

composition

=

in

r

completeness

The

sups

F : I ~ A,

sup

for a n y o b j e c t

-

functors.

for e v e r y d i a g r a m

Here

17

in

sups and

3ol.

B.

[-, A]

A

of

~ A ~ ~ Ens

A_~

The [F -,

G preserves functor B]

~

o

the f u n c t o r s

preserve

F : A_A~ B__ is left a d j o i n t

and o n l y if B

For a n y o b j e c t

infs.

to

G

: B~

infs.

F : A ~ B -~ Ens

preserves

preserves

infs

A_~

-

Proof.

First,

functor

IF -,

functors

[-

preserves

B]

B]

,

:

B

assume

that

sup

in the remark : {07 ~

that

where

let

(i))

[F-,

i ~

sups.

from

The

the i n f - p r e s e r v i n g

- * B ~ , hence

A__~

B]

preserves

t(i)

[F(F(i)), B]

(F(A),

also

v(i)

such

~ B

Hence that

B],

i

be n a t u r a l

there

with exists

[F(u(i)),

we have

a = f(O),

F =

= [F(u(i)),

2, we a s s o c i a t e

B].

infs

for

(A, u). in

I.

B].

We

Fou).

: F(E(i))

in S e c t i o n

sup

[F(r(i)),

B], v), w h e r e

aF(uCi)) -- tCi), F (A)-

9

F~ :

and

that

(F o r} =

f : { O} -* [F(A), f = ~,

preserves

F : I ~ A_A and s u p p o s e

([F(A),

Indeed,

t~i)

F

By composition

-~ Ens

the d i a g r a m

inf is

claim

B~

in B__. Let

Consider Its

arises

that

-

infs.

Conversely, all

assume

18

a unique

in

this

i. As a mapping

a unique

B]f = t(i). a ~ [F(A),

map writing B]

such

as required.

> B

[~),s]

4-

[F(u (i)),81

[F(rli) ), B1

{o)

-

LEMMA from

3.1.

A__ to

Let

B__. Then

The p r o o f

B

infs.

contains

b)

B_Bis the

induced

embedding

hence

(a) The

so does

A~

with

C.

We c l a i m

F =

m u s t be in Let

of

: F(A)

may easily exists

F

: A_~

T' does.

C~

[F -, B]

that

: A ~ ~ Ens

F;

of

immediately

then

sup

[F -, F(A) ]

full,

infs,

by Proposition

~ : J ~ B

with

is in

B. A s s u m e

in

3.1.

inf

~ =

(C, v)

F : I ~ A

= (F(A),

Fou)

in

B_B u {C],

so that

B.

C

(b). ~ C

For each

I.

in

B~

-~ 4(3)

such

be shown

that

be n a t u r a l j

in J,

Therefore that

y : F(A)

x(j) -~ C

in

i, w h e r e

we have

there

v(j)t(i)

3 A,

Lemma.

from P r o p o s i t i o n

(FoF)

true

sups;

and

, b y the above

C

that the

in C.

preserves

any d i a g r a m

such

C

preserves

functor

: F(F(i))

an u n i q u e

of

is faithful

this remains

F(i) ) -~ 4(9) x(j)

image

to show that

B_~ by

any object

such

infs

F

former

(A, u),

t(i)

if

be omitted.

subcategory

under

follows

that

in C

functors

if and o n l y

an e m b e d d i n g

F(A~ ~

latter

We w a n t

sup

B

Since

(c) C o n s i d e r in

Given

largest

the

(b) This

infs

and will

F(A~ , the

B is c l o s e d

[ -, A]o

be i s o m o r p h i c

Then

B

Proof.

T'

preserves

of all

a)

c)

T

3.2.

consist

preserves

and

is routine

PROPOSITION let

T

19-

exists

i

v(j)t(i)

:

a unique

= x(j)F(u(i) ).

is natural

in j. Hence

such

v(j)y

that

is

It there

= x(j).

-

20-

Therefore

v(j)t(i) hence

: x(j)F(u(~)) = v ( j ) y F ( u ( i ) ) ,

t(i) = y F(u(i)),

w i t h this property,

One easily verifies that

and this completes

y

is unique

the proof.

x(j) F(A)

F(u (-~)

9

-.

I "

A(j)

y

F(r(:~ )

I

->

v(j)

c

t(s COROLLARY. cular,

Every inf-dense embedding preserves

the canonical

sups and, dually, preserves

B

C

is

C = inf B.

C

of

~

Now

_B contains

Ens]

there is a diagram

F(A~

Thus

that

C = B. from the c o r o l l a r y to

A__ be any small category.

[~, Ens].

functors

H~

induce embeddings of

A_A into

A ~

o , Ens] [A__

and

[_ A ~ , EnS]inf

from

H : ~ and

2.1.

[~, Ens]in f

In view of Proposition

canonical embeddings

:

Proposition

We shall write

for the category of all inf-preserving a subcategory of

and

F : I ~ A__ such

(F.F). It follows from the proposition

The rest follows Let

H ~ : A _ ~ [A ~

F : A ~ C_. is an inf-dense embedding.

be constructed as above.

in

[A__, Ens] ~ preserves

the canonical embedding

Suppose

for any object that

H : A ~

In parti-

infs.

Proof. Let

embedding

sups.

A to Ens, 3A, the [A_, Ens]~

[~, EnS]inf~

-

respectively. by

H~

and

We shall often denote these induced embeddings H

also.

PROPOSITION functor

A ~

3.3.

Given a small c a t e g o r y

[AO, Ens]inf

inf-preserving

embedding

We refrain Proof.

2 1 -

is a sup-dense,

sup-preserving,

and

into an inf-complete category.

from spelling out the dual statement.

That the embedding

c o r o l l a r y to Proposition

is sup-dense

follows from the

2.1. That it is inf-preserving

from the corollary to Proposition will

_A, the canonical

follow from Proposition

then follows

3.2. That it preserves

3.1. if we show that [A_. ~

sups

EnS]in f

consists of all functors

T : A ~ ~ Ens

such that

Ens preserves

[A__ ~

is inf-complete will

infs. That

follow from Proposition

3~

and

Now, by Yoneda's Lemma, [H O-, T] preserves

EnS]inf

infs if

[H O- , T]

. AO 9

IC for the same reason.

[H ~

T] ~ T.

and only if

T

By Lemma 3.1, does. This completes

the proof. Unfortunately sup-complete.

I do not k n o w whether

It is sup-complete

[A__ ~

EnS]in f

in m a n y examples(see

8 and IO).but is not known to be so in general.

is always Sections

However,

it is sup-complete,

then it is actually a left reflective

subcategory of [A__ ~

Ens],

A subcategory inclusion

functor

B

as we

of

~

in

B,

i.e.,

every map

a map

shall see.

is said to be left reflective

B ~ ~ has a left adjoint.

to saying that every object

C

p : C ~ B

f : C ~ B'

with

when

of with

B'

in

if the

This is equivalent

~ has a best approximation B

in

B

such that,

for

~,

there exists a unique

-

map

b

: B ~

B'

such that

f = bp.

assumptions

it t u r n s o u t t h a t

only assume

that

p

Under

C

fairly mild

m u s t be an e p i m o r p h i s m :

is c l o s e d u n d e r

t h a t e v e r y m a p of e

2 2 -

has the form

subobjects

in

me, w h e r e

m

~

One need

and

is m o n o

and

is epi. LEMMA

3.2.

Let

t h a t the d i a g r a m

sup

F =

that

pv(i) Proof

e

F : I ~ B in

(c, v)

_BB be a s u b c a t e g o r y

~.

= u(i)

has

Then

sup

F =

of

(B, u)

the u n i q u e m a p

fv(i) :F(i) -~ B'

is

exists

a unique

b

bpv(i)

= fv(i),

f : C ~

B'

natural

in

: B ~ B'

i.e.,

where

t

i,

in

of

C

B' is in

for

such t h a t

B

p : C ~

is a b e s t a p p r o x i m a t i o n

Suppose

C__, a n d a s s u m e

i

in

bu(i)

and

B

such

in

~.

B.

Then

__

I.

Hence

= fv(i),

there

i.e.,

b p = f. f

C

)

B'

%. ..

r .

v(i)

I

", p

ib

k

I k

F(i)

>

B

u(i) The r e a d e r w i l l F

: _BB~ ~

was

called

sup of some d i a g r a m stipulated to c a l l

F

that

we obeerve replaced by

I

properly

index category

recall

from Section

sup-dense FoE,

if e v e r y o b j e c t o f

where

F : I ~ ~.

m u s t be small. sup-dense

I. L o o k i n g

2 that a functor

We

~

is the

It w a s n o t

then

now find it c o n v e n i e n t

if the same is true w i t h

a g a i n a t the c o r o l l a r y

t h a t the r e s u i t r e m a i n s v a l i d "properly sup-dense".

small

to P r o p o s i t i o n

if " s u p - d e n s e "

is

2.1,

-

PROPOSITION subcategory

of

3.4.

~

Assume that

C__ and that

is a left reflective if

~,

then

small category

subcategory.

Let

C

sup-complete, C

F z I ~ B in

has a best approximation

with small

I

in

and let

in

Finally, because

H~

Actually, in Proposition

a slightly

PROPOSITION sup-dense

~,

is small.

where

B~

I in

thus

_BB.

of

If

C

in

v B

is

By the lemma,

B_B is left reflective.

C__is sup-complete and

in

there

B_B

F : I ~ B

C__. Let

p : C ~ B

_BB, then also

sup F

=

p. 129]

is properly sup-dense

in

to Proposition

[A_A ~

Ens],

2.1.

stronger result than the first statement

may be proved by the same methodz

3.4'.

embedding

a left adjoint

in

is, by the corollary

3.4

C__. By assumption, transformation

supF = (C,v)

EnS]inf

if and only

and a natural

B__. [See Mitchell, [A__ ~

for any

Ens].

