Coherence in Categories

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

Author: Saunders Mac Lane (Editor) | G. M. Kelly | M. Laplaza | G. Lewis

59 downloads 1167 Views 7MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

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

281 G. M. Kelly, M. Laplaza, G. Lewis, and S. Mac Lane

Coherence in Categories Edited by Saunders Mac Lane, University of Chicago, IL/USA

+ Springer-Verlag Berlin. Heidelberg. New York 1972

AMS Subject C]~qsifications (1970): 1802, 18A25, 18D15, 18D25

ISBN 3-540-05963-6 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-05963-6 Springer-Verlag New York • 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1972. Library of Congress Catalog C~d Number 72-87920. Printed in Germany. OffsetdnwAcJulius Beltz, Hemshach/Bergrtr.

PREFACE Categorical

arguments

are full of diagrams

very many cases these diagrams theorems

which

The first

tensor product natural

Coherence

conditions

such coherence

functor

® which

sufficient

are

always

commute,

to insure commu-

but only up to a

isomorphism

One of the diagrams

involving

~

(A®B)OC

a

[ (A@B) ®C] ®D

[A® (B®C) ] eD that both possible

The first coherence

commutes,

then all

will also commute.

(larger)

bibliography Calculus,

reassociations

theorem asserts diagrams

homotopies

275-282),

by S. Mac Lane and by D.B.A. others mentioned

, of four factors

involving

this a s s o c i a t i v i t y

later

in algebraic

by J.D.

topology

and was also established Epstein.

Stasheff (Trans.

independently

Their two papers

in this preface)

are those cited

(and the in the

of the article by G. M. Kelly on Many Variable

Functorial

in this volume.

The next major

coherence

question

are

that if this pentagon

This result was first obtained

in his studies of higher I08(1963),

a is the pentagon

> (AOB)@(C®D)

Ae [ (BOC) eD]

which requires

,

this associativity

A®[B®(C®D) ]

A.M.S.

and in

result dealt with the case of a

is associative,

a:A®(BeC)

equal.

theorems

state that a large class of diagrams

or theorems which describe tativity.

commute.

of arrows,

was that for diagrams

IV involving both a tensor product functor and an (internal) hom functor; categories of vector spaces or of modules over commutative rings present such diagrams.

The general case is that of a closed category;

the work of Eilenberg-Kelly brought out the great utility of their study but involved very many complicated commuting diagrams.

This

emphasized the importance of getting a good coherence theorem.

Lambek

obtained a preliminary coherence result which recognized the connection between these questions and the cut-elimination theorem of Gentzen-style proof theory, and then Kelly-Mac Lane combined some of Lambek's ideas with the notion of the "graph" of a generalized natural transformation to obtain a much more extensive coherence theorem covering many of the diagrams arising in closed categories. The present volume on coherence collects a number of closely related papers which take the next steps in this development.

First,

one wishes to have not just an effective treatment of closed categories but one for "relative" categories

(those based on closed

categories) and of functors between them.

The first paper in this

volume, by Kelly-Mac Lane, shows that the resulting coherence problem, and the corresponding problem for natural transformations between functors,can be treated by a straight forward extension of the Kelly-Mac Lane closed category method.

In similar spirit Geoffrey

Lewis treats a coherence for a "closed functor" between two closed categories; here interesting new phenomena arise because it is by no means the case that "all" diagrams commute; the problem is rather that of giving necessary and sufficient conditions for commutativity. Finally, Kelly,in his study here of the cut-elimination theorem, shows that this classical result really provides for the preservation of coherence under the addition of adjoints; the original case being the coherence of tensor products and the addition to O of a right adjoint, the internal hom, one for each functor - ® B.

This general

understanding is formulated against the background concepts provided

by Kelly's paper "An Abstract Approach to Coherence";

there it is

indicated how each coherence problem can be described in terms of the "actions" of a suitable structural category called a "club" club is a monoid in a comma category).

(Each

Moreover, this development

in turn requires a generalized functor category, where the objects are not only the usual functors A functors Ax,¶°xA

>B

>B

of one variable, but also

of n variables, for all n.

This concept,

which is of independent interest, is set forth in Kelly's first paper in this volume "Many Variable Functorial Calculus I" Monoidal functors provide another fascinating coherence problem, treated by Lewis in his paper in this volume "Coherence for a

Closed Functor"

A monoidal functor @:V

~V'

between two cate-

gories V and V', each with an associative tensor product, is a functor between the underlying categories together with a natural transformation

~:~Ae~B This the

transformation case

of the

need not

forgetful

> ¢(AeB)

.

be an isomorphism,

functor

U:K-M~od

as

~Ab,

for

instance

in

(K a c o m m u t a t i v e

ring) where there is a map ~:UA®UB

> U(A®KB)

not necessarily an isomorphism.

One assumes, with the given asso-

ciativities a and a' in V and V', that the diagram

commutes. commute?

The coherence question is:

Do all bigger such diagrams

Moreover, does the same hold when we add identities

I and

I' for each ® product with natural isomorphisms b:I®A ~ A,b':I'@A'~A ', 7:AOI~A, and when we add to the data above a morphism ~°:I'

~I

(~,~) for the monoidal functor

such that

Vl I'8~A

~IS~A both commute.

b'

> @A

~ASI'

g>~(ISA)

@ASOI

r'

>@A

g>~(A®I)

For the answers to these questions and the correspond-

ing questions for closed categories, we refer to the paper by Lewis. Distributivity involves a different sort of coherence question, which has been raised by Hyman Bass been studied by J. Benabou

(in conversations)

(results not yet published).

and which has Given are

two functors, direct sum • and tensor product 8, with a natural homomor phi sm d :AS (BeC)

> (ASB) • (A®C)

One again asks for a certain number of basic identities which will suffice to get the coherence of all diagrams involving this distributivity d -- and involving as well the associativity isomorphism for both 8 and 8.

The first paper by Laplaza in this volume tackles

this problem.

Here again, it is not that all diagrams commute;

Laplaza's second paper in this volume, building on his first, gives a definitive necessary and sufficient condition for commutativity. There are many further problems, in particular those noted by Kelly in his papers here.

We trust that this volume may focus

interest and encourage further progress.

Saunders Mac Lane June 6, 1972

TABLE OF C O N T E N T S

G. M. K e l l y and S. Mac Lane: Closed C o h e r e n c e

for a Natural

Transformation

...

1

M. Laplaza: Coherence

for D i s t r i h u t i v i t y

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

29

G. M. Kelly: M a n y V a r i a b l e F u n c t o r i a l Calculus.

I ............

66

G. M. Kelly: A n A b s t r a c t A p p r o a c h to C o h e r e n c e

............... 106

G. Lewis: Coherence

for a Closed Functor

.................. 148

G. M. Kelly: A Cut-Elimination

Theorem

....................... 196

M. Laplaza: A N e w Result of C o h e r e n c e

for D i s t r i b u t i v i t y

.... 214

CLOSED COHERENCE

FOR A NATURAL

TRANSFORMATION

*

G. M. KELLY The University

of New South Wales,

Australia

and SAUNDERS The University

Received

transformation

k:F

an arrow kA:FA

>G

Chicago,

February

i. For two functors

MAC LANE

of Chicago,

Ill., USA

23, 1972

INTRODUCTION

F, G:A

~A',

states

the definition

that k assigns

of a natural

to each object A of A

> GA such that the diagram k FA ~

GA

FB

commutes that

for every morphism

"all" other

the diagrams A

~ B

)C

stated with

kB

f:A

appropriate

> GB

> B of A.

diagrams

This one condition

commute;

implies

more explicitly,

all

built up from k, F, G, and a string > .-suitable

transformation.

~E

of composable

precaution,

This paper

is a coherence

is concerned

theorem when A and A' are not ordinary based on some closed

morphisms

category

V

of A.

This fact,

theorem

with proving

categories,

for a natural

such a coherence

but categories

(and when F and G are correspondingly

i

~-functors). The resulting

theorem will

[i] for a closed category.

include our recent coherence

That theorem has made unnecessary

theorem the many

* This paper is based on research supported by grants from the National Science Foundation and the Louis Block Fund of the University of Chicago.

-

individual

verifications

grams needed

as in

to establish

gories.

Our present

diagrams

involving

2

-

[3] of the commutativity

the fundamental

result

similarly

avoids

not only the closed

A and A', M-functors

transformation

>G.

our previous

coherence

The surprising

proof

several V-functors

M-category,a in

requires

[-functor

of commutativity

of these

A00p

x ~0

gory A_Op ® ~ and a V-functor place

the definition

category

and not their properties latter

A has an underlying

hom-functor

theorems

transformations

involve

as we aim to make redundant.

that a M-category valued

coherence

>M;

could be

-- say for between

them.

of V - n a t u r a l

of a

as established

"do not know"

category

~0' w i t h a ~-

still less that there

Hom:A_Op ® A

so

just such verifications So we

ordinary

of

case,

us to use only the basic definitions

etc.,

[3]; for the proofs

natural

and a M-natural

to the present

data over a closed

with various

Our purpose

>A'

fact is that the methods

[i] apply exactly

for more complex

cate-

V but also extra data

F,G:A

that our proof will also show how similar established

of closed

the even more horrendous

category

such as E-categories k:F

properties

of large dia-

>Z-

is a Z-cate-

For this reason,

transformation

given

in

we re-

[3], which

J

used ~0 and the functor hO °P x ~0 easily

seen to be equivalent. We consider

monoidal. torial

>V, by a more basic one that is

first such a k:F

The coherence

argument,

>G:A

in this case

pentagonal

diagram

when V is merely

is proved by a purely combina-

like the now classical

tivity of a certain

>A'

proof implies

([6],

[7]) that commuta-

coherence

for associa-

tivity. We then pass to the case of a closed M, by adding adjoint

[X,-]

of -®X.

The difficulty

that now, when we form a composite make up our diagrams,

in the way of a similar >B

B may be arbitrarily

A and C, so that the familiar We find our salvation

A

the right

>C

of the morphisms

much more complicated

kind of proof by induction

in a cut-elimination

proof

argument,

is

that than

is precluded.

directly

-

generalizing via Lambek

[5].

and inherited

This argument asserts

(a c o m b i n a t i o n

e after a given m o r p h i s m [B,C]

• A

"cut").

>C, f:A

l~f >[B,C]

For

A central

and commutativities),

already constructed,

of a given m o r p h i s m A ® B

evaluation

(Gentzen's

there are four basic types:

of a s s o c i a t i v i t i e s

tensor product of two m o r p h i s m s >[B,C]

[4]

that the m o r p h i s m s w h i c h con-

to the above o p e r a t i o n of c o m p o s i t i o n

morphism

from Gentzen

can be built up from certain basic ones w i t h o u t

the case of a closed category,

A

-

that in our earlier paper,

stitute our diagrams recourse

3

a

an adjoint

or a process of applying ~ B as in

® B

e >C.

(Actually we need a more complex process of evaluation w h i c h acts on two given m o r p h i s m s [B,C] These

f:A

® A ® D

>B

and g:C ® D

l®f®l >[B,C]

® B ® D

four types are used as follows:

[B,C]

® A ~ C ®

with given domain and codomain. phism m i g h t

e >C

>[B,A]

in

regarded

provided

(=formulas)

are

struction,

The basic

as d e s c r i b e d

essentially

idea of the

process,if

possible,

that is, that they never

[nonconstant,

is the same process,

corresponding

to the con-

the proposed domain and codomain,

"proper";

involve an i n t e r n a l - h o m of the form present argument

~ S ®.--

is thereby reduced

[i] is to show that this c o n s t r u c t i o n

as shapes

g>E.)

by one or more of the four basic

The problem

does yield a unique result,

® D

These data suggest that such a mor-

struction of simpler p r o s p e c t i v e morphisms. proof

to form

There is a p r o s p e c t i v e m o r p h i s m

[R,S]

in fact be c o n s t r u c t e d

types of construction.

>E

constant].

Our

with just one more type of conto the c o m p o s i t i o n

of a string,

above.

This d e v e l o p m e n t m a y cast some light on the surprising use in category theory of the p r o o f - t h e o r e t i c We recall

that Lambek in

m e t h o d of cut elimination.

[i] started this use;

our work has

-

4

-

also been stimulated by knowledge of the results of M.E. Szabo published)

and of an early form of the results of

G. Lewis

(not yet

(now pub-

lished in this volume).

2.

STATEMENT OF RESULTS

Let ~ be a monoidal category, with the usual tensor product ® and the usual natural isomorphisms a (associativity), and b = bA:A ® I ~ A, as specified for example in [i].

c (commutativity) A E-category

is a set of objects A, B,-.-, a function which assigns to each pair of objects A, B a "hom-object"

(A,B) or A(A,B) which is an object of ~,

together with two families of morphisms B

M = MAC:

(B,C) ® (A,B)

>(A,C),

j = JA:I

>(A,A)

such that the familiar diagrams

{'IC,D)~(B.C)} ®(A,B)-~-->(c,D)®[(13,C)®(A,B)] I@M > (C,D)®(A,C) (2 .i) (B,D)®(A,B)

> (A,D),

b~

l®J A (A,B}®I

JB 01 ~(A,B) ~ (A,A)

I® (A,B)

~ (B,B) ® (A,B) (2.2)

(A, B), always commute. written

(A,B)

Let A' be a second such category,

(A',B')' or

(A',B').

A ~-functor F : A

with hom-objects

>A'

is then a pair of

functions which assign to each object A of A an object FA of A' and to m

each pair of objects A, B of A a morphism FA,B:(A,B)

)(FA,FB)

of V

such that the familiar diagrams (B,C)®(A,B)

<

(FB,FC) ®(FA,FB) always commute.

M

> (A,C)

I 3A >(A,A) (2.3)

M'

> (FA,FC),

(FA,FA)

These definitions apply also when V is a closed u

-

category

5

-

(add [-,-]).

If F and G:A tion k:F

>A' are two ~-functors,

>G:A----~A'

a V-natural

transforma-

is a function k which assigns to each object A

of ~ a morphism k A : I - - ~ ( F A , G A )

of [ such that every diagram

F

(A,B)

> (F~,FB) (2.4)

(GA, GB)

k*

> (FA, GB) A,GB

commutes,

where k ~ = k~, ,B and k* = kA,B, are defined from k by (A',FB)

k~

> (A',GB)

bc~ •

~M'

I®(A' ,FB)

kB®l > (FB,GB)®(A' ,FB),

k*

(GA,B')

(2.5)

> (FA,B') (2.6)

(GA,B') ®I

l®kA > (GA,B') ® (FA,GA) .

When V_=Set with ® = ×, then I is a one-point as a morphism F A

> GA in A'.

set and we may regard k A

Then k~,,B_ is just composition with

*

k B on the left, kA,B, first commutative

is composition

with k A on the right,

diagram above is just the familiar definition

naturality as given in our introduction

>(FA,GB),

then a natural

fined to be a family of such morphisms M'o(v®F)

k A = VA,AOJA:I

Then k may be constructed > (FA,GA).

of these two definitions

transformation

> (FA,GC)

(2.4) is can be de-

(2.7)

from v as

An easy diagram

of naturality.

diagram

VAB such that the diagrams

= voM = M'o(G~v) : (B,C)O(A,B)

all commute.

of

-- or as in [3], page 466.

If the diagonal morphism of the commutative called VA,B:(A,B)

and the

chase shows the equivalence

The definition by k is more

-

convenient

monoidal usual d:X

-

for our purposes.

The whole ~-natural

6

situation

transformation naturality.

k and the accompanying

If in addition

sense that -®X:~ ~[Y,X®Y]

W = IV, A, A', F, G, k~ consisting

>~

has a

shall call the whole

situation

data will be called a

the category

right adjoint

and counit e:[X,Y]

® X--~Y

of the

V is closed,

in the

[X,-], with unit

for the adjunction,

a closed , naturalit[,

we

we abbreviate

both terms to naturality. We can now define a certain terms of "shapes". formulas

These

shapes,used

for the functors

[ , ]; thus the shape

typical

obtained

in

I is the formula

S and T were defined

recursively

the N-shapes, (FA,B'),

vious

to include

(GA,GB),

A and B'

We retain

etc.

Formally,

four inductive S5

(-,-)

and

we then define rules (x,y)'

These N-shapes

number

n = {i,

Variance: The ordinal

associated

are N-shapes,

also a new one, (A,B),

(A',B'),

by adding

to the pre-

where each of x and y is any

(nine cases). Each N-shape

n~ equipped

contra~,

Type:

in n) are defined set v(T)

with two functions n--~{V,

sum n ~ m

as before. defined

is to have

is now to be a finite

A, A ~

.

(change the variances With each N-shape

just as before except

rule V2 for the shape 1 is replaced

rules

the shapes

Sl to $4 for shapes one new such rule:

2,...,

an N-variable

the previous intended)

our N-shapes

sum n + m and the twisted

of all variables

I,

- and -' instead of variables

where an N-variable-set

n--~co,

from ® and

that I, i, T ® S and

and introduce

$5 we call the terms.

its set of N-variables,

In general,

built up using also

We will write

one of -', F-, or G-

ordinal

this notion,

in

the well-formed

composition

~.

by requiring

functors

naturality,

for the functor constantly

for the identity ~

are shapes.

[l],were

by iterated

and 1 is the formula

[T,S]

or "formal"

by the

T is that

(evidently

-

V2:

v(1)

is the ordinal

VS:

v(-,-)

7

-

l, covariant,

type V m

is the ordinal

2, contra,

co, both of type A,

v(x,y)' is the ordinal

2, contra,

and co, where x = -' has

type A' , x = F- has type A and x = G- has type A. _

For N-shapes point-free

T and S an N-graph

involution

on the disjoint

property

that mates under

opposite

variance

N-graphs,

evident descriptions two N-graphs

jects all N-shapes N-analog shapes

a closed

and morphisms

of the category

called

(those formed without

way a monoidal

category

First,

construct

object

the shape

(-,-),

The composition

as before.

[1]).

of

With the [~,~]

of

~0 of N-graphs,

with ob-

(This category

~0 is the

Moreover,

square brackets)

the inte@ral

N-

form in the same

~00"

Over ~0 we then obtain lows.

all N-graphs

using

with the

~ ® ~ and a bracket

category

G in

+ v(S),

~ v(S).

are defined

of a tensor product

~, ~ we obtain

is to be a fixed-

~ have the same type and

sum v(T)

N-graphs,

>S

union v(T)

the involution

in the twisted

and compatible

~:T

a "typical"

a ~0-category

~

closed naturality, with one object

and with composition

as fol-

-, with hom

and unit morphisms

the

evident N-graphs (y,z) ®(x,y) where we have replaced under

the intended

letters.

Take ~

>(x,z),

the blanks

I ---~(x,x) ,

(2.8)

in the shapes by letters,

graph represented,

in the familiar way,

to be the ~0-category

with mates by the same

with three objects,

written

-', F-, and G-, with the corresponding

shapes

introduced

in ~0' and with the definition

in S5 above)

of composition descriptions F 0 and GO;

as hom-objects

M' and unit j' like those

yield ~0-categories

for example,

(the nine terms

just given

~0 and A~,

for ~0"

These

and there are ~0-functors

F 0 sends the sole object written

- in ~0 to

!

the object denoted

F- in A0, and the arrow F0_ - is the graph

(x,y)J

Finally,

>(Fx,Fy).

(x,y)

the graph

I

>(Fx,Gx)

from the shape

I to

-

the shape

(F-,G-)

naturality striction

8

is ~0-natural.

W0, consisting to integral

-

Thus we have defined

of ~ ,

~,

N-shapes,

~,

a closed

and k0:F0----~G 0.

we similarly

obtain

Upon re-

a monoidal

natur-

ality W00. The closed category objects

and N-graphs

gory ~ of graphs in a different naturality

precaution

as morphisms;

as used in

closed

N(~)

needed

jects of ~.

were

IS I

S

ables defines

(in V).

that each

these will provide

case;

theorem:

(ordinary)

the natural

for ob-

functors.

This

shape T in n variITI =

ITIv of n

constructions

transformations

of

between

over which we then get a

an obstacle

because,

and ~' are not ~ priori given as categories not interpret,

say,

the shape

(-,-)

~0' because

be used to simplify by regarding

our coherence

by their objects.

at all,

We do not want to

result,

suitably

as the discrete

result

is only concerned

reference

to morphisms

with diagrams

(ordinary)

formulated,

We eliminate

can

the obstacle

categories

The reason we can do so without

coherence

so that we can-

A has an underlying

the proof of this fact.

A and A' henceforth

while V is a category,

as a functor.

the fact that each ~-category

category

(shapes)

which

N(~) were the natu-

Z a functor

for ~ ,

of

W#.