C__. Consider any diagram

be the best approximation (B, PioV)

of [A~

object of

we assume that

is left reflective

In particular,

is sup-complete

then also supF = (B,u)

Conversely,

B__

In the converse direction,

subcategory

be any

supF = (C,v)

then

subcategory of a sup-complete

o [A_ , Ens]inf

A,

exists a diagram such that

is a properly sup-dense

_BB is sup-complete.

if it is a left reflective Proof.

B

B_B ks sup-complete,

is a left reflective

category

2 3 -

Assume

and that

G : C ~ B.

that

F : B ~ ~

B is sup-complete.

is a proper Then

F

has

-

4.

2 4 -

A sup-complete sup-dense, sup-preserving extension. LEMMA

4.1.

Given

a subcategory

A_ of C_~ let

B

be the

subcategory of ~ whose objects are subobjects of products of objects from

A_. Then

B__ is closed under products.

The proof is routine and will be omitted. Let there be given a small category canonical embedding

H O : A ~ [AO, EnS]inf

inf-complete category subcategory of

is in

A.

We recall the

of

A into an

(Proposition 3.3). Now let

o , Ens]inf [A_A

Obviously,

B

_BB be the

which consists of all subobjects

of products of representable functors A

A_.

H~

= [ -, A], where

is closed under subobjects,

hence

under kernels. Moreover, by the lemma, it is closed under products. Since [~o, EnS]in f of Proposition 1B

is inf-complete, the assumptions

are satisfied, hence

__B is also inf-complete.

(Actually, a closer examination of the argument

shows that

_BB is closed under infs with small index categories in [__A O, EnS]inf. By Proposition 2.1, or rather its dual, cogenerates of

B~

_BB, hence

H~

we see that

Ho

is a cogenerating subcategory

Moreover, any object of

~

has a representative set

of subobjects, as we shall verify presently. We may therefore apply the Special Adjoint Functor Theorem

(Proposition 7.1

below) and deduce that the inclusion functor has a left adjoint, i.e.,

B

B ~ [A__ ~

Ens]

is a left reflective subcategory

of [~o, Ens]. Since the latter category is sup-complete (Proposition IC), so is

_BB (see Proposition 3.4). In view of

Proposition 3.3, we thus have:

-

PROPOSITION 4.1.

25-

Given a small category o

the subcategory of [4 , Ens]in f of products of functors is a sup-dense, A

A

in

A.

Then

Proof.

B__.

It remains to show that any object set of subobjects.

subcategory of [A__ O, Ens] _BB. Therefore

Now

H~

_B

t h e inclusion functor

any pullback

Consider any m a p

B

in

B

has

~

2.1), hence

A O , Ens] [_

preserves

3.2, or rather its dual.

B remains a pullback in [A__ O, Ens].

b : B' -~ B

and only if in the pullback

of

is a sup-dense

(Corollary to Proposition

infs, by the c o r o l l a r y to Proposition In particular,

embedding of

from spelling out the dual of this result.

a representative

so is

B__ be

A ~-~--)[ -, A]

and sup-preserving

into the inf-and sup-complete category We refrain

let

which consists of all subobjects

[ -, A],

inf-preserving

A~

in

B__. This is mono if

B'

B

P

-

/ f

V ~

B'

in

B

[AA~

we have

u = v. Since this remains a pullback in

Ens], we see that

o

[ A , Ens]. Moreover, that,

b

mono

~

implies b mono in

it follows from the proof of Proposition

for each object u(A)

A

of

A_~

~ # B' (A)

P (A)

v(A)

in

B (A)

-. B ' (A)

is a pullback in Ens.

IC

-

N o w assume b(A)

that

b

: B' (A) -~ B(A) t(A)

is a subset

B(A) .

B*(a) and b

set,

: B'(A)

t : B' -~ B*

Then

is i s o m o r p h i c

our p r o o f

is complete.

We r e m a r k

that

in

A.

sup-dense,

is c o g e n e r a t i n g ,

short

of i n f - d e n s e n e s s .

small

category

there

for each

It gives

rise

B*(A)

C__ is sup-

to

= b(A)(B'(A) }

in

: A~ -~ Ens

is a functor,

subobject

As the

A__-~ B which

Thus

a sup-

A_~

every

, in a d d i t i o n property

and

falls for

inf-dense

and i n f - c o m p l e t e .

define

subobject

B* ~ B,

B* c l e a r l y

I do not yet k n o w w h e t h e r exists

A in A,

a 9 A ~ A'

isomorphism.

the e m b e d d i n g

being

that

B*

Then,

where

to a special

B* (A) ~B (A) , for each

A - ~ C_~ such

B.

-~ B*(A),

is a n a t u r a l

A

in

For any m a p

= t(A) B'(A)t(A') -I.

: B' -~ B

where

is m o n o

is a m o n o m o r p h i s m .

an i s o m o r p h i s m of

26-

form

a

to just every

embedding

-

5. The completion We now abandon in forming

for

Isbell

a small

T, with

there exist

D

x = T(f) (y).

T

that

and

The main obstacle

need not be small

we call

subcategory

Letting

is small.

if

T'.

(1960),

D_D,

A

[T, T']

this property:

in

-

is not small.

is that

functors

Following there exists

A

the assumption

[_AA, Ens]

we admit all

set

when

27

~

T z A ~ Ens

of

A,

For all

y r T(D),

A

proper

called in

a dominating

A

and

x ~ T(A)

and

f : D ~ A

such that

R : {0} ~ T(A)

such that

(o) = x

(see the Remark

in Section

by a commutative

diagram:

2), we m a y illustrate

if

this p r o p e r t y

{o) 9

y %

\

T (A) <-

T (D)

T(f) Let D. T/D.

T

:~

As an object

necessarily t (A) (x)

--

functor with dominating

is determined it is determined

is in

~

and

into another

by its value

A in functor

A_.

If T'

I

t

not

t (A) IT (f) (y)) = T' (f) (t (D) (y)).

of course

determined

that

by its restriction

t/D = t'/D

set

by its restriction

then

is completely

This means

D

transformation proper,

T

on maps,

D ~ A, where

is a natural

t

be a proper

function,

As a function

all maps

Thus

Ens

implies

t/D.

t = t'.

for =

T

~

T'

- 28By [T, T'],

we

shall m e a n

IT/D, In this way,

T']

the class

of all p r o p e r

into a c a t e g o r y

Actually,

this

category

choice

of a d o m i n a t i n g

small,

we c h o o s e the

We shall all p r o p e r that,

canonical

o If

is an o b j e c t

of

~,

such of

set

A

We n o w state

to

functor

and

[A_A, Ens]. on the

If

~

with

~ any

discussed

is

[A~ Ens]

of

Ens. We o b s e r v e

[

A ~

T = [G -, B], w h e r e they were

by

for the c a t e g o r y

the c a t e g o r y

-~ A_A a s s o c i a t e d

to Ens

_AA, so that

{A}. T h e r e f o r e

[ A O, Ens]

A

sense,

functor.

to be

from

A_~ the

recall

, A] we

9 A ~ -* Ens

still have

[A_O, Ens]inf. o

and the

functor

G : A-~ B in the p r o o f

a generalization

T

.-

and

A_O

-~ Ens

B

of

of the c o r o l l a r y

latter.

PROPOSITION

5.1.

that

B]

B.

of

A-~

will

: XT

2.1.

functors

dominating

2.

PropDsition

sets

[_AA, E n S ] i n f

embeddings

o

small

also w r i t e

from S e c t i o n

to the

set for each

A

from

in a t e c h n i c a l

as before.

with

FT

functors

same m e a n i n g

The reader diagram

depends,

all d o m i n a t i n g

for any o b j e c t

T' (D) ].

that m a y again be d e n o t e d

inf-preserving

is proper,

set

in D [T(D),

C ~

is turned

then has

the

[G-,

Let

associated

XB with

o _BB-~ [A , Ens]

Let

: oA

and [G-,

there be g i v e n

~ Ens

is proper

FB : X - ~ A B]. Then

is an e m b e d d i n g

a functor for each

be the c a t e g o r y the

functor

if and o n l y

if,

G

: ~ ~_BB

object

B

and d i a g r a m

B ~-~->[G-, for each

B] B

in

-

B,

-

sup

(Got B) =

(B, I),

We m a y call equivalent

where

= x

in

X B-

of this pr op os it io n.

This d e f i n i t i o n

to Isbell's

term

one w h e n

"properly

A

the

is small and c o r r e s p o n d s

left adequate".

Right adequate

are d e f i n e d dually.

Proof.

First,

assume

that

B~-~

and full, we w i s h to show that sup

b

x

if it sa ti sf ie s

the e a r l i e r

Let

for each

left a de qu at e

agrees w i t h

functors

l(x)

G : _AA-~ _B

conditions

29-

t(x)

9 G[FB(X)) ~ B'

: B~B'

such that

[G-,

(GOFB)

naturally

in

=

B]

is faithful

(B, I).

x, we seek a u n i q u e

bx = t(x). b ........

B

-->

B'

G(r B (x)) Defi n e

t' (A)

: [G(A),

One e a s i l y v e r i f i e s t' = p(t),

then

By assumption, [[G-, Put

B],

that

the m a p p i n g B']]

b = k~(t)),

is uni q u e w i t h

then

[G-,

b']

This completes

B']

is n a t u r a l

by

t' (A) (x) = t(x) .

in A. Wr it e

B'

] ~ [ [ G - , B], [ G - , B']]. XB b ~ - - ~ , [ G -, b] : [B, B'] -~

has an inverse,

i.e.,

bx = [G(A),

b

t' (A)

~ : [GOFB,

[G-,

B] -+ [G(A),

call

it

A.