This program meets

assume

The diagrams

the associated

category

the morphisms

The objects

this was just the usual

We will make the corresponding

in the present

naturality

[i].

of the category

> ITI between

for each closed

in

lie

on V and on the

of V but formulas

>T

as

which commute will

~#, which depends

not objects

on the observation

variables

The diagrams

still the shapes;

The morphisms

W 0 has N-shapes

it is the exact analog of the cate-

to state any coherence

ral transformations

functors

[i].

category

have vertices

depended

in the naturality

W, and will be the analog of N([)

that category

commute

~

determined

loss is that our

in V and makes no

in A or in A'.

Let W be a naturality

with A

(and hence V and A')

non-empty.

-

Let T be an N-shape equal

-

with variable-set

to the category

to the variance

9

n.

For each i in n take B. 1

(type i) or the category

of i in n.

(type i) °p according

Now use T to define

a functor

ITI : ITLw:£ 1 ×...× --n B --~Z by recursion shape T. of

IT~

in the evident way suggested

We can in fact take over our previous

([i], page 103 foot)

new terms.

upon adding

Thus the shape

I(-,-)I:A_ °p x ~

(-,-)

A is discrete.

ing hom-object

functor

-

-

×

G

>ISI

components,

[i].

naturality

conditions,

(discrete)

categories

with one precaution.

described value

in

[i]).

ject of V.

transformations

When the graphs variables

The resulting

Similar

transformation

Xi;

in

>S

and certain

are to satisfy for variables

the usual from the

from V.

is again defined

The comas in

[i],

~ and ~ of g and f are not comwhich occur

To get a definite

composite,

in closed

loops

(as

one must choose a

[i] we took X i = I, the base ob-

"variables"

the variables

functor

seven terms of S5 above.

fixed object D in the non-empty

in ~' to FD.

as a functor.

A and A', but not for those

Now we also have

by specializing

considered

which are vacuous

for each such variable

func-

is the correspond-

is the composite

of an N-graph Ff = ~:T

there are certain

for our

>v,

These components

posite gf of two natural

rules

L(-,-)I

T and S, a natural

now consists as in

~(-',-')I

I(F-,G-)~

-

apply to the other

Given two N-shapes

patible,

Similarly

>A'

definition

this is trivially

>A_ '°p × A ' ~.op

here G is the object-map definitions

the appropriate

>(A,B);

for A', and

A °p × A

recursive

for the

leads to the functor

> Z given by A,BI

torial because

f:ITl

by the notation

category

in A and A'; we select a A, and we define

in V to I, those

composites

a composite

in A to D, and those

of natural

transformations

then depend on the choice of D, but for fixed D the composition

will is

-

i0

-

still associative and determines a closed category ~# with objects the N-shapes T, S,.-. and morphisms T mations ~TIw

~IS~w.

>S all the natural transfor-

It will turn out that the dependence on D does

not matter (we will actually use only the "compatible" part of ~#). Next we construct a closed naturality W# = {~#; F#, G#:A#---~A~; k#:F#---~G#~.

Here the ~ - c a t e g o r y A# is

to have one object, the "variable" -, with hom-object the shape (-,-) regarded as an object of V#; composition is the natural transformation with the graph already described in (2.8) but with components given in the evident way by the actual composition operation M in the given ~-category ~.

The unit map j for A# is constructed in a similar way.

Corresponding constructions yield a ~ - c a t e g o r y ~ -', F-, and S-.

Then F # : A # - - - ~

on objects, while ~#(-,-)

with three objects

is the Z#-functor which is (-)~ ~F-

>A~(F-,F-)

is that morphism in ~

which

has the graph F0_ - already described and the components FA,B:A(A,B) ----->A'(FA,FB) , for each pair of objects A, B of A.

Thus F#, and similarly G#, is a

sort of transliteration of F to act on N-shapes; it is routine to verify that it is a ~#-functor. transformation with graph I I

Finally, k#:F#---~G# is the natural

> (Fx,Gx) and with components

>~(FA,GA) the components k A.

Thus we have defined a closed

naturality W#; moreover, the process "take the graph" defines a morphism F:W#---~W 0 of naturalities which is strict (i.e., one that preserves all the naturality data on the nose). applies for monoidal shapes.

Again the same

We have in effect reproduced much of

the structure of W within W# (though W# is much "smaller"); the diagrams which commute are to be those in W# -- i.e., certain diagrams of arrows in Z#" Given any naturality W, involving a closed category ~, the class of central morphisms of V is the smallest class of morphisms

-

containing all identities

ll

-

for objects of V, all instances of a, a "I

b, b -I, c, and closed under ® and composition.

The allowable mor-

phisms of [ are defined in the same way, but requiring also closure under

[ , ] and including all instances of the unit d and counit e of

the basic adjunction; d:X

i.e., all

> [Y,X®Y] ,

for objects X and Y of V.

e: [Y,X[®Y

'

>X

These closure conditions are exactly the

five conditions AM1 - AM5 of

[i], except that in [I] they were applied

only to a category in which the objects are shapes.

We now introduce

also the N-central morshisms , by adding to the closure conditions for central morphisms all instances of M, j, M', j', F, G, k, k* and (e.g., for every pair of objects A, B of ~, FA,B:(A,B) N-central).

>(FA,FB)

is

We similarly define N-allowable by adding these same con-

ditions to the definition of allowables. construct a closed category AllOWN(~)

Observe that we can then

with objects all the objects of

and with only these morphisms which are N-allowable. naturality then restricts to a naturality

The given

(with the same A, A', k)

based not on V but on this closed subcategory AIIOWN(~).

Briefly,

the N-allowable morphisms of V are those, and only those, necessary for the description of the associativities, formations,

etc. at issue.

In particular, N-allowable N-graphs, tions.

functors, natural trans-

the N-allowable morphisms of ~

are called the

those of [# the N-allowable natural transforma-

With this language, we will prove that all the main theorems

2.1 - 2.4 of

[i] hold in the present case

(replacing "graph" by "N-

graph" and "allowable" by "N-allowable"l.

In particular,

proper N-shape by the previous four conditions P S l tial restrictive condition constant is improper), PS5.

Every term

is that

(the essen-

IT,S] with T not constant and S

adding the condition (-,-) or

PS4

we define a

(for x,y = -', F- or G-):

(x,y)' is proper.

-

The principal

coherence

Theorem 2.4 N. N-shapes

12

-

theorem then states

If W is any naturality,

then any two N - a l l o w a b l e natural

while T and S are proper

transformations

f, f ' : I T I w - - - ~ I S I w w i t h the same N-graph are equal: In other words,

f = f'

any d i a g r a m of N - a l l o w a b l e maps b e t w e e n proper

shapes commutes when the d i a g r a m of graphs commutes. It will be clear from the proof that there is nothing to change if, instead of k:F

>G,

we are given only F : A

>~'

or only A, or

only A and A'. D

m

3.

THE C E N T R A L M O R P H I S M S

The proofs of our coherence simply by e x t e n d i n g

the proofs of

those shapes w h i c h are terms. the p a t t e r n and n u m b e r i n g of consider

[i] to take care of the n e w case of

Indeed our argument will

follow exactly

[1] w i t h a m i n o r technical variation:

the central m o r p h i s m s

We n o w define:

theorems will be c o n s t r u c t e d

(§ 4 there)

An N - s h a p e

before the "monoidal case".

is c o n s t a n t w h e n its c o n s t r u c t i o n

involves no terms and no shapes i; it is integral when no involved; the form

it is prime w h e n it is a term,

[ , ] are

a shape I, or an N - s h a p e of

[T,S].

It is p l a u s i b l e

to express any N-shape T as an iterated tensor

product of its prime factors; T.

We

Specifically, PROPOSITION

we call this the ~rime f a c t o r i z a t i o n

of

an easy r e c u r s i o n proves 3.1.

There

is a unique way of a s s i g n i n g to each

N-shape T an integer n ~ 0 and a list X l , ' ' ' , X n of prime N-shapes (the prime factors of T) in

such a way that for each T exactly one of

the following holds : 1 ° n = 0 and T is a constant

integral N-shape,

2 ° n = 1 and T = X 1 is prime, > 3 ° n = i, T = R ® S, R has prime factors YI,''',Yk, factors

Z1,-.-,Zm,

n = k + m and

S has prime

-

13

{XI,''',X~= This prime differently

factorization

(and more

"substituting"

prime

simply) shapes

the shape

(-,-)

including

the constant

I here does

-

{Yl,''',Yk,Zl,''',Zm} is exactly

expressed;

like that of

in an integral

shape.

prime

factor

not count as a prime

functor

objects

[I, I ® I], but that

is the identity

3.2.

permutation

of ~ l , ' ' - , n ~

variable

in Y~i"

the same condition,

AM5 used

in

in

factors. morphism

in K.

If XI,.-.,X n

those of S, then there

is a

permutation

then there

the same

of { l , . . . , n ~

is a permutation

hi # i implies

which

~ of

that Yi and Yki

shapes. x of K for which this

satisfies

the three closure

conditions

[i] to define

central morphisms

and therefore

central morphisms

the

can be described

factors.

The set of all those morphisms

holds

or ~# and let

the graph of x exactly

that ~' = k~; moreover,

PROOF. condition

n of prime

If ~' is another

are equal constant

(the first)

such that X i = Y~i and such that each

of X. has as mate under 1

{l,'--,~such

of the prime

of T and YI,..-,Yn

variable

satisfies

factors,

In either case,

> S be a central

Then T and S have the same number factors

that in

(when K = ~) or the

and a central m o r p h i s m

Let x:T

are the prime

~

(when K = ~#).

way as a permutation

PROPOSITION

of

factor.

"take the graph"

of K are N-shapes,

the following

Observe

® I ® 1 ® [I,I ® I] there are three prime

0 be the functor which

previous

[i], but is

there we used the process

Now let K stand for one of the categories F:~-~

•

of K.

As for the uniqueness

AM1,

AM3,

and

contains

all

of ~ or ~, the graph

m

of x will determine no variables [I,I],

~ except

-- i.e.,

for those prime

the possible

[I ® I, I], etc. This proposition

constant

factors prime

Thus we get the asserted serves

to replace

X i which contain

factors

such as

permutation

Propositions

~.

4.3 and 4.6 of N

[i]; it has the same corollaries

(4.4 and 4.5).

We will write

T

S

-

for a

(usually unnamed)

N-shapes

14

-

central morphism

T and S centrall~

con@ruent.

of K, and we then call the

Also two morphisms

f, f' in

N

will be called centrally

congruent

(in symbols,

are central

morphisms

the n prime

factors Xl,''',X n is centrally

the parentheses

at the left;

which does not involve

category oidal

translates

"variables"

in Lemma

4.7 of

The result

is a central

theorem

isomorphism

prime

3.3.

factors

is a unique central each variable

THEOREM

monoidal mon-

we observe,

is isomorphic

as

to I.

of {1,...,n~

morphism

f:T

>S

in ~ .

central

For any such f we call Y~i the prime factor X. of T.

of Propositions

in Y~i"

implies

If W is a closed naturality,

in ~0 is the graph of a unique

via f, w i t h the prime

in K for which the

of X i with the same variable

>S

respec-

such that Y~i = X i •

f:T--~S

for K = ~0 or ~#, hence

3.4.

while T and S

Xl,-..,X n and YI,''',Yn,

~:T

proofs).

(3.1)

replacing

Let W be a closed naturality,

~ is a permutation

This applies

analogs

(symmetric)

provided

prime

(say)

c.

situation,

shapes,

(3.1)

all start

congruence

[6] for a

[i], that each constant

with the prime

while

graph mates

to

is

are N-shapes

Then there

there

at once to the present

by arbitrary

PROPOSITION

tively,

congruent

tensor product

the commutativity

coherence

Any shape T with

(or, T N I if n = 0)

in this iterated

for that matter,

The familiar

f') when there

x and y such that f' = xfy.

T -- XI®...®X n where

f

each central

natural

N-graph

transformation

factor of S associated,

With this notion we have the

4.10 and 4.11 of

[i]

(with slightly different

-

4.

15

-

THE MONOIDAL

We come to the combinatorial In the ordinary ~:T

>S

monoidal

between

case

(integral)

transformation

these properties

fails

(Theorem

3.1 of

f:T---~S

for integral

graphs.

part of our coherence

and there was one and

of graph ~.

N-shapes,

For example,

theorem.

[i]), every graph

shapes was central,

only one natural

only for N-central

CASE

while

The first of the second holds

the N-graphs

(mates indicated

by the same letters) (x,y) ~ >(fx,Fy),

are N-central, and Vx,y;

(x,y)

>(Gx,Sy),

since they are respectively

however

the formally

an arrow A

> (Sx,Fy),

>(FX,FX),

but I

Now define

(4.2)

from GA to FB, given only

Similarly

I are N-central,

(4.1)

the graphs of Fx,y,Gx,y,

(There is no way to "get"

>B).

>(Fx,Gy),

similar N-graph

(x,y) is not N-central

(x,y)

~(Gx,Fx)

I

>(Gx,Gx),

I----~(Fx,Gx),

is not.

a strin~ of length n = 1 to be an N-graph

,

~ :I ----~ (Ul,Vl) , where the variable

u I has mate v I under ~ .

u I stand for any one of the symbols ditions

(4.3) Here these

"labels"

-, F-, G-, -', subject

v I and

to the con-

that u I and v I have the same type and that if one is - so is

the other.

Similarly,

define

a string of length n > 1 to be an N-

graph of the form

c:(Vl,U2)®(v2,u3)®...®(Vn_l,Un)

>(Ul,Vn)

(4.4)

-

with

each

label

i = l,''',n, are of

-.

indices

1 <

the

and w h e r e

Finally,

vi,--.,u j or

v i of

(if any)

type

as u i a n d m a t e d - implies

in a s t r i n g

u i = G-,

equal

i = j-i < n-i

-

u I = - or v n

a block

i ~ j with

same

16

vj

to -.

(4.3)

= F-,

For

first

variant (4.1)

case

F-,

so m i g h t

above

are

string

which

string

may

vI

(i=j)

a covariant

be c a l l e d

be p i c t u r e d

v3

u2

u3

vided

of

s

and

a

wish

of m o r p h i s m s

N-graph

category

PROOF.

Suppose

first

associative

such N-central inclusions this must

s,

of a f a c t o r later

be

-. •

a contra-

the g r a p h s

while

without

(4.2)

is a

blocks.

A

u1

>\

will

vn be composable

pro-

to be equal. is a s t r i n g

N-central

all v a r i a b l e s

take

identity

law

factors since

~ which

that

applications

the

i = j < n

with

no b l o c k s

natural

is

transforma-

~#).

s to be the

in w h i c h

mate

Again,

strings

which

of a u n i q u e

n = 2, w e

order

1 <

un

Every

(in the

of the

G- w i t h

u4

If n = I, w e m a y

ert~es

with

no b l o c k s ,

(u i & v i) c a n be a d j u s t e d

s to be n-2

variables

blocks

variable

2 with

We w i l l

e a c h v i = u i.

take

intermediate

"bad m a r r i a g e " .

length

is the g r a p h

take

of a p a i r

Vn_ 1

LEMgLA 4.1. N-central,

all

consists

as

a sequence

the m a t e s

(4.4)

all v a r i a b l e s

(-',Gx)®(x,y)~(Fy,Gz)

\\\

It r e p r e s e n t s

tion

has

a block.

v2

that

a,

are

strings

has

and

example,

( - ' , G x ) ~ ( F x , F y ) - -~ ,

The

or

to u i b y

are

s to be

for M,

the

composed. possible

an M,

of

the

same This

type

~(-,-),

(x,y) u~ > ( x , y ) .

I by b -I f o l l o w e d by

j:I

of c o m p o s i t i o n Mc;

the o n l y

followed

are

For

by very

s results

while

by

j:I

rules

(2.2)

if n,

familiar

prop-

no m a t t e r

would

>(A,A); give

the

the only

variations

later

then

larger

is m o r e o v e r

further

-;

be

since

the r e q u i s i t e

-

uniqueness

remains

the case when

Then by the d e f i n i t i o n cannot

except

k* in order

some v a r i a b l e s

of a string,

be of type

s by iterated case,

-

of s.

There

right

17

-.

the v a r i a b l e s

that we first make

M'

u i = v i for every

-' no such a d j u s t m e n t

is needed,

and

i.

in w h i c h

we are given

make

these

arrows

k~ :(A',FA)

>(A',GA)

k ~ and k* m a k e s tions

composable

}FA

by applying

no d i f f e r e n c e ,

since

of k* and k ~ via k shows

would

a diagram

(v i, Ui+l),

sider

longest

variables

equal

tion of the

i = j

is in effect we can a d j u s t

p < q;

to -.

=

This

chase

choice

between

from the d e f i n i -

and v i = F- c a n n o t

G-

(a "bad m a r r i a g e " ) .

of the

Writing

has the

it is b o r d e r e d

GAp and on the right the

"bordering"

(Ul,V n) of our

string

occur,

There

are

-.

remain

as it only

Now if v i is - in

it is no shape),

containing

the m a t c h i n g

so con-

v i and w i t h all

variables

as A i, this

by

por-

form

on the

left by

(Vq,Uq+l),where

term Up is a c t u a l l y string

to

as

(4.5)

(Ap,Ap+l) ~ (Ap+ l,Ap+2) ®. • .~ (Aq_ l,Aq) with

one

> (A',B')

so is ui+ 1 (otherwise

portion

string

are F- or b o t h

> (GA,B')®(A' ,GA)

some but not all of the v a r i a b l e s

the shape the

>B';

>(FA,B').

M'

The case u l•

be a block w i t h

cases w h e n

kbor

~M '

(FA,B') ®(A' ,FA) commutes.

situation

k, either

k*®l~

always

applying

that l®k ~

(GA, B' )O(A' ,FA)

if b o t h

and GA

or as k*:(GA,B')

as in the p r e v i o u s

adjustments,

similarly

In case u. is F- and v. is G- the l l A'

natural

In case u i and v i are both

are G-.

arrows

the d e s i r e d

in A', m u c h

preliminary

-.

u I and v n on the

In this case we o b t a i n

use of the c o m p o s i t i o n

to m a k e

are not of type

(4.4),

and

(Vp_l,U p) w i t h Up = FAp or

Vq=FAq

one w h i c h

similarly

or GAq. appears

for q = n.

In case

p = i,

on the right However,

the

side

-

18

-

discussion below will apply equally well to these extreme borders. Now the portion

(4.5) of the string in case V = Set amounts to giving

a list of composable morphisms Ap---->Ap+I---->Ap+ 2 Applying F, G, or k:F

> ...

> Aq.

> G we get the following commutative diagram

GAp

~ FAp+ 1

~ •••

) GAq

so that composition gives a unique morphism FAp at our choice.

> FAq

,

FAp----->GAq

,

GAp

> GAq

,

We cannot get GA

> FA , but the "no block" conP q dition means that this will not be required. Translating this argument in the standard way to the V-case, F, G, k* and ~ ,

this means that application of

followed by the composition M', will yield a unique

graph or natural transformation

from the shape

(4.5) to any one of the

three shapes (FAp,FAq),

(FAp,GAq),

(GAp,GAq)

We choose that one which matches the given border u Vq

FAq or GAq.

As before

and P (or use the argument of Lemma 4.2) the reP

= FA

or GA

P

sult is unique. We next aim to show that every N-central morphism in K is a tensor product of strings.

By definition,

the N-central morphisms are

formed from the central morphisms and the various morphisms F, G, k*, k~; M, M'; j, j', k,

(4.5)

by applying successively composition and tensor product.

Since @ is a

bifunctor,

we can rearrange the order, applying first @ and then com-

position.

Thus every N-central morphism is equal to a composite of

central morphisms and "instances" of the morphisms instance,

(4.5); here by an

say, of F, we mean a morphism centrally conjugate to

-

19

-

I~FAB®I:X®(A,B)SY for some N-shapes

X and Y.

only on the prime

factor

LEMMA 4.2.

such instance its domain,

acts effectively

(® is a functor)

on two factors,

Indeed,

by an overlapping

j followed M will by

of

(2.2));

I is equal

prime

(4.5).

Each

factors

factors

of

for

The order of applicacan be interchanged

involved

do not over-

with any M', any j with

of j in the composition; so that it applies

by F or G becomes (2.2)

disappear

by an overlapping

j' by

we assert

last in the com(2.3),

(i.e., become k*, k ~ or M'.

M' disappears

j followed

central), On the other

(again by the analog

by k*,

J'FA

k* > (FA,FA)---~--~(FA,GA)

to k, by an easy application

ments move

of

can

(2.6).

as for M and M', on just one,

by an overlapping

j' followed

we have listed

(2.5) and

in our composition

(or dropped)

and j cannot be followed j' followed

in the

of k.

an instance

that it can be moved

taken

of

they actually

of instances

for j, j' and k.

instances

any j' or with any instance

though

as in

any M may be interchanged

Now consider

(4.5),

on at most two of the prime

if the effective

for example,

hand,

(4.5),

an arbitrary composite

either

in K is a composite

of the morphisms

"basic" morphisms

tion of two different

position.

morphism

that to get this ordering

F, G, k*, or k ), or on none,

lap;

of its domain.

in terms of k and the others,

Consider

acts effectively

(4.5).