[G -, b] = t', that is to say

b] (x) = t' (A) (x) = t(x) .

this property;

= t', h e n c e

for

if also

b' = A(t')

= b.

the first p a r t of the proof.

b 'x = t(x),

- 30Conversely, We w i l l

assume

Let

B], [G-,

B']]

and onto. t'

[G(FB(X)),

: [G -,

B] ~

B], we h a v e

easily verifies

that

there

exists

Thus

t' = [G -,b],

t' = [G -, b'].

[G -,

B'].

Then,

this b

is n a t u r a l : B ~

in

B' such

the m a p p i n g

by uniqueness.

B e f o r e we can a s s e r t

One

is onto.

for all

This c o m p l e t e s

~ B~

x. By a s s u m p t i o n ,

is one-one,

bx = b'x

x

that

and so our m a p p i n g

Then

for any

t'~FB(X) ) : G(FB(X) )

a unique

To see that

= b

[B, B']-~ [[G-,

s h o w that the m a p p i n g

is o n e - o n e

b'

sup(Got B) = (S, I)

that

assume x

that a l s o

: G(A) ~ B, h e n c e

the proof.

that the c a n o n i c a l

embedding

__ A ~ [A_ O , Ens] is left adequate, we n e e d two lemmas.

A~

LEMMA

5.1.

to Ens.

If

dominating

5.2.

to the i d e n t i t y Proof. natural

in

It follows

T

T

and

T'

is p r o p e r

be

isomorphic

then so is

functors

T', w i t h

from

the same

set.

The p r o o f

LEMMA

Let

Let

is r o u t i n e

Any

functor

functor t(C)

C. Then,

and w i l l be omitted.

of

C

for a n y m a p

is o n e - o n e

which

is i s o m o r p h i c

is an e m b e d d i n g .

: C ~ F(C)

that the m a p p i n g

[F(C) , F(C') ]

F 9 C ~ C

be the g i v e n

isomorphism,

c : C ~ C',

F(c)

c ~-~F(c)

and onto.

: [C, C']

= t(C')c

t(C) -I.

-

PROPOSITION embeddings adequate,

5.2.

A-~

sup-preserving.

For any category

[A_A O, Ens]

sup-dense

31-

and

A ~

A,

[A__ O, EnS]in f

and inf-preserving,

Moreover

[A_~

Ens]

the canonical are left

the second is also

is sup-complete.

The last statement has also been asserted by Benabou(1965). Proof. Consider Take any Lemma. H~

T

in [A__ ~

By Lemma 5.1,

Ens],

then

[H ~

T]

H ~ : A_A~ [__o Ens].

[H ~

3.2, H ~ preserves

the other embedding It remains

For each

i

F (i) (A)

I, F(i) set

z

infs. The statements

D.. --I

and verify that

-~ Ens

For any

sup r'CA

is A

Indeed, some

i

in

I.

A, write

and write u(i) (A) = u'(A) (i). T

lC,

Moreover,

x ~ T(A), (Otherwise

T(A) -~ T(A) - ~xj

F' (A)(i) =

is a functor.

is a functor

then

and

T

is in [A__ ~

is

set

Ens] at all.

D = Ui~ I _Di-

x ~ v(i) (A) IF(i) (A)), for

there would exist a mapping

such that

u

it will follow that

is proper with dominating

let

let us say

1C.)

sup F = (T, u), if we make sure that T

proper,

in

= (TCA ,

transformation.

We claim that

concerning

is sup-complete.

F' (A) : I ~ Ens

As in the proof of Proposition a natural

By the corollary to

[~o, Ens]

(See the proof of Proposition

Let

5.1

Ens] be a diagram with small index category I.

in

with dominating

By Proposition

are proved similarly.

to show that

F : I -~ [A ~

T] ~ T, by Yoneda's

is proper.

is left adequate and sup-dense.

Proposition

Let

the embedding

gv(i) (A) = v(i) (A),

leading

-

to a c o n t r a d i c t i o n . ) Therefore

there

y ~ F(i) (D),

Hence

exist

and

D

32-

x = v(i) (A) (z) , w h e r e in

f : A -~ D

Di,

the d o m i n a t i n g

such

that

z ~ F(i) (A) . set of

F(i),

z = F(i) (f) (y), h e n c e

x = v(i) (m (z) = (v(i) (m r(i) (f)) (y) = [T(f)v(i) (D)) (y), by naturality,

and so

This

the proof.

completes

Unfortunately, [A_ ~ , E n S ] i n f a kind

we c o u l d

denote

technical

the p r o d u c t

LEMMA

5.3.

: _B -~ C

Let

G

If also

F' p r e s e r v e s

t' (B)

:

F(B)

circle

be

left a d e q u a t e

to

then

so that FoG

5.1

to a u n i q u e

t'[G(A) ) = t(A) .

~ F'oG

from P r o p o s i t i o n

(FoGoFB)

every natural

can be e x t e n d e d

: F -~ F'

sups,

Then

and

implies that

: F(G(FB(X)) ) -~ F' (B),

there

exists

-~ F' (B)

m a p of

establish

m a y be u s e d

Since

the i d e n t i t y

and

first we r e q u i r e

Fol).

shown

such

that

that

t' (B)

t' (B)F(x) is n a t u r a l

G(A) , then

B = G(A)

sup

F ~ F'. (GoF B) =

a unique

= F' (x) t(FB(X)). in and

B. Take FB(X)

x to be = A,

t' [G(AI ) = t(A~. Next,

tha t

shall

= ~F(B),

It is e a s i l y

hence

: A-~ B

t'

We recall

F' (x) t[FB(X))

(The small

t : FoG-~ F'oG

sup

we

But

F sup-preserving.

transformation

(B, I), h e n c e

Instead,

[ A O, Ens]

of functors.)

natural

Proof~

y' = v(i) (D) (y)~ T(D).

show that

completeness.

result.

with

transformation

not

inf-complete.

of r e p r e s e n t a t i o n

a somewhat

F, F'

are

x = T(f) (y'), w h e r e

suppose

that both

t z FoG ~ F'oG

F

and

is a n a t u r a l

F'

preserve

isomorphism

with

sups and inverse

u.

-

Extend u't'

t

to

t'

z F ~ F

u't'

hence

I : F o G -~ FoG.

= I. S i m i l a r l y 5.3.

[A, Ens] ~ Proof.

to

u'

: F' ~ F.

= u'(G(A) )t'(G(A) ) = u(A) t(A)

PROPOSITION from

u

Then

and

extends

u't'

and

: F ~ F'

(u't')'G(A)),

Thus

33-

or

Take

t'u'

Every

proper

for e x a m p l e

F z [4, E n S ] i n f O ~ Ens

be

But so does

= I,

[4, E n S ] i n f

I - F ~ F,

and our p r o o f

inf-preserving

o

the

= I

to Ens latter

functor

is r e p r e s e n t a b l e .

category.

inf-preserving

is complete.

Let

and proper.

As o

before,

consider

the c a n o n i c a l

defined

by

= [ -, A].

H(A)

Proposition

5.2),

for the m o m e n t FoH

hence

that

is an o b j e c t

by Yo~eda's

Then

Putting H

and b o t h

and

F' p r e s e r v e

5.3

is r i g h t

and d e d u c e

that

then

adequate infs.

[~, E n S ] i n f

infs

Ens.

(see

Let us assume

it follows

Therefore

F' = [FoH,

Now

Lemma

Foil : A - ~

[~, E n S ] i n f o

of

H : A ~

preserves

is proper,

F o H ~ F'oH. F

H

so does

FoH

Lems, a.

embedding

-], we

that

F o H ~ [FoH, thus have

(see P r o p o s i t i o n

We m a y

F ~ F' = [FoH,

H -],

5.2)

therefore

apply

-].

F

Thus

is

representable. It r e m a i n s a dominating

to show that

set for

o [A~ E n S ] i n f , and functor

D

We c l a i m

that

F. T h e n

D

any o b j e c t

of

~ _AA ~ Ens, ~

FoH

ks proper.

D

is a s u b c a t e g o r y D

let us say w i t h

= UD in D ~ D

Let

is itself

of

a proper

dominating

is a d o m i n a t i n g

be

set

set for

_~. FoH.

-

Indeed, Since

F

take

any

A

has d o m i n a t i n g

z ~ F(D),

and

Now

g r [D, H(A)] isomorphism

dominating

set

f : E -~ A

D

such

[D, H(A)]

such

that

between

and

h

h

exist

that

corresponds

we m a y pick

to some

x ~ F(H(A)).

, there

ED,

corresponds g

_AA and

set

g : D -~ H(A)

the Y o n e d a

and

in

34-

x = F(g) (z).

to some

-~ D(A). E

D in D,

in

g'E D(A)

Since ED,

D

is c l e a r l y

~ D(E).

has

h'r D(E) ,

g' = D(f) (h'). Again,

~ [D, H(E)]

h'

The r e l a t i o n

this:

g = [D,. H(f) ] (h) = H(f)h,

as i l l u s t r a t e d

b y the c o m m u t a t i v e

diagram:

D

H(A)

~-

H(E)

H(f) Thu s

where

x = Fr

r

= (Fr162162162

y = Fr

(z)

r F(Hr

as required.

),

and

= F ( Hr E

is

under

in

)r E,

-

~. The relationship

between

35-

different

forms of completeness. , ,

We aim to investigate completeness functors

by Benabou

PROPOSITION

The

Proof.

6.1.