First observe

k* and k ~ among the be defined

(A,B)

and instances

in the list

PROOF.

We say that this instance

Every N-central

central morphisms order given

>X®(FA,FB)®Y

of the definition.

These argu-

j, j' to the end of our composition.

Next,

kA:I

F, G, k*, k~ , or M.

>(FA,GA)

can't be followed

If it is followed

posite must be that used either

by an overlapping

by an overlapping

in the definition

(2.5)

M', this comor the

- 20

definition (a central

(2.6)

of k ~ or k*.

morphism

for listing

and)

Thus M' following

an instance

k ~ and k* in our basic

have transported

all

-

(remaining)

k A may be replaced

of k ~ or k*.

list

(4.5).

instances

by

This is the reason

With this argument,

we

of k to the end of our com-

position. Consider

next instances

overlapping

F or G, application

lowed by M'

If M' is followed

of M or M'. of

(2.3)

If M is followed

replaces

it by F

by an

(or G) fol-

by k*, we may apply the commutative

diagram (B' ,C' ) 0 (GA,B')

M' > (GA,C')

~

iOk~, B '~

k~,c,

M' > (FA,C')

(B' ,C' ) 0 (FA,B') now M' is preceded

by an instance

M' followed

by ~ .

composition

indicated

THEOREM morphism

f:T

The same argument

With these arguments

4.3. >S

of k*.

in

(4.5), proving

we have achieved

for

the order of

the lemma.

If T and S are N-shapes,

of K is centrally

applies

congruent

then each N-central to a m o r p h i s m

of the

form iOhlO...Ohn:ROTiO'''OT n where

the Ti, S i and R are N-shapes,

each N-graph Fh.

is a string

I:R

~ROSIO"''OS n > R is the identity,

~i:Ti-----~S i with no blocks,

and

so that

l

F(I • h I 0 " ' ' 0

h n)

PROOF. of instances

=

1 O ~l

Consider (4.5)

0'''0

~n"

an N-central

last in this composite

a string of length one and effective factors

si as desired.

central

morphisms)

clearly

overlap

presented

in the order now established.

j, j', and k applied

tensor

morphism

domain

Arrange

-- either

The instances

of

each have a graph which I; they yield

the instances

into groups whose effective

in strings

as a composite

some of the

of M and M'

domains

is

overlap.

strings of M, preceded

(by They

by none of

-

F, G, k* or k ~, or strings stances

21

-

of M', possibly

preceded

by certain

in-

of FAB: (A,B)

> (FA,FB) ,

k*:(GA,B')

Any easy induction lecting

these

strings

N-shapes.

completes

4.4.

Col-

the proof of the theorem. of this section we obtain a final

Let W be any closed naturality,

with the same N-graph

natural

are equal,

is an N-central

PROOF.

o i has no blocks.

maps.

Two N-central

in ~0 there

> (FA,GB) ,

k~: (A',FB)--->(A',GB).

the main results

for N-central THEOREM

>(FA,B'),

shows that each such string

By combining theorem

GAB:(A,B)

By Theorem

transformations

and T, S any two

f, g:T

and to any N-central

morphism

f:T

>S

in ~

4.3 we may apply a central

>S

in

N-graph

~:T

>S

with N-graph

~.

congruence

to f

(and also to g) so that the new f has the form 1 • h I 0"''® h n with graph 1 @ a I ®''-® none of the prime (u,v)----~(u,v) strings gJ

in Ff.

>xgy

a n as described. factors

of length

Moreover,

of the identity

we can arrange

l:R

>R

central

congruence

= 1 ® k I ®-..~ k m where each Fkj = ~j

Ff = Fg,

so F(xgy)

(Proposition

arises

3.2) which

are strings

i; then the list al,''',~ n includes

Now a further

applied

is just a permutation

volved ,so that n = m and ~j = apj.

4.1.

to g gives

of prime

But factors

p of the strings

Therefore,

same graph so must be equal by Lemma

all the

is a string.

from Ff by a permutation

that

This

in-

kj and haj have the in turn implies

f = g,

as required. The same theorem holds integral Theorem

N-shapes. 3.1 of

morphisms

A similar

stronger

of

[i].

These theorems

include

[i], but they are stronger

N-central

context

for W a monoidal

and arbitrary

naturality

and T, S

a result analogous because

(not necessarily

theorem could have been stated

to

they apply to integral)

N-shapes.

(and proved)

in the

-

5. Again,

W; the objects

N-constructible

mor~hisms

Hf:A

f:A

and g:C ® D

>[B,C]

>E

where is defined

in the class,

(CM6)

all unblocked

string to be an N-central The first

CMI - CM5 of

®l > C®D

l®f

strings,

five closure

in the class,

in the class,

g ~S

e

for

(5. i)

with evaluation

e:

>C,

where we define

whose graph

conditions

all

morphisms

,

composite

>[B,C] ®B

morphism

>C

the

of morphisms

all the composite

to be the following

[B,C] ®A

conjugates

f:A ® B

for a

class of morphisms

f ® g of morphisms

of morphisms

[B,C] OA®D

blocks.

all central

~0 or ~ We define

of K to be the smallest

all tensor products

adjuncts

and finally

OF CONSTRUCTION

of K are N-shapes.

all central morphisms,

in the class,

>B

PROCESSES

-

K will be one of the closed categories

given naturality

containing

22

an unblocked

is a string with no

are exactly

[1]; we have added a sixth condition

the conditions

for strings which we

call CM6. Note especially N-central

ones which are strings

in this definition however,

that the central morphisms (CM6)

(CMI) and the

play two quite different

-- and in the subsequent

it does follow that every N-central

proof.

By Theorem

morphism

roles 4.3,

is N-construct-

ible.

6.

CONSTRUCTIBILITY

The arguments and

"allowable"

"N-allowable";

of §6 in

replaced

To start with,

trivially

to give

PROPOSITION

6.2 N.

MORPHISMS

[i] now carry over, with

by the new terms

essentially

string"

OF ALLOWABLE

by adding

"N-constructible"

everywhere

one new case

the proof of Proposition

Proposition

"constructible"

6.2 of

6.2 of

and "unblocked

[i] extends

[i] holds with one added

-

case

-

(v) : h is c e n t r a l l y

Observe

that,for

conjugate

such a string

just as in

we define

[i] w i t h the added proviso,

(-,-) or

tion 6.4

to an unblocked

(x,y);

N-constructible

R5, that r(x,y)

Case 7.

h@I>s~u,

= ~)

"unblocked

itself an unblocked

in case 7:

h mates no v a r i a b l e s

k

(6.1)

not the kAOf the naturality). added cases,

f ® g.

"unblocked string,

string".

asserts

string and k of type ®, so

Then k(h ® i) is c e n t r a l l y conjugate s ® l > Z®U N A®B

This has the form

to a prime

the prime

factor

Z

factor of A or of B,

Then A -- Z ® A', and so U - A' ~ B and k(h ® i) is

centrally conjugate X®A'®B

to a composite

f~g>C®D;

in the middle associates

(we use this property of our strings) let us say of A.

the string

S.

h is of type u n b l o c k e d

isomorphism

(for

this holds

There are no "closed loops" because in its codomain

Then

and hence N-con-

statement of our p r o p o s i t i o n

X~U

the central

where one of h

string".

Both h and k are of type

The final

Case 8.

Proposi-

we are given h and k

that the graphs of k and h ® i are compatible;

trivially

= 1 for any

k>v

(here k is any morphism,

k(h ® i) is o b v i o u s l y structible.

a rank r,

In the crucial

N-constructible)

The new proof simply needs the following or k is of type

and Z is prime.

and wish to show the composite T~U

N-constructible

>Z.

for each N-shape

Lemma 6.3 still holds.

(to show composites

string s:X

s, X and z are N-integral

To carry out inductions,

term

23

to s , l ~ Z®A'®B -- A ® B - - ~ - ~ C O B ~ - ~ C ® D .

(1 @ g)o(f, X~A'

® i) = f' ® g, where

s®I~zQA,

N A

f>c.

f' is

-

24

-

It is enough then to show f' N-constructible. trivial,

this follows

Since g is given non-

from the induction a s s u m p t i o n

by an easy calcu-

lation of ranks. Case 9.

h is of type unblocked

Then our composite

k(h ® i) is centrally

X®U

s®l > Z O U ~

conjugate

to

[B,C]®A®D- g ( < f > ® l ) > V.

Now Z is both prime and N-integral, congruence

string and k is of type < >.

hence

is a s s o c i a t e d

by the middle

either to a factor of A or to a factor of D.

corresponding

case subdivision,

Case 9a.

Z associated

There is a

as follows.

to a factor of A - Z • A'.

Then also

N

U

[B,C] ® A' ® D and h ® 1 is c o n j u g a t e

[B,C]®XeA'®D

l®sOl

so k(h • i) is c e n t r a l l y c o n j u g a t e

to

> [B,C]@Z@A'@D,

to a composite go( 0 i) where

f' is the m o r p h i s m X®A'

Rank c a l c u l a t i o n N-constructible. Case 9b. U T D' ®

[B,C]

s®l>z®A,

-- A

and the induction a s s u m p t i o n The c o m p a t i b i l i t y Z associated

argument

f >B.

show this f' is is similar.

to a factor of D T Z ® D'

® A and k(h ® l) is c e n t r a l l y c o n j u g a t e

[B,C]®AOX®D'

< f > ® l > C®X®D'

g'

Then to

> V

where g' is the morphism. C®X®D'

10s®I>coZOD,

T COD

g >V,

N

so that g' - g(s • i); it is N - c o n s t r u c t i b l e Case I0. case h = ~ f : T Therefore,

by induction.

h is of type ~ and k an unblocked >[B,C]

k(h O i) is

for some f:T • B

>C

which

string.

In this

is N-constructible.

-

T®U

25

-

h®l>[B,C]®U

Since X is an N-integral

-- X

S>p

-- V.

shape, which cannot have the prime factor

[B,C], this case does not occur. A table of all possible choices have now covered

all cases,

so that P r o p o s i t i o n

The point of this proof process

still applies

e s s e n t i a l l y whenever

for h and k will

6.2 N is proved.

is to show that the

in exactly

show that we

"cut elimination"

the form used in our previous

there are additional

types of c o n s t r u c t i b l e

m o r p h i s m s which have integral domain and prime and integral so that they do not

The proof

The N - c o n s t r u c t i b l e

is like that of

[g,f]

is

N-allowable,

(three times)

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

is N - c o n s t r u c t i b l e

a misprint

If two N-graphs

of ~ 5

and one added case

for

[i], on line -9 of page [f,g].