Recall

If

we associate

F = FT : X ~ A its dual,

(Actually,

the quoted

might

we have

Now

A__ is inf-complete

D

the subcategory Then

Y

shall

see,

inf

that

It remains it suffices

X

whose

inf

Hov),

2. By Proposition =

5.1,

(T, I).

: ~ ~ Ens, where

preserves

(A, u)

in

set for

the natural

and let

Y be

UD in D T(D)-

(A, u),say.

for suitable

T ~ H(A), F =

is not small.

form the set

(A, v),

and so

X

T,

and inf F/y = F =

infs. We

A__, hence T ~ H(A).

is that

objects

to show that inf

to extend

H

this argument

this implies = [H(A),

and the

deals not with T, but

and

inf F =

is a small c a t e g o r y

(HoF)

X = XT

(H,F)

[T, H -]

be a dominating of

With any functor

in [_AA, Ens]inf.)

The only trouble with let

inf

proposition

functor

then every

embedding

as in Section

is taken

However,

of

A ~ Ens is representable.

the c a t e g o r y

Hom

try to argue

is inf-complete

is right adequate.

or rather

the

inf- and sup-

result has also been

that the canonical o

w i t h the isomorphic

A

functor

H : A__-~ [A, EnS]inf T : A ~ Ens

following

,

(1965).

inf-preserving

diagram

between

on the one hand and the r e p r e s e n t a b i l i t y

on the other.

announced

proper

the relationship

r

As we v, hence

as required.

(A, v). To this purpose

transformation

u :F / y ~ %

-

to

v

: F ~ ~. PROBLEM.

F : X -~ A, u

CONDITION

I.

of

F/y.

v : F-~ ~

?

is s u f f i c i e n t :

For all

there

x

in

X

that

exist

y

If x'

~1

in

Yl "~ x

:

X,

Pl

and

: x' ~ Yl'

"

P2

x'

first c o n d i t i o n

a l l o w s us to d e f i n e

v(x)

d e p e n d on the c h o i c e

~

y.

of

We n o w c o n t i n u e w i t h Condition

I

Condition

II, a s s u m e

~i =

is s a t i s f i e d

and

the p r o o f of P r o p o s i t i o n since

that

(Yi' x, a i),

T

is proper.

~i : Yi ~ x, Yi

x = T(a i) (yi). We seek

Y2

~ T(Di)' x'

and Hi

u(y),

does not

6.1.

To v e r i f y

i = 1 or 2, x ~ T(A), 9 x' -~ Yi

a. : D. ~ A , 1 1 such

~1~I = ~2~ 2. Since

may

~

= F(~)

t h a t the d e f i n i t i o n

that

Y

~1Pl = ~2H2.

assures

and

in

~2 : Y2 ~ x,

and the s e c o n d c o n d i t i o n

where

X,

W h e n can

p a i r of c o n d i t i o n s

II.

t h e n there e x i s t

The

a lower b o u n d

of some c a t e g o r y

~ : y ~ x.

CONDITION

such

is a s u b c a t e g o r y

to a lower b o u n d

following

and a m a p

Y

: F/y -~ A y

be e x t e n d e d The

Let us ask m o r e g e n e r a l l y : Suppose

u

36-

is i n f - c o m p l e t e

form the p u l l b a c k s

a n d the a r r o w s

below.

emanating

and

T

preserves

(For the m o m e n t ,

from it.)

infs,

we

disregard

R

-

37-

D1

T (D I)

9

bl /"

1

/,"T (b 1 )

d

A' \

R --- p - - ~T (A')

A

b2~ x~ D2 Now let

T(A)

S T (D 2)

R c T ( D I) x T(D 2) be defined by R = ~(z I, z2)

Also let

E T(al)(zl)

= T(a2) (z2)}.

Pi : R -~ T(Di) be given by

Then there exists a unique T(bi)P = Pi"

Since

Pi((Zl' z2)) = zi"

p : R-~ T(A')

such that

(YI' Y2 ) ~ R, we may put

x' = P((YI'

hence Yi -- Pi((Yl ' Y2 )) = (T(bi)P)((Yl' Putting

Y2 )) = T(bi)(x').

Pi = (x', Yi' hi)' we then have

glPl = (x', x, albl)

= (x', x, a2b 2) = ~2P2 ,

as required. This completes

the proof of Proposition

Of course one can also prove this directly, recourse

to Proposition

6.1. without

5.1, by taking a suitable

inf.

Such a procedure has in fact been proposed by Lawvere for the general Adjoint Functor Theorem, be obtained as a corollary,

which will here

as is also done by Benabo=.

Y2 ) )'

-

COROLLARY. complete,

(Adjoint F u n c t o r

then a functor

F

B,

the

functor

Proof easy.

Conversely,

It r e a d i l y

that

B

of the c o n d i t i o n

G(B)

the

of

is a f u n c t o r

functor

is [B, F -]

A_, b y P r o p o s i t i o n

and a left a d j o i n t

F. Proposition

6.1

inf-completeness

functor has

A-~-'~

a small

: F ~ AI,

D

in

D,

for all

has

an u p p e r b o u n d

exists

Therefore

f : D-~ A

Suppose

6.2.

the a s s o c i a t e d that

for e v e r y u p p e r b o u n d y : F ~ D I,

such that

x(i)

every proper

is r e p r e s e n t a b l e .

If sup

is i n d e e d proper,

is proper.

so that,

when

= f y(i),

Io

Ens

Conversely,

in w h i c h

This m e a n s

there

a sup if and o n l y Proof.

is proper. D

in

A_~

proper

subcategory

PROPOSITION functor

a pseudo-converse,

F : I ~ A

[F, AI]

and a m a p i

admits

is r e p l a c e d b y a form of s u p - c o m p l e t e n e s s .

L e t us call a d i a g r a m

x

G

be inf-

is p r o p e r .

the c o n d i t i o n ,

b y some o b j e c t

A

for e a c h o b j e c t

: A ~ Ens

m a y be r e p r e s e n t e d

of

Let

has a left a d j o i n t

The n e c e s s i t y

assuming

follows

B

infs and,

[B, F -]

(sketched).

Theorem.)

: A ~

if and o n l y if it p r e s e r v e s of

3 8 -

Then

inf-preserving

a diagram

F : I ~ A

if it is proper.

F =

(D, y),

the

with dominating assume

that

N o w it is e a s i l y

functor set

{D}.

the f u n c t o r seen

it is r e p r e s e n t a b l e ,

A ~--~./~ [F, A I]

A~--~->

that it p r e s e r v e s

by Proposition

6.1.

[F, AI] infs. Thus

6.1.

-

there exists an object n a t u r a l l y in

B

of

A

A. By Proposition

COROLLARY.

39-

such that [F, A I] ~ [B, A], IA, this means that

sup F = B.

An inf-complete c a t e g o r y is sup-complete

and only if every diagram

I ~

if

with small index c a t e g o r y I

A

is proper. Propositions

5.1 and 6.2

together almost establish

the equivalence of inf-completeness not quite. To rescue something make a definition.

Dually,

T*

U+:

in [~o, Ens].

very proper

A O-~ Ens, where

eyery proper functor

gives rise to a functor [U, H~

:

A ~

T z ~ ~ Ens

T*(A) z

= [H(A), T]

A ~ ~ Ens

U+(A)

=

We shall call the functor

T

if the functors

are all proper.

We call the diagram

if the index c a t e g o r y [F, AI]

A

I

F : I ~ A

is small and the associated

Given any category

functor

A, the following

are equivalentz

(I)

Every very proper diagram

(2)

E v e r y v e r y proper inf-preserving

is representable.

very proper

is very proper.

PROPOSITION 6.3. on

.....

(Each of them exists because the preceding

one is proper.)

conditions

U

Ens, where

T, T*, T *+, T *+*,

A'--'~-~

but

from this situation we shall

Recall that every proper functor

gives rise to a f u n c t o r in [A_~ Ens] ~

and sup-completeness,

I ~ A

has an inf. functor

A - ~ Ens

-

4 0 -

(3)

Every very proper diagram

I -~ _A

(4)

Every very proper inf-preserving

has a sup. functor

A ~ ~ Ens

is representable. Proof.

In view of duality considerations,

suffice to show write

T

(I) .

(2) .

Assume category

functors

from

(1) and let

X = ~

T

be in

proper for each

T

in

dual of Proposition Proposition 6.1,

X

F = inf F/y,

will follow from

F = FT 9 X -~A,

T

(HoF)

5.1, inf

(HoF)

(I) if we show that

to

put

= H(B). Therefore

then

t(y)

to

b y the

As in the proof of

[H(A)y,

F/[

Y

such that

is v e r y proper.

inf F = B. Since T ~ H(B),

and so

F/y is very proper. U(A)

= [~,

~ [A,F(y)]

t~-~.~H.t (HoF)/y].

[H(A), T], by Proposition

F/y].

H

preserves

(2).

Let

U 9

oA

-+

Consider

~ [H(A), H(F(y))]

t(y) goes to

It is easily verified that F/y]

is

if the latter exists. That it does exist

Under this isomorphism,

of [ ~ ,

= T.

has a small subcategory

be the functor defined by .t ~ U(A),

(see

, by Lemma 5.1. Therefore,

We must still show that

any

as in Section 2.

[T, H -] -~ T in [A, Ens] ~

Assuming this for the moment,

Ens

T . Form the associated

H : A - ~ T__ is still right adequate

Lemma 6.1 below). Moreover,

inf

all very proper

A to Ens.

and diagram

Now the functor

infs,

We shall temporarily

for the opposite c a t e g o r y of

inf-preserving

inf

(3).

it will

in

T_.