~:T

>S

and ~:S

>R

are

they are compatible.

This follows as before sition 6.4 N.

from the

It shows that our

"compatible"

"artificial"

clauses

composition

in Propo-

of non-

is of no consequence.

7.

THE COHERENCE

In the present context, those shapes which are either prime

[1], using

in the previous proof

THEOREM 2.2 N.

compatibles

of K are exactly

is easy.

Incidentally, 127,

morphisms

ones.

that every N-central m o r p h i s m AM6, w h i c h

codomain--

"collide" with evaluation or adjunction.

T H E O R E M 6.5 N. the N - a l l o w a b l e

paper,

factors alone

reduced N-shapes

(i.e., no factors

is centrally conjugate

carries

because

a morphism

to one with X and Z reduced.

can be d e s c r i b e d

I or else an iterated

every N-shape over,

THEOREM

s:X

as

®-product of

I); as in Lemma 7.1 of to a reduced one.

[i],

Lemma 7.2

~ Z of string type is conjugate

-

THEOREM

2.1 N.

-

Given N-shapes

for deciding when an N-graph PROOF.

26

~:T

T and S, there is an a l g o r i t h m

~S

is N-allowable.

By the results of §6, we need only test when it is

N-constructible.

This we do by the a l g o r i t h m of

[I] plus the evident

finite test that a graph is a string with no blocks. LEMMA 7.3 of notion of "proper"

[i] applies shapes

LEMMA 7.4 N. and T proper,

to N-shapes without change.

is essential

If h:T

~S

The

to

is N - a l l o w a b l e

in K, with S constant

then T is constant.

This is proved trivial because

inductively

as before,

with one new case,

the codomain of a string is never constant.

LEMMA 7.5 carries

over at once.

We now come to the essential

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

w h i c h assert that the "construction"

of

of N - a l l o w a b l e m o r p h i s m s

[i]

can

always be done to follow a given c o n s t r u c t i o n

(type ®, ~, < >, or

"string")

shapes are proper.

of their graphs

Proposition N-allowable

-- provided

certain

7.6 does this when h:P ® Q

® N is given

with N - g r a p h Fh = ~ ~ ~ and P, Q, M, N are proper.

previous proof applies unchanged, is not n o n - t r i v i a l l y

The

since if h ~ s of type string,

Fh

of the form ~ ® ~.

Proposition

7.7 holds as before.

Proposition

7.8 N considers

a N-allowable morphism

h:[Q,M]~P~N

between proper N-shapes,

with

[Q,M]

h itself has the form q(

@ i).

>S

(7.1)

not constant,

form ~(<~> ® i); under s u p p l e m e n t a r y

change,

>M

hypotheses,

and with F h of the one is to show that

The previous proof applies w i t h o u t

since h cannot be of type string when its domain

includes

[Q,M]. We can now consider lated

in ~2 as Theorem

the main coherence

2.4 N.

theorem,

already

formu-

We are to show that two N - a l l o w a b l e

-

morphisms

h, h':T

>S

27

-

of the same N-graph between

proper

equal.

But,by Theorem

6.5N,h and h' are N-constructible,

reduced

to the various

five cases

for h and for h'.

shapes are so we are

To the previous

proof we thus need add only one new case, when h is of type string, while h' is some other N-constructible

w i t h the same graph.

visible

cannot be a graph for a mor-

form of a string graph clearly

But the

phism of type ®, or type 7/, or type < >, or a central morphism. Therefore string.

h' is also of type string, The hard work was all done

and by Lemma in

4.1 is the same

[i] and ~4 above.

-

28

-

REFERENCES [i]

G. M. Kelly and S. Mac Lane, Coherence in Closed Categories, Pure AppI. Alg. 1(1971)

J.

97-140.

[2]

G. M. Kelly, An abstract approach to Coherence, (in this volume).

[3]

S. Eilenberg and G. M. Kelly, Closed Categories, on Categorical Algebra,

La Jolla,

1965

in: Proc. Conf.

(Springer-Verlag,

1966)

pp. 421-562. [4]

G. Gentzen, Untersuchungen ~ber das logische Schliessen I, II, Math.

[5]

Z. 39(1934-1935)

176-210 and 405-431.

J. Lambek, Deductive Systems and Categories Io Syntactic Calculus and Residuated Categories,

[6]

S. Mac Lane, Natural Associativity and Commutativity, University Studies 49(1963)

[7]

Math. Systems Theory 2(1968)

Rice

28-46.

S. Mac Lane, Categories for the Working Mathematician, Verlag, Berlin, Heidelberg,

287-318.

New York 1972.

Springer-

COHERENCE

FOR DISTRIBUTIVITY

Miquel L. Laplaza University of Chicago and University of Puerto Rico at Mayaguez Received November

5, 1971

INTRODUCTION A familiar

situation

~, and two functors, are associative, to ~.

in category theory is given by a category,

®, ~:~ X ~

commutative

A coherence

>~,

that within natural

and such that ® is distributive

result for this situation is a consequence

and some suitable conditions

on the natural

ing to give a more precise description ®, e:~ X ~

the

of the above structure

isomorphisms.

We are go-

of this situation. >~,

two functors,

fixed objects of ~, called the unit and null objects. have natural

relative

is to characterize

diagrams whose commutativity

Let ~ be a category,

isomorphisms

U and N

Suppose that we

isomorphisms,

eA,B,c:A®(B®c)

(A®B) ®C,

YA,B :A®B

>B@A,

e' :Ae(BSC) A,B,C

> (ASB) SC,

y'A, B:AeB

>BeA,

~A:U®A

>A

~:N~A

>A

,

~

~:N®A

>N

,

~:A®N

,

pA: A®U

>A,

(i)

A~N--+A, >N,

and natural monomorphisms, 6A, B,C :A® (B~C)

>(A®B) ~ (A@C) (2)

6#A,B,C : (A~B) ®C ----~(A®C) 8 (B®C) which are defined A coherence

for any objects,

A, B, C of C.

result for the structure given to C by the family

of i s ° m ° r p h i s m s ''~ A , B~, C : k A

~A' y A,_~m] was given by S. Mac Lane

(see

-

[4] and

30

-

[1]), and when we say that C is coherent

~A.B.C'

kA' ~A' YA,

to that result,

or for

although

~'A,B,C'

IA, @~,

on C, answering

a question

paper was given

in

combinations

proposed

[3]. Roughly

[5].

a reasonable

in the

for the above

An announcement

by taking

morphisms

conditions

of this

for vertices

of C and for arrows (i) and

result we have to impose

the coherence

structure

we intend to characterize

that can be obtained

by ® and 8 of objects

that are called

in

theorem

speaking

(also by ® and 8) of the natural to obtain

we want to refer

in [i].

We are going to give a coherence

diagrams

¥'A,

we are going to use the conditions

form given by G. M. Kelly

commutative

for

the

the combinations

(2) with

identities;

some conditions

which hold

the

on C

in some usual

situations. The paper can be summarized X = IXl, x2,.--,Xp,

u, n} be a set and construct

on the set X with functors and

~:~(X)

= U, m(n)

> ~, extending

(I) and

(2).

~(X)

and identities. coherence

given later)

(i)

> O b ~,

by

®, S and the morphisms

of ~(X) will be the elements ~''+I'

detailed

of the free

over X and the arrows will be all

. and + over formal

The coherence

conditions

"free" category

such that for any map, m:X

the map m and preserving

The objects

generated

the

Let

= N, there exists one and only one functor,

algebra with two operations, the elements

words:

® and • and w i t h the natural morphisms

(2); this is a category

such that m(u)

in the following

result

states

symbols

of type

(i),

that if C satisfies

in 91 and is regular

(2)

the

(in the definition

then the image by ~ of the set C(X) (a,b)

has at most one

element. We have to remark

that the construction

will be given almost completely, cept of "free"

category

of the category

but we are not going

C(X)

to use the con-

given above.

From now on C will be a category

with the structure

given above,

M

whose objects

will be denoted

by capital

letters.

We shall use the

- 31 -

parenthesis with the usual conventions on sums and products and the symbols ® will be omitted as often as possible. The core of this work was done in the Department of Mathematics of the University of Chicago where the author spent one year as Postdoctoral Visitor and he wants to thank Professor S. Mac Lane for his illuminating direction and patient revision of the different versions of this paper.

~l.

The Coherence Conditions We will say that the category C is coherent when C is coherent

in the sense of [i] separately for I~, y, ~,~} and {~', y', ~', ~'}and the following types of diagrams are commutative

6

A (BeC) •

>

for any vertices:

ABeAC

!

!

A YB,C

YAB,AC

(I)

6 A(C®B)

A,C,B ~ A C e A B

6# = 6 (AeB)C (YA,cOYB,c) A,B,C C,A,BYAOB,C :

, 6# YAC,BC A,B,C

[Ae(BeC)]D

I~

=

6#

, B,A,C(YA,BSI C) : (AeB)C

6#

>ADe(BeC)D

A,B~C,D

~#

6# AeB,C,D >(ASB)DeCD

(II)

(III)

>BCeAC

>ADID (BDIgCD)

iAD~B,C,D

A,B,C'ID

[(ASB)~C]D

> CAeCB

AD, BD, CD

6#

A,B,DelcD

> (AD~BD) ~CD

(iv)

-

32

> ABeA (CeD) A[B~(C~D)] ~A,B,C(DD

-

IABe6A,C,D

,D A[(BeC)eD]

>ABe (ACeAD)

B,AC,AD

6 AtBeC'D~.A(BeC)eAD

6 AtBtC~IAD)(AB~AC)~AD

A[B(C@D)] 1A.6 > A(BC@BD) ~ >A(BC)~A(BD) i B,C,D A,BC,BD i~ (~A,B, C~D AB,C,D

{AOB) (CD)

n

>

(AB)Ce (AB) D

A (CD)eB (CD)

A,B,CD

(vii)

aA,C,DSaB,C,D

~A@B, C, D

A [{B@C)D]

A'B'C'ID > (AC@BC]D

6# AC,BC,D >(AC) De (BC)D

> A(BDeCD) .6# IA B,C,D

6 > A(BD) eA (CD) A, BD,CD

~A ,B~C D

(viii)

I~A, B, D~A, C ,D

% .