H(t (y)) = (H.t) (y) . is an isomorphism

But the latter

is isomorphic

IA. One easily verifies n a t u r a l i t y

-

in

A, h e n c e

so is

T*,

also

(I)

U ~ [H -, T] ~ T*.

according

This means

(2), a n d that

T

= [F, AI].

an object

B

Proposition subject

of

is v e r y p r o p e r , "very proper",

completes

the p r o o f

hence

that

preserves

T ~ [B, -]. in

where

A, h e n c e

infs,

there

is

Therefore sup

(3), a n d our p r o o f

F = B, b y is c o m p l e t e ,

to the two l e m m a s b e l o w . Given a functor

of .~.

is l e f t a d e q u a t e , The p r o o f

S

t h e n so is

If

Let

u* = [H -, u]. transformation. u'v*

S * ~ T e.

=

u

- S -~ T,

It is e a s i l y

In t h i s way, that

if

also S

B

is

A__~ _BB ~ C__

J.

then

if a n d o n l y if

Moreover,

where

functor

: __ A - ~ Ens,

(uv)* = i* = i,

Now assume

: A ~ ,

and will be omitted.

S = T

is v e r y p r o p e r

Proof.

J

If the c o m p o s i t e

is t r i v i a l

L E M M A 6.2.

T, T*,

such that

(2) ~

be a very proper diagram.

is v e r y p r o p e r ,

T evidently

, naturally

IA. Thus

a subcategory

S*,

F z I ~ A

: A ~ Ens

in_AA

L E M M A 6.1.

then

let

Since

[F, A I] E [B, A]

S,

This

T

(2). Assume

and

Since

to the d e f i n i t i o n

U, b y L e m m a 6.2 b e l o w . .

T(A)

41-

then

u*

T

: T ~

~ ~ .- A__

is v e r y p r o p e r .

u* S

is a n a t u r a l is the

and similarly S * + ~ T*+,

inverse of

v * u * = I.

S*+* ~ T *+*

is v e r y p r o p e r .

This means

Thus etc

that

S *+ ....

are all p r o p e r .

By Lemma

5.1,

T ~+,

are all p r o p e r .

Thus

is v e r y p r o p e r .

...

Ens,

9 S* -~ T*, w h e r e

seen that v

S* --- T*

T

also

U,

-

~.

Theorems

without

We w i s h to investigate

42-

properness

when

conditions.

the properness

Proposition

6.1 and the c o r o l l a r y

be removed.

The proof of the following

Mitchell's

proof of the Special

[Mitchell,

page

sentable,

Let

7.1.

functor

in either

Functor

T : A~

Ens

from

of

of the following

A

hence

repre-

two cases:

a right adequate

small

has a representative

these assumptions,

Then every

is proper,

A_A contains a cog@nerating small

F : ~

can

Theorem

_AA be inf-complete.

CASE

Under

subcategory

subcategory

C.~

~,

set of subobjects.

every inf-preserving

functor

has a left adjoint. Proof.

and

Given any object

a = F[A ' _]

: X ~

associated

with

(Actually,

we should put

it absorb

l(i)

sition

5.1.

or rather

for all

In Case

its proof,

monomorphism I.

= i

k

A

of

A

, let

~ A_ be the c a t e g o r y

the functor

the inclusion

where

in

Adjoint

A__ contains

2.

6.2

has been adapted

CASE I.

and every object

i

to Proposition

on

126].

PROPOSITION inf-preserving

conditions

[A, -]

: C - ~ Ens,

~ : X - ~ C, functor.) i

in

I = X [ A ' _] and diagram as in Section

but we m a y as well

In Case

I,

inf a =

let (A, I),

I, b y the dual of Propo-

2, we apply the dual of Proposition and obtain

: A - ~ A'

inf ~ =

such that

(A', w), w i t h a

w(i)k

= i

for all

2.

2.1,

-

either

In

be

case,

the c a t e g o r y (As a b o v e ,

Put

inf Take

Yi = T(i) (x) verify

Case w(i)a

this

a

2, t h e r e

~ T(A).

B

For

with

the

has been

I -~ C -~ A -

that

absorbed

in

F.)

A.

In C a s e

ia = Vx(i) a

-

functor

put

= u ( Y i ) : B -~ A(i)

in io

a unique

rTlC

i - A - ~ C = A(i),

Vx(i)

is n a t u r a l such

is in

any

Write

exists

T

exists y

Section

2.)

that

preserves a unique

in

In C a s e

Therefore

Y.

infs, z

: B - ~ A'

hand

~

(0)

exists

T(i) f = Y i "

inf

(Tot)

: {O} -~ T(B)

(Here

I, t h e r e

a n d o n the o t h e r

Thus

functor

r =

I,

there

exists

= u ( Y i ). such

and

In

that

= u ( y i).

for all

such

associated

where

- B -~ A

Since there

x

and

inclusion

~ T(C).

that

a unique

the

(B, u),

any

y =

and diagram

T/C.

r =

let

43-

Now

= y,

a unique on

R = f = T(a) 9., a n d

so

set

T.

T(a) T (B)

/9

(c) / !

z

I '^ /

/

see

map

Tou),

that

f .-

hence

T(u(y))

the R e m a r k

in

{O) -~ T(A)

~i = T ( i ) R ,

= T(i)T(,}9..

x = T(a)(z),

for

T (A) <

(o}

such

the o n e h a n d

~i = T(u(Yi))9

[B} is a d o m i n a t i n g

= (T(B),

z ~ T(B).

z = y,

-

44

-

In Case 2, form the pullback: k A > A'

T

r

a II

a

P I

P Then

k'

preserves

B

k'

is easily seen to be a monomorphism.

Since

T

infs, the square in the following diagram is

another pu llback: T(A)

/(a'

Again,

I

I T(a)

there exists a unique

T~w(i) )f = ~i"

Therefore

such that

T(a') (t) = x

such that

~i = T(i) R = T(w (i) )T (k) R .

~i = T~u(Yi) ) ~" = T(w(i)) T(a) 9o

T(k) R = T(a) 9o

: {0} -~ T(P)

f 9 {O}-~ T(A')

On the one hand

On the other hand

Thus

T(k) > T(A')

with

Hence there exists a unique T(a')

t = R

t ~ T(P).

and

Since

of

B, it follows that any representative

of

B

is a dominating

set for

T.

T(k') P

t = 9. .

is a subobject

set of subobjects

-

Before ideas if

stating

of Isbell

m = m'e'

He p r o v e d

map

A__ has f

He c a l k e d

e' epi

we s u m m a r i z e

a monomorphism

implies

that

e'

some m

extremal

is an i s o m o r p h i s m .

this:

PROPOSITION of

our n e x t result,

(1964).

and

4 5 -

7A.

If

A__ is i n f - c o m p l e t e

a representative

of

A

has

and e v e r y o b j e c t

set of subobjects,

a canonical

decomposition

then e v e r y

fe

is epi

e

and

decomposition product

where

f = fm f

--

f m

is an e x t r e m a l

is u n i q u e

of e x t r e m a l

monomorphism,

up to isomorphism.

monomorphisms

and this

Moreover,

is again

#

the

an e x t r e m a l

monomorphism. From

this we deduce

LEMMA complete objects. e epi,

7.1.

(Diagonal

and e v e r y If

then

the

there

Lemma.)

object

mg = he,

following:

has

m

exists

Assume

that

A

a representative

an e x t r e m a l a unique

g

d

is inf-

set of sub-

monomorphism such

that

and

m d = h.

> J

e

d 4

~

J f / f j

h Proof L

decompositions.

canonical

g = gmg e

Let

Then

x

and

h = hmh e

(mg m) ge = h m (hee)

decompositions

an i s o m o r p h i s m

'>

such

of that

fo

be the c a n o n i c a l

= f, say,

By u n i q u e n e s s ,

Xge = hee

and

are

there

two

exists

h m X = mg m.

-

Take d

d = -mU x

-1

he,

then

46

-

Since

md = h.

m

is mono,

is unique with this property. ge

.>

L

~[

gm

I

->

x I I

m

! !

>

m

h

PROPOSITION

is also sup-complete CASE I. consisting of

A

O

A

Let

CASE 2.

A_ be inf-complete.

_A

A

contains

two cases:

subcategory

Ao, and arbitrary

sums

a generator

A. Moreover,

Ao, and arbitrary

every object in

A

- -

has representative Proof.

sets of subobjects

For any set

X, let

the direct sum of copies of x ~ X. Then

G : Ens ~ A

right adjoint

=

A__

A.

exist in

F = [A O, -]

Then

in either of the following

contains a left adequate

O

[X, F(A)]

m

of a single object

exist in

sums of

> h

e

7.2.

%

= ~xeX

= [ x~X Ao

denote

one for each element

is a functor, which has as a

the so-called

" A___~ Ens.

Ao,

G(X)

and quotient objects.

for@etful

functor

Indeed,

x], F(A

= TIxex [

A] ~

e

O'

=

-

It follows

47

-

that there e x i s t c a n o n i c a l

e : GoF -~ 1

and

m

: I -~ FoG

natural

transformations

with well-known

universal

properties . Consider

any d i a g r a m

r : I-~ A, w he re

In v i e w of the c o r o l l a r y show

that

F

subcategory exist x(i)

D

is proper.

D

in

of

= (G(V),

(FeF)

=

i

Since

map

f : G(V) -~ A

in

such that

e(A)

: G(F(A) ) -~ A

(b)

F(e(A)):

let

x r FCG(A))

universal

sups

also sup

(see

(GoFoF)

: G ( F ( F ( i ) ) ) -~ A

is

a unique

= x(i)

e(F(i)).

facts:

is epi.

of the fact that

A O is a ge ne ra to r.

(a), bu t is shown directly.

= [Ao, A] = [GC{O}),

p r o p e r t y of

preserves

f G(v(i))

-~ F(A)

(a) is an e a s y c o n s e q u e n c e from

such that

is epi.

F(G(F(A)}~

(b) is not d e d u c e d

G

i, there exists

We shall use the f o l l o w i ng (a)

n : D -~ A

3A), he nc e

x(i)e(F(i))

e a s i l y seen to be n a t u r a l

x : F -~ A I , there

I.

(V, v). No w

to P r o p o s i t i o n

e(v)).

and

in

to find a sm al l

given

y : F ~ D I,

is smal~.

6.2, we n ee d o n l y

Thus we w a n t

so that,

for all

sup

the c o r o l l a r y

A

D,

= n y(i) Let

to P r o p o s i t i o n

I

A].

e, there ex is ts a

Then,

unique

Indeed,

by the f : {O} -~ F(A)

such that X = e(A)e(f)

= [A O, e ( A ) ] ( G ( f ) )

= F(e(A))(G(f)).

-

48

-

G(f) G(F (A))

---G({O})

~

y

e(A)I, A

In view of the preliminary

spadework done by Isbell,

Case 2

will be a little easier to deal with than Case I. We shall therefore

consider

where

is an extremal monomorphism

m

Since

e(F(i))

it first.

is epi

the Diagonal Lemma y(i)

: F(i) ~ D

verified

that

quotient

(see

y(i)

objects

of

and

= x(i)o

in

i, h e n c e

to be a r e p r e s e n t a t i v e G(V).

> G (V) i e j, J

I P P

P D

y(i)/ ..:~ /

r(i)"

with

pf %

I I

m~.,~

J

x(i)

e

image

D,

is epi.

and obtain a unique map

m y(i)

is n a t u r a l

D

f = me

(a) above), we may apply

(Lemma 7.1),

such that

Thus we may take

Let

> A

It is e a s i l y y 9 F-+ D Iset of

- 49In Case

1

we shall consider

but after applying

the forgetful

the same square as above, functor

F.

!4

F(f)

//m (W)

s (i) /

i ~

m

////~i)

F(m~) ~ ~

~ F (A)

F(x(i)) Write

F(f)

with image

= me, S.

where

Since

m

is mono and

e

F(e(F(i) )) is epi

(see

there exists a unique mapping

s (i)

It m a y be verified

is natural

Now c o n s i d e r By the universal m'

: G(W) ~ A

= re(W) s(i),

set

m'y(i) D

s(i)

the canonical map property of

such that then

will be complete that

that

such that

m(W)

= m.

= m s(i)

if we can assert that

= x(i).

in

(b) above), m s(i)

= F(x(i)).

i.

: W -~ F(G(W) ).

m, there exists a unique

F(m')m(W)

F(m')t(i)

is epi in Ens,

Write

t(i) =

= Fix(i)).

Our argument

t(i) = F(y(i))

such

For we m a y then take the dominating

to be the set of all

G(W),

where

W

is any

-

quotient valence

set of

FIG(V)),

5 0 -

defined b y an equi-

let us say,

relation.

To p r o v e more general

the a b o v e

assertion,

let us c o n s i d e r

the

situation:

F(B)

>

F(A)

F(C)

P (v)

It is a s s u m e d seek

w

: A-~ B

Since

CAo}

the m a p p i n g and onto.

u

s u c h that

F(u)x

x = F(w)

-+

[F(A),

it is onto,

u w = v.

2, see also

= F(v)

and

subcategory F(B)]

Benabou

F(uw)

. We

u w = v. of

A_~

is o n e - o n e

there e x i s t s

Therefore

is one-one,

For case

and

is a left a d e q u a t e

x = F(w). F

: B-~ C

F : [A, B]

Because

s u c h that Because

that

w

: A ~ B

= F(u) x = F(v).

T h i s completes the proof.

(1965),

Th~or~me

5.

-

~. CQmpletions [A__ O, EnS]inf

when

of 9roups.

We wish to investigate

A_A is some known small category.

our first example we take one object,

5 1 -

A

to be a group

we may as well call it

(= elements of

given any two objects

r

[ik, ik+l]

for any or

and

I

I

connected

...

if,

in = j

'

k = O, 1, . ., n - 1,

[~+I'

and a

i, j ~ I, there exist objects

i ~ = i, i I ' i2,

such that,

has

G) are all isomorphisms.

F : I ~ A_. We call

(*)

G :A

G, and the maps of

Now consider any small index category diagram

In

~]

one of

is nonempty.

What do the lower bounds of a connected diagram look like? Consider

Suppose

(G, s) is a lower bound of F.

two neighbouring

+IJ

or

case we have a map

F

indices

~'

~+I'

so that

I, % 1

is not empty. In the first

Lk : ~

-~ ik+ I.

By naturality, -i

F(tk)

s(ik+ 1) = s ( ~ ) ,

hence

In the second case we m a y take s(ik+ I) = F(t k) s ( ~ ) . determined by

s(~).

s(ik+l)

tk " ~ + 1

Thus in either Applying

= F(L k) "~ ~

case

s(i k)and

s(ik+ I)

this to the sequence

is (*), we

obtain

(**)

s(j)

=

r(t

n-

1 )+-1 . . .

r ( ~ l )+1 r ( , o )~1

s(i)

= g s(i).

-

PROPOSITION

8.1.

Let

A__ be a g r o u p

is a lower b o u n d

of a c o n n e c t e d

inf

If

F =

(G, s).

connected

diagram

Proof. Given

two

(*), we of

G

of

i

have

-1

~ G.

t(j)

= s(j)h,

(We r e g a r d

F : I ~ ~,

F

is a c o n n e c t e d

j

which

are c o n n e c t e d

= g s(i),

where

then a dis-

g

t(j)

(G, s) j

fixed,

diagram. b y a sequence

is the e l e m e n t = s(j)h,

where

is the o n l y e l e m e n t

h

hence

as

(G, s)

has no info

Clearly

i

If

than one element,

(**). T h e r e f o r e

t(i)

that F.

by

G.

that

and t(j)

determined

h = s(i) such

assume

indices

diagram

has m o r e

F : I ~ A

First

then

G

52-

is in fact

as a n y e l e m e n t

of

G

the inf of

I.)

g

s(j)[ G

~"

G

F(j) Next

consider

m a y assume 11 and

iI

Let I

of

I

is the u n i o n

such that

12

for all

h/

that

a disconnected

in

(G, s) G

as follows -

11

[il, and

be a n y

and define

in

[i2,

i I]

We

categories

are e m p t y

12 9

lower b o u n d a new

F z I ~ A~

of two n o n e m p t y and

i2]

i2

diagram

of

F.

lower b o u n d

Take

any e l e m e n t

(G, t)

of

F

-

t(i)

Clearly

This

= s(i)

if

i

is in 11

= s(i)h

if

i

is in 12

there

for all

i

is no e l e m e n t

in

I.

completes

then

inf

8.2.

If

F =

But then also

G

F =

Surely show

If

that

t(i)

is n o t the

= s(i) g

inf of

P.

G, e v e r y

has o n l y one e l e m e n t

inf

= (T(G),

(TOP)

li(i)

1

= 1

and

F : I -* G,

for all

i

TolI),_ as is e a s i l y

has m o r e

(G, s). Then

I

is connected,

(T (G), Tos)

functor

infs.

G

that

such

is a g r o u p

where

than one e l e m e n t

ToP,

any other

let

i

b y the

sequence

lower b o u n d

b e a fixed (*). Put

in v i e w of

j

f = TIs(i) -I) u(i)

of

T(P(to)u,

u(~+ 1) = T(P(Lk )-+1) u(i~),

+I)

of

any index,

(**),

= T(F(tn_l) -+1) ... by naturality

index,

(X, u)

u

(i)

.

: X

I.

verified.

and that

by Proposition

is a lower b o u n d o f

in

8.1.

we w i l l

it is the inf.

Again,

Now,

A

that

Consider

Then,

G

(G, s)

(G, 1i) ,

N o w assume inf

of

Therefore

: ~ ~ Ens p r e s e r v e s Proof.

g

the proof.

PROPOSITION

T

53-

To F. connected -~ T(G).

- 54-

for

k = O, I,

the above,

..., n - 1.

(F(G),

f

Fos)

this r e p e a t e d l y

to

we obtain

T~sCj))

Moreover,

Applying

f = uCj).

is clearly unique with is indeed

inf F,

this property.

and so

F preserves

Thus infs,

as to be shown.

f F (G) <-

X

,(scj )

-- F (G)

COROLLARY. = [ A~

Ens]

sentations

If

is the

~

is a group

category

of the opposite

G

,

then

o Ens [~-9. 9 J inf

of all permutational

group.

repre-

-

5 5 -

9___.Completions of categories of algebras. an equationally operations,

~

a small subcategory of

[ -, C]

Proposition

3.2,

containing

A

preserves

~

be

defined algebraic category with finitary

the subcategory such that

Let

~

A_*

be

~ which consists of all algebras

C

Ao

-~ Ens

of

~.

preserves

Let

infs.

In view

is the largest subcategory

such that the inclusion

functor

of

B of ~

B

sups~

We will show that, under fairly general circumstances, o [A_~, Ens]inf of

is equivalent

R-modules,

to

A_*.

If

~

is the category

this result was first pointed out to the

author by Ulmer. The proof in general will rely heavily on known results by Isbell and Lawvere. PROPOSITION 9A (Isbell).

Let

C

be an equationally

defined algebraic category with operations Let

~

be a small subcategory

free algebra

in

subcategory of

generators.

~

is that

which contains the

Then

if

C

F

is a left adequate

~

is the category of groups, contains the free group

generators,

because multiplication

[See Isbell

(1960), 2.2

the

in two

is a binary operation.

and I.Io]

PROPOSITION 9B (Lawvere).

Let

~

be an equationally

defined algebraic category with operations at most and equations

n-ary.