[A (BSC) ]D

(VI)

A, B, C~eA, B, D

(AB) (CeD)

[ (ASB) C ] D

(v)

A,B~C D~ (ABeAC)D

AB ,AC,D > (AB)De (AC)D

-

A (CeD) ~)B (C~)D)

33

-

6A,C,De6B,C,D>

(ACeAD) ~ (BC(gBD)

AC~AD, BC, BD

[ (ACeAD) eBC] eBD #A,B,C~D ~l,-I AC,AD,BCelBD

~

[ACe (ADeBC) ] eBD

(A~)B) (C~)D)

(Ix)

(IAC~YAD, BC }~)]BD

[ACe (BCeAD) ] eBD

aAC , BC,ADelBD 6

AeB,C,D [ (ACeBC) eAD] eBD

6#A~B~Ce6~,B,D >

(A~B)C~(A~B)D

>N

N (A~B)

5

<

,

(X)

~- NA~gNB

k ~B

N

,

(AC~)BC) (I)(AD~gBD)

~k

N

NeN

(xl)

- 34 -

!

*

,

,

-- eA®B"

A,B,N

(XII)

>N®N

(A®B)~

e~ = x.-.,J

>N

(XIII)

AN = P : : U N

>N

(XIV)

= IAYA, N:AN

>N

N[AB) -

<~N,A,B

(xv)

> (NA)B (x-~'z)

k

N<

B

A(N8)

NB

> (AN)B I~

(LA'N'B

OCVZZ)

A~XB AN

A'IB .

.

NB ,

N *

= PA(1AOPi):A(BN)

A(N@B)

6

~AB~A,B,N

I

,

(XVIII)

> AN@AB

A,N,B

I~AelAB

A ° k B'

AB ~

>N

A5

N@AB

(xix)

-

35

-

W

>BA

(xx)

~B(1ABe~A) 6A,B, N : 1A®~:A(BON) )AB

(xxi)

e~B(1ABeIB )6#A,N,B = ~A@IB:(AeN)B >AB

(XXli)

I~A(IAelBA )6# N,B,A = I~®IA:(NOB)A

U (AeB)

6

> UAeUB

U,A,B

/ (xxiIi)

IA~I B AOB

(~AeSB)6A,B, U = ~AeB:(AOB)U. >AOB

,

(xxiv)

The commutativity of some types of diagrams imply the commutativity of others, and we are going to indicate some of those relations. A detailed study of the minimal conditions assuring the coherence of ~ for {~, ¥, 1,P1 or Is', y', I', ~

is contained in [1].

We will prove the following set of relations, in which the number of the diagram denotes the condition

of commutativity of all

the diagrams of that type:

i)

(II)

> (

(I)~---~(III)

2)

(II)---~(

3)

Coherence

4)

Coherence of C for {~, y, I,~ I /% (II)

5)

Coherence of C for {~, y, I,~} A (XV)--~ (

6)

(II)A (XV) ~

7)

Coherence of _C for ~ '

(IV ,k=-~,(V) of

),

C for ~ ,

(

),

y, k , ~ /% (II)

(XI)~--> (XII) , ~

> Any two of ~(XVI),

', I' ,

) ~

>(

(VI) ~/----~z (VII)

> (

(VI) ---> (VIII)

), ), ),

(XIII)~/--~,(XIV)

, i} ^ ( x v ) - - ~

(XVII), (XVIII)~ imply the other 2

l

-

8)

(XV) A (I) A (If) ~

36

-

Each one of ~(XIX),

(XX), (XXI), (XXII~implies

the others, 9)

Coherence of ~ for ~ ,

¥, 1,8}A(II)---~/(

(XXII I)~-~i (XXIV)

).

The proof of all the above relations uses the same method: the construction of a diagram in which the commutativity of all the subdiagrams with the exception of two follows from the hypothesis of the relation so that the commutativity of any of these two diagrams are equivalent conditions.

We are going to indicate the construction of

these diagrams and to identify by its number each of the subdiagrams involved.

The symbols (cob) and (nat) in the inside of a subdiagram

will indicate that the reason for the commutativity is the coherence of [ for ~'~, ¥, l, ~} or the naturality of the elements involved in the construction of the subdiagram. Proof of i):

It is given by the following diagram in which

the outside is of type (I)

A(BSC) (zz)

(B~C)A (nat)

i

> BA@CA (ImI)

I (nat) > CA~BA

(C~B)A

A(CeB) .....

~

~AC

k

) ACOAB

-

Proof of 2):

37

-

It is given by the following diagram in which the

outside is (V)

A [Be (CeD) ]

> ABeA (CeD) i

~ (nat)

(II)

"9 ABe (ACeAD (II)A (nat) ABe (CeD) A

> ABe (CAeDA)

(nat) [BS(CSD) ]A

>BAe(CSD)A

•

> BAe (CAeDA)

!

(nat) I

(nat)

(IV)

[ (B(BC)eD] A----> (BeC) AeDA " (II)

~ (nat)

> (BA~CA) eDA

(If) A (nat) A (BeC) SDA

> (a~eac) eDa

(nat) A[ (BeC) eD]

> A(BeC) SAD

Proof of 3):

> (ABeAC) eAD

It is given by the following diagram in which the

outside is (VI)

A [B (CeD) ]

A (BC) eA (BD)

> A(BCeBD) , (II) A (nat)

A [ (CeD) B]

~L > A (CBeDB)

(coh)

/

> A (CB) eA (DB)

(II)

(nat) [ (CeD) B] A

(nat)

> (CBeDB) A

(vii)

(CeD) (BA)

) - (CB) A® (DB)A

~ (coh) A /k (nat) > C (BA) OD (BA)

(nat) (CeD) (AB),

> C (AB) eD (AB) (if)

(AB) (CeD)

> (AB)ce (AB) D

-

Proof of 4):

38

-

It is given by the following diagram in which the

outside is (VIII)

A[ (BeC) D]

\

> A (BD~CD)

> A (BDeCD) !

(nat) A (II)

.I.

(nat)

> A(DB®DC)

A [D (BeC) ]

> A(DB) ®A(DC)

(vi)

(AD) (BOC)

(AD) Be (AD) C

(coh)

~ (coh)

(nat)

(DA) Be (DA) C

(DA) (BOC) (vl)

> D (AB) OD (AC)

> D (ABeAC)

D [A (BeC) ] (nat)

~

[A (BOC) ] D

(II) > (AB) De (ac) D.

> (AB®AC)D

Proof of 5):

It is given by the following diagram in which the

outside diagram commutes by the coherence of C for {~, y, ~, PI

UN

Proof of 6):

It is given by the following diagram in which the

outside is of type (II)

N (AOB)

(

> NASNB

N

>N@N

(XV)

(XII) (A~B) N

> ANSBN

- 39 -

Proof of 7):

It is given by the following diagram in which the

outside is commutative by the coherence of C for ~ ,

> N(AB)

(NA)B

k,~,

y

,,>!~)N

5 (xvl) NB

Z (XVIII) AN

(XVII)

// (AN)g

Proof of 8):

> A (NBI

-

>

{BN)

It is given by the following diagrams in which

the outside are of type (I), (II), and (II) respectively

> ANeAB

A NOB)

k

\

(xix)

(nat)A AB Alcoh' ) ~

(nat)

AB~N (XXI) A (BON)

> ABeAN

-

40

-

~

A (N~B) (X!×)

AB

--N~AB

i

(nat)

~

(nat)

BA<

(nat)

N~BA

A

(N~B) A

>

NAeBA

/~AB~AN

,

(xxz)

~

~< (nat)

ABSN (nat)~ (XV)

(nat)

sA

. /

BAeN (XXII)

(BEN) A

> BA®~a

Proof of 9): outside is

[XV)

..,

(XX)

A (BEN)

B

It is given by the following diagram in which the

(II)

uiA xxii ur (coh)

~ASB

(nat)m (cob)

(AeB) U

> AUeBU

An immediate consequence of the above relations

is that for C

to be coherent it is sufficient to check that C satisfies the following conditions:

l~ c is coherent ~or {~ y ~ ] 2)

All the diagrams of type

and ~or { ~

(II),( IX),

(X) and

y

~ ~} (XV) are commutative.

-

3)

41

-

For one type contained in each one of the sets,

~XIX),

(XX),

(XXI),

(XXII)}, ~(XXIII),

~(I),

(III)~,

(XXlV)~, all the diagrams

are commutative. 3)

For two of the types contained

in ~(XVI),

(XVII),

(XVIII~

all the

diagrams are commutative.

~2.

Definition and evaluation of the paths:

F o r m u l a t i o n of .the

coherence problem Let X be the set IXl,X2,''',Xp,n,u~,

A the free ~+,.~-algebra

over X and G the graph consisting of all the following formal symbols, for x, y, z ¢ A,

x,y,z x

:x(yz)

• (xy) z

:ux~x

~x:XU

7x,y:Xy

~x

~yx

X

:nx

_6)x : x n

their formal inverses, 6

,

~' :x + (y + z) x,y,z

,

A':n + x x

~x

,

,

~:x

~x

,

,

7x,y':x + y

% n

+ n

~y

% (x + y) + z,

+ x

,

~ n

indicated by the upper index -i, and

x,y,z

:x(y + Z)

6# :(X + y) z x,y,z 1

:X

~X

• xy + XZ

,

~XZ

,

+ yz

•

X

Note that we use the symbol

~ to indicate the edges of the graph to

distinguish them from the arrows of the category denoted by

>.

Let H be the free [+,l>-algebra over the edges of G and take on H the unique extension of the graph structure of G in which the

-

projections

are ~+,.~-morphisms.

in its expression:

of type I x are called ties.

only elements

the elements

instantiations

of G of type 1 x

involving

We can define

~i cT. --

Y2

N2•"

..