~.

For example, assumption

n

of

at most

involving at most

n

variables.

Let

n-ary F

be

-

the s u b c a t e g o r y in at m o s t F O ~ Ens

n

of

~

which consists

generators.

of all free a l g e b r a s

T h en the functor

is an e q u i v a l e n c e

[~o, E n S ] p r o d

5 6 -

of

of all p r o d u c t

~

C

~

[ -, C]

w i t h the c a t e g o r y

preserving

functors

from

F ~ to Ens. For example, assumption most

if

is that

~

F

consists

three generators,

multiplication implicit

is the c a t e g o r y

because

involves

in L a w v e r e

PROPOSITION

of all free groups the a s s o c i a t i v e

three va ri ab le s.

(1963)

9.1.

and

Let

C

equations

involving

w i t h at m o s t equivalent

n

to

Proof. L(C)

= [

_

LoL'

~ 1

, C]

[A_~

M(B)

Then

n - a r y and Let

A__ be a

all free a l g e b r a s

[A__ ~

[F ~

EnS]inf

EnS]prod

: _F~ -~ Ens. P r o p o s i t i o n L'

is

be d e f i n e d b y 9B asserts

: [F O, E n S ] p r o d -~ C

that

such that

L'oL ~ I. functors

- [A O, E n s ] i n f ~ A__ ~.

write

which contains

L : C ~

a functor

and

at m o s t

defined

A_*.

We n o w c o n s t r u c t M'

C

[This re su lt is

n va ri ab le s.

generators.

Let

there exists

of

in at

law of

be an e q u a t i o n a l l y

category with operations at m o s t

the

(1965)].

algebraic

small s u b c a t e g o r y

of groups,

= [ -, B]

M

o : A * -~ [A__ , En S] in f

For an y a l g e b r a

: A__~ -~ Ens.

E n S ] i n f , its r e s t r i c t i o n

T/F

If to

B

T F

in

A.~

and

we

is any functor

in

is an o b j e c t of

:

-

[~o, Ens]in f

[~o, EnS]prod"

c

We claim that Indeed,

M'tMC ) naturally

in

[A O , Ens]inf,

MoM' ~ 1 take any

=

and B

in

=

57-

We write

=

L' ( T / F ) .

M ' o M ~ I. A_*, then

cB )

B. On the other hand, and put

M' (T)

8, take any

T

in

T' = M(M' (T)). Then

T'/x = '[M' (T)) -- '(L' (T/f)) -= T/_F 9 Now

T, T'

adequate

~ ~ Ens preserve = A__

subcategory of

infs,

and

_A ~ , by Proposition

we m a y apply Lemma 5.3, or rather its dual, T ~ T' = M(M' (T)). is natural

in

_F ~

is a right

9A. Therefore and obtain

It m a y be verified that this isomorphism

T, hence

MoM' E I.

This completes

the proof.

-

58-

i0~ Completions of categories of m~dules: write R

_~

for the c a t e g o r y of all right

is an associative

ring with

stood to be unitary.) and

~

of

MR

I.

We shall

R-modules,

(All modules

where

are under-

We shall consider subcategories

which satisfy the following three

conditions: I.

If

B

is in

B

and

C

is in

~,

then

[B, C] = 0.

II.

If [B,C] = 0

for all

B

in

B,

then

C

is in

~.

III.

If [B,C] = 0

for all

C

in

~,

then

B

is in

B.

Well-known examples, the following groups;

B

zB

when

R

is the ring of integers,

all torsion groups,

all divisible groups, ~

~ all torsion-free

all reduced groups,

all Abelian groups w i t h o u t nonzero divisible It is not difficult submodules extensions, of pair

C

to show that

if and only if i.e.,

implies B, ~

C in

C' in

C C

~.

(1962)

B

and

subgroups.

is closed under

C' an essential extension

In this case one might call the It follows from the work

that a torsion theory is completely described

by a certain set of right ideals of

R.

I am told that

torsion theories have also been investigated by Spencer E. Dickson. A pair of subcategories conditions category

I A__ of

i.e.,

is closed under essential

a "torsion theory".

of Gabriel

are

to

III

MR :

B, ~

of

_~

may be constructed Let

B = B(A~

satisfying from any sub-

consist of all

- 59-

modules let

B

such that

~ = ~(A_)

[B, C] = 0

[B, A] = O

consist of all modules

for all

B

in

the ring of integers and rationals, ~(~)

for all

then

B(A~

B(A~.

C

A

in

A, then

such that

For example, when

R

is

_AA consists only of the group of

is the c a t e g o r y of torsion groups and

is the c a t e g o r y of torsion-free groups. We shall show

that, under various conditions on c a t e g o r y of that

_~

A, ~(A~

= A*, the sub-

w h i c h consists of all modules

the functor

[ -, C]

: A__~~ Ens preserves

C

such

infs.

(See Section 9.) We begin by listing some elementary properties of B

and

C.

PROPOSITION IO.I. MR

satisfying

Let

conditions

B, ~

be a pair of subcategories

I to III. Then the following

statements are true:

(1)

is closed under factor modules

and under sups.

(1")

C

(2)

Every module

M

has a largest submodule

(2*)

Every module

M

has a finest factor module

is closed under submodules and under infs.

S/~ M

(3)

M

in C .

is in

modu le in (3*)

M

~M in B.

is in

module in

B

if and only if it has no nonzero

factor

C. _C if and only if it has no nonzero subB.

-

(4)

If

B

and

M/B

are in

B

then

(4*)

If

C

and

M/C

are

C

then so is

(5)

~M=

M o s t of this is easy.

to s t a t e m e n t s (4)

B

(4) and

Suppose

the i n c l u s i o n

where c

C

is in

Kerf.

Let

B

form N

is in

fm : B ~ C

and so 7M

B,

M/~M

attention

N But

f = O.

It is in

c ~M, b y M/yM

hence

ge = f o

mapping

~M-~ M-~ ~/u

. T h e n the image of

C Mo

f 9 M - ~ C,

epimorphism,

yM. On the o t h e r hand,

(4). T h u s C.

: B-~ M

m u s t be zero,

the c o m p o s i t e

~M C

m

any mapping

such that

g = O, h e n c e

~M c N

is in

B . Let

be the c a n o n i c a l

g 9 M/B ~ C

f : B-+ M/~M

by

in

C. T h e n

where

B,

are

and c o n s i d e r

hence

and

N/~M,

M.

We shall c o n f i n e

mapping,

On the one hand,

m u s t be zero, be in

and M / B

e : M-~ M/B

M / B is in (5)

M.

(5).

B

then there e x i s t s Since

in

so is

yM.

Proof.

be

60-

B,

by

let f

has

the

(I)o H e n c e

(3). T h e r e f o r e

is finer,

B

by

f = O,

(2 *) , h e n c e

~M. COROLLARY

I.

is c l o s e d u n d e r

Let

A

be a s u b c a t e g o r y

submodules.

Then a module

if and o n l y if it has no n o n z e r o a module module

of

M

is in M

We o m i t

has

C(A)

of M

factor m o d u l e

MR

which

is in in

A,

if and o n l y if e v e r y n o n z e r o

a nonzero

factor module

the s t r a i g h t - f o r w a r d

proof.

in

A.

B(A) and sub-

-

COROLLARY of

M

2.

Let

satisfying

B, ~

61-

be a p a i r of s u b c a t e g o r i e s

conditions

I to III

~ R

Then

C

is a

"

left r e f l e c t i v e

subcategory

and

B

is a r i g h t r e f l e c t i v e

subcategory. Proof. Let

We shall p r o v e

p = M ~ M/u

be

the s t a t e m e n t

the c a n o n i c a l

t h a t it is a b e s t a p p r o x i m a t i o n Indeed, f = me,

suppose

m mono,

e epi, w i t h

In v i e w of

(2"),

such that

e = xp.

m x

there

y

exists

such

M

in

where

image

C'.

~o

f = mXpo

that

~.

we c l a i m

(See S e c t i o n

C

is in

~.

C'

is in

C, b y

a unique mapping

Therefore

is the u n i q u e

epimorphism;

of

f : M ~ C,

concerning

Since

Write (1").

x = M / y M ~ C' p

is epi,

f = yp.

f M

~

I

p

C

e

M/u

m

........

)

C'

x PROPOSITION

10.2.

Let

A

Then any submodule

of a p r o d u c t

~(~).

holds when

The c o n v e r s e

A

I

be a s u b c a t e g o r y

of

of m o d u l e s

A__ is in

from

is c l o s e d u n d e r

MR

9

essential

extensions. Proof. m

First,

: M ~ P = ~iEI A.z

consider is mono,

a module where

the

M

and a s s u m e A. 1

3.)

that

are o b j e c t s

-

of

A

.

Let

Pi

and c o n s i d e r in

each

m f = O,

Consider K of C.

fore

there

exists

with

AK

A'

K

in

that

any By

C

where

M

C(A)

K

essential

gK

into

is A.a.

C(A~.

under

essential

and a n y n o n z e r o

is not

to

B

is in

in

homomorphism

fK

B

is c l o s e d

in

a nonzero

is a s u i t a b l e

A

(3*),

A_. E x t e n d

B(A~.

There-

fK : K ~ AK,

: C ~ A' K, w h e r e

extension

of

A__, for instance K

the

injective

Put

P' = ~

modules

of

hull

C,

and

Then

P ' K m = gK" Indeed,

this w o u l d

of

A'K,

epimorphism. that

f = O. T h e r e f o r e

assume

submodule

f : B ~ M,

epimorph[sms,

P i m f = O, as it sends

hence

Conversely, extensions.

be the c a n o n i c a l

any homomorphism

_B(A)_ . Then

Thus

: P ~ Ai

62-

~,

hence

where

K

let

P'K

there

ranges

We shall

be c o n t a i n e d

over

that

Ker gK = K*, in

Ker gK*"

A_. sub-

be the c a n o n i c a l

a unique

prove

of

all n o n z e r o

: P' ~ A ' K

exists

Ker m = ~K

also an o b j e c t

m m say.

: C ~ P'

such

is a m o n o m o r p h i s m . If

K* ~ O,

But then

0 = gK* K* = fK* K* ~ O,

b y choice

of

fK*'

and our proof

of

z the g r o u p Abelian

Therefore

Ker m = O,

is complete.

EXAMPLE. subgroups

a contradiction.

Suppose Q/z

, wh~re

of integers.

groups.

A

consists Q Then

o n l y of

is the g r o u p ~(A~

Q/z

or of all

of r a t i o n a l s

is the c a t e g o r y

and

of all

-

LEMMA

IO.I.

is c l o s e d u n d e r best

Let

LEMMA assume

~(~)

x

C

Proof.

Suppose

: M ~ A

Let

whenever

approximation Then

x

is r o u t i n e

10.2.

that,

be a s u b c a % e g o r y

submodules.

approximation The p r o o f

A

63-

in

of

the m o d u l e A

.

Then

~

be a s u b c a t e g o r y

M

has a

x is epi.

a module

: C ~ A

in

C

in

A_~

of

~(~)

_~

and

has a b e s t

x is an i s o m o r p h i s m .

~. In v i e w of P r o p o s i t i o n

3.2,

it s u f f i c e s

~ ~ ~(~)

preserves

F : I ~

(A, u)

A

with

in

which

and w i l l be omitted.

s h o w t h a t the i n c l u s i o n

holds

_~

sup

F =

in

sups.

. We c l a i m

to Let

this still

~(~).

Indeed,

put sup

r =

(M, v)

in

_~

. Now we have

best approximation

e : M ~ C = M/yM

of

M

By a known

sup

in

C

(see P r o p o s i t i o n

a unique

x

: C -~ A

theorem,

3.4.)

.

Therefore

that

x e v(i)

(C, eiov)

there e x i s t s

= u(i)

x

is a b e s t

x

is an i s o m o r p h i s m .

was

F =

for all

approximation

of

Therefore

i

in

C

in sup

to be shown. x

A

4-

C

u
r(i)

e

v(i)

> s

I. A F =

in

a

~(~).

By L e m m a

such 3.2,

. By a s s u m p t i o n , (A, u)

in

A,

as

-

PROPOSITION which

10.3.

is c l o s e d u n d e r

of the f o l l o w i n g CASE

I.

A

CASE

2.

If

C A S E 3.

Let

: K ~ C

A

and

M/A

Let

in

C

of

t h a t in all t h r e e c a s e s

the

(~) E v e r y e p i m o r p h i s m

e

A'

in

: K ~ A'

in

A_~ so is

e

A,

Write

a x = f.

then

k ' e = fk = a x k = O.

Therefore

K

is in

Moreover,

x

is epi,

in

assuming

k'

of Lemma

~(A~

and

x. We s h a l l

10.2

x

this to exist.

: C ~ A Put

see p r e s e n t l y

assertion holds=

: K ~ A',

with

f : C ~ A'',

A'

in

where

is an e p i m o r p h i s m w i t h (e). S i n c e

x

: C ~ A

there exists a unique

that

M.

A2 A''

can is

A.

F o r the m o m e n t w e a s s u m e approximation

A__* in a n y

are p r o j e c t i v e .

following

to a h o m o m o r p h i s m

Suppose

in

be a n y m Q d u l e

for the k e r n e l

of

_~

extensions.

s h o w t h a t the a s s u m p t i o n

A,

an e x t e n s i o n

are A

in

be extended

C(A) c

is c l o s e d u n d e r e s s e n t i a l

its b e s t a p p r o x i m a t i o n k

Then

of

three cases:

We w i l l

is s a t i s f i e d .

be a s u b c a t e g o r y

submodules.

All modules

Proof.

~

64-

: A' ~ A''

B(A),

Since

by Lemma

7.1.

in

: A ~ A''

is m o n o ,

K = Oo

A_~.

was a best

for the e x t e n s i o n

k'

a n d so

a

A'

Thus

in

such (e),

e = O. x

is m o n o o

-

65-

A

~a > A''

C

4 I I

Ik ,

k

I !

I

> a'

K

It r e m a i n s

A''

to v e r i f y

Case

1.

Take

Case

2.

Let

= C/K',

C/K ~

an

A'' K'

Hence

e

3.

be the k e r n e l

10.3.

that

~

assumption,

Let

Suppose

A

(B, v)

in

i~

Then = t(i).

in

~ A'. so is

f,

where

A.

take

Then

K/K'

and

C/K'. summand

A''

be a s u b c a t e g o r y

holds,

of

C.

= A'o

in

A__ ~. T h e n

of ~ A__*

B

for example,

is c o m m o n

However,

to

a diagram

A_~. S u p p o s e

there e x i s t s

t(i)

a unique

since

and has o n l y

B(A). when

in v i e w of P r o p o s i t i o n

there e x i s t s

F =

to

e, and

is a d i r e c t

in c o m m o n w i t h

R + R,

sup

K

h u l l of

the proof.

The a s s u m p t i o n and c o n t a i n s

K/K'

of

A, h e n c e

is s u p - d e n s e

the zero m o d u l e

Proof~

of

In this case,

m a y be e x t e n d e d

LEMMA

fv(i)

to be the i n j e c t i v e

are in

This c o m p l e t e s

assume

(e) in t h e three cases:

extension

(C/K')/(K/K') Case

e

B

A__~ and F : I ~ A

A_A is small 9A. B(A~o

By

such t h a t

: F(i) ~ A, n a t u r a l l y f : B ~ A

is in

B(A~,

such that f = O, h e n c e

-

t(i)

= O.

Thus

sup

also

sup

We may write F = F =

(O, O) (O, O)

this

OO(i)

in

A.

Since

in

A*,

and

f

-~

B

66-

so

O .........

and assume that

sups,

~ A

/ (i)

10.4. A

Let

~

be a subcategory of

is sup-dense

in

_~;

A_* and that ~ ( A ~ c A__*.

C(A_) = A_*. Prool.

is in

= O.

r(i)

PROPOSITION

Then

~

O(i)

r(i)

O(i)

B = O.

j\

v(i)

where

_AA-+ A__* p r e s e r v e s

O

A

= t(i),

Consider

~(A_),

any module

hence in

closed under infs in

M

in

A__*. Then

A__*. Now, by Proposition

_~. Moreover,

~M

M/~M

3.2,

A__* is

m a y be regarded as

p the inf of a diagram epimorphism

and

By Lemma 10.3,

M

~ M/~M, where p is the canonical O O the zero map. Therefore ~M is in A*.

~M = O, hence

M

is in

~(A~.

Thus also

A__* c _c(~. SUMMARY.

We have shown that

following assumptions:

A.* = ~

under the

-

67

-

(I)

A is sup-dense

(2)

A is closed under submodules.

(3)

One of the three cases of Proposition

In view of

in

A ~.

(2), we m a y then say that

those modules

M

has a nonzero Proposition

in

~.

in

Proposition

[~o, Ens]in f

left adequate

when

R + R

Moreover,

and products,

Proposition

7.2,

contains

[A_~ ,~ EnS]in f

to

(I) will hold if

If even

R + R + R

allows us to conclude

subcategory

and Lemma 6.1o

small,

9.1

R + R.

is equivalent

Actually,

modules

I

M

IO.I.)

is small and contains A~

submodule of

(See Corollary

We have already pointed out that A

holds.

A__~ consists of all

such that every nonzero

factor module

10.3

to

is

that

~(A).

is in

~,

~(A__) contains

{R + R}, by Proposition ~(A)

the

9A

is closed under sub-

hence under direct sums. By

~(A)

is sup-complete.

Thus, when

R + R + R, and satisfies

will be sup-complete.

(2) and

We present

~

is

(3),

a few

examples of this last situation. EXAMPLE

I.

Let

I

be the injective hull of

(regarded as a right R-module) of all submodules EXAMPLE

2.

of Let

the group of integers,

and suppose

I + I + I. R

R

A_consists

(Case i of Proposition

be the group of reals and and suppose

Z

A_A consists of all

10.3.)

-

submodules of

of

Z + Z + Z , EXAMPLE

A

R/Z .

consists

3.

EXAMPLE suppose

Let

R

~

that

stipulated

sure that it is in

Let

consists

right R-modules.

R

be right

10.3.)

and suppose

that

(Apply Case R

2

is

A. In this case, generated".)

semihereditary

of all finitely generated

(Either Case

a copy

I of Proposition

right R-modules.

is the same as "finitely 4.

A_A contains

be right Noetherian,

10.3. We have

to make

"Noetherian"

-

and again apply Case

of all Noetherian

of Proposition Noetherian

(Observe

68

2 or Case

, and

projective

3 of Proposition

10.3

will apply.) EXAMPLE

5.

A

consists

of the Abelian

groups

O, Z, Z + z, Z + Z + Z. (Case 3 of Proposition

10.3.)

POSTSCRIPT Since these notes were following

important

Structure

of categories,

Here

typed there has appeared

paper on the same subject:

is a list of closely

results: Isbell

Lambek C o r o l l a r y to Proposition Proposition 3.2 Proposition 6.1 Proposition 7.1 Lemma 7.1

John R. Isbell,

Bull.Amer.Math. Soc.,72(1966), related

2.1

3.3C 3.4 3.10 3.12 2.4

the

619-655.

-

69-

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Offsetdruck: Julius Beltz, Weinheim/Bergstr

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