We can speak of the existence

of edges of T is not

Fix now p objects, the m o r p h i s m

01,

of graphs defined

02,

= gx @ gy,

'

of diagrams

involving

of such diagrams

°°"

, 0p in C and let g : T

on the vertices

for x,y ¢ A;

symbol onto the arrow of C determined

because

>~

This definition

depends

p,

ii) g(x + y) =

on G by taking each formal replacing

each subscript

by its

that a diagram with elements

paths contained

=

upon the 0. and allows us to define 1

the value of a path as the product define

be

by the conditions,

image by g and such that for x, y ¢ ~, g(x + y) = gx • gy, g(xy) gx ® gy.

ele-

(and will not be) defined.

i) gu = U, gn = N, gx i = 0i, for i = i, 2,''', gx • gy, g(xy)

identi-

of all the instantia-

N m - l ~ Ym

ments of T, but not yet of the commutativity the product

or simply

of G

now the ~aths as the sequences, ~i~

Yl

are in-

only elements

of identities

We will denote by ~ the graph consisting

tions of ~.

where

-

An element of H is an instantiation

if, with at most one exception, volved

42

of the value of the steps and to in ~ is commutative

if any two

in the diagram and with the same origin and end have

the same value. An ideal coherence the sense of ~i,

result would

state that if C is coherent

then for any choice of the 0 i any diagram of elements

of T is commutative,

that is, for any choice

of the 0. the value of

-

1

any path only depends

upon the origin

this is not true in some simple cases; gory of unitary modules • the direct

in

and the end of the path. for instance

over a commutative

ring,

if C is the cate-

® the tensor product,

sum and if 01 is not the null module,

ixl+x l:x I + x I

But

then the value of

• x I + x I is the identity map of 01 • 01 and the value

of 7' :x I + X l ' - X Xl,X 1

1 + x I is the map defined

by

>

that

-

is not the

identiry.

In this

43

-

sense the c o h e r e n c e

tive a n s w e r but we are g o i n g to p r o v e a reasonable coherence

restriction

result

that

on the v e r t i c e s

that holds

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

of the d i a g r a m s

referred

We shall u s e the s y m b o l The e x p r e s s i o n

3.

a

to in the

set of all

to get a

by the g r a p h G w o u l d be

Introduction.

~ > to i n d i c a t e

will denote

paths with

the e x i s t e n c e

R e @ u l a r i t [ and some p r e l i m i n a r y We shall

the

0>b

to i m p o s e

for any c h o i c e of the 0 i.

N o t e t h a t the free c a t e g o r y g e n e r a t e d the free c a t e g o r y C(X)

p r o b l e m has a n e g a -

steps

in T .

of a p a t h f r o m a to b.

concepts

i n d i c a t e by N the set of n a t u r a l n u m b e r s and by S finite

sequences

shall r e p r e s e n t the e l e m e n t s s e q u e n c e of the e l e m e n t s ,

of e l e m e n t s

of S.

of S [N] by p u t t i n g

identifying

In g e n e r a l ,

IN]

we

into p a r e n t h e s i s

the

the e l e m e n t s of S w i t h the

se-

q u e n c e s of S [N] w i t h o n l y one e l e m e n t . A l l the d e f i n i t i o n s

included

of the c o n c e p t of r e g u l a r i t y ,

in this part,

are a u x i l i a r y t o o l s

w i t h the e x c e p t i o n to be u s e d

in the

p r o o f of the p r o p o s i t i o n s . The r a n k of the e l e m e n t s rank:~ i) ii)

> N, u n i q u e l y d e t e r m i n e d

For x c X, r a n k x = 2 For a, b ¢ A, r a n k The

i) ii)

size,

For x ¢ X, For a,b

of A is d e f i n e d

by the f o l l o w i n g

conditions,

,

(a + b) = rank(ab)

siz:A

by m e a n s of the map,

> N,

is d e f i n e d

= r a n k a + r a n k b.

by the c o n d i t i o n s ,

siz x = 2,

c A,

s i z ( a + b) = siz a + siz b,

It is v e r y e a s y to p r o v e

siz(ab)

that for any e l e m e n t ,

=

(siz a ) ( s i z

y, of A,

< r a n k y = siz y,

and t h a t r a n k y = siz y, t h a t are p r o d u c t s The norm,

if and o n l y

of e l e m e n t s II:A

~N,

if y is the sum of e l e m e n t s

of

of S.

is u n i q u e l y d e f i n e d

by the c o n d i t i o n s ,

b).

-

i)

F o r x c X,

ii)

For

-

IXI = I,

a, b ¢ A,

The

44

la + b~ =

additive

~ab~

=

decomposition,

~a I + Adec:

~b~. A _ _ _ > A [ N ] , is d e f i n e d

by the

conditions, i)

F o r x ¢ X, A d e c

ii)

For

iii)

y,

x = x,

z ¢ ~, A d e c ( y z )

If A d e c

a =

= yz,

(a I, "'" ,dr),

Adec

b =(b I "'',bs),

then

m

Adec(a

In a s i m i l a r Mdec:~ i)

>~[N],

way,

For

iii)

y,

a =

M d e c (ab )

=

de c o m p o s i t i 0 n ,

by the conditions,

x = x,

z ¢ A, M d e c ( y

If M d e c

(a I, " ' ' , a r , b I , " ' ' , b s) .

the multi~licative

is d e f i n e d

F o r x e X, M d e c

ii)

+ b) =

+ z) = y + z,

(a I, • '',dr) , M d e c

b =

(b I " ' ' , b s ) ,

then

(a I , "'" ,a r ,b I , "'" ,b s) .

The additive

pattern

of t h e

top,

Apt:A

>A,

is d e f i n e d

by the

conditions, i)

F o r x, y ¢ A, A p t ( x

ii)

For

x ¢ A,

if A d e c

In a s i m i l a r Mpt:~ i) ii)

>A,

For x,y For

For i) ii)

Apt

if M d e c

then,

Apt

x = x I.

the multip!icative

pattern

of t h e

top,

by the conditions,

¢ A, M p t ( x y )

=

(Mpt x) (Mpt y),

x = x,

then,

M p t x = x I.

1 any elements

a = Apt bAAdec

Mpt

x = x,

way,

is d e f i n e d

x e A,

Proposition

+ y) = A p t x + A p t y,

a = Mpt bAMdec

a a n d b of A w e h a v e a = Adec a = Mdec

b--->a

= b.

b~a

= b.

the

following

relations:

Proof: It w i l l If A p t

sufficient

to prove

one of the relations,

a = A p t b = x I, t h e n ,

a = Adec

a = Adec

is p r o v e d .

be

Suppose

now that,

b = b,

say

i).

and the relation

-

45

-

Apt a = Apt b = x + y Adec a = Adec b =

Then

it is i m m e d i a t e

t h a t if

,

(Cl,''',ct).

Ixl = r, then,

a = a' + a", b = b' + b"

with A p t a' = A p t b' = x,

From these

A p t a" = A p t b" = y,

A d e c a' = A d e c b' =

(Cl,''',c r)

A d e c a" = A d e c b" =

(Cr+l,''',ct).

facts,

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

by i n d u c t i o n

on

IApt a I is i m m e d i a t e . Let A

be the free

commutativity e l e m e n t n,

for

That means

.

A

. relatively

a [+,.~-morphism,

t h a t the s u p p o r t

If x , y If x , y

c A,

and

to +, null

condition,

is a s t r i c t a l g e b r a and the i d e n t i t y m a p called

the support,

is d e f i n e d

If x c X, S u p p x = x c A

iii)

of

i d e n t i t y e l e m e n t u, and the a d d i t i o n a l

of X d e f i n e s

ii)

- a l g e b r a o v e r X, w i t h a s s o c i a t i v i t y

and +, d i s t r i b u t i v i t y

na = an = n for a ¢ A

i)

+,.

Supp:A----~A*

by the f o l l o w i n g c o n d i t i o n s :

,

S u p p ( x + y) = S u p p x + S u p p y,

¢ A,

Supp(xy)

-- (Supp x ) ( S u p p y).

An e l e m e n t a of A is d e f i n e d

to be r e g u l a r

if Supp a can be

*

expressed

as a sum of d i f f e r e n t

p r o d u c t of d i f f e r e n t

elements

t i o n can be e a s i l y c h e c k e d , another

e l e m e n t s of A

of X.

e a c h of w h i c h

is a

In a n y c o n c r e t e c a s e this d e f i n i -

but we shall p r e s e n t

s i m p l e c a s e ~ n w h i c h the r e g u l a r i t y

later

(Proposition

of an e l e m e n t can

3)

immedi-

a t e l y be a s s e r t e d . Proposition

2.

Suppose a to b.

Then,

0 >b,

t h a t is, a s s u m e

a is r e g u l a r

if a n d o n l y

the e x i s t e n c e

if b

of a p a t h f r o m a

is r e g u l a r .

Proof: It is e a s y to p r o v e

that,

a

%b

>Supp

a = S u p p b, and hence,

-

a

0 >b--~Supp

46

-

a = Supp b, and this r e l a t i o n

immediately

p r o v e s the

proposition. Define

the e l e m e n t a l

components,

Ecomp:A

> ~(X),

the p o w e r

set

of X, by the c o n d i t i o n s : i)

If x ¢ X, E c o m p x = { x } ,

ii)

For a, b ¢ ~, E c o m p ( x

PROPOSITION

= Ecomp x UEcomp

y.

3

Suppose

that a is an e l e m e n t

at m o s t once

appears

+ y) = Ecomp(xy)

of A such that any e l e m e n t

in the e x p r e s s i o n

of a.

Then,

of X

a is regular.

Proof. The first t h i n g ments also

of A such that, regular

immediately

elements

to p r o v e

Ecomp x~Ecomp

if x and y are r e g u l a r

by i n d u c t i o n

the p r o p o s i t i o n

ele-

y = @, then xy and x + y are

and this is r o u t i n e ~ h i s

the p r o p o s i t i o n

a = x + y, then,

is that

fact a l l o w s

on

hypothesis

lal, b e c a u s e implies,

us to p r o v e if a = xy or

Ecomp x

E c o m p y = ~. Observe find a path, xx:

that if a is not a r e g u l a r

a

where b involves

as it has b e e n n o t e d

type of e l e m e n t

~4.

0 >b,

The c o n c e p t

element,

a situation

in the c o u n t e r e x a m p l e

originates

an

"incoherent"

it is p o s s i b l e

of type x + x or

included

diagram

to

in ~2,

this

in some usual

cases.

of r e d u c t i o n

Let a be an e l e m e n t

of A.

A reduction

of a is a p a t h a--8-~a'

N

such that, i)

Every

step in the p a t h

is an i n s t a n t i a t i o n

of ~*, ~*,

~', p' or an

identity. ii)

a' = n or there Note

is no o c c u r r e n c e

that the c o n d i t i o n

ii)

of n in the e x p r e s s i o n

is e q u i v a l e n t *

the o r i g i n speaking

of an i n s t a n t i a t i o n

a reduction

a by m e a n s

of I , P

of k , ~

to say that a'

is not

*

, ~', or p'.

of a is any p a t h o b t a i n e d , k' and ~'.

of a'.

Intuitively

by e l i m i n a t i o n

of n in

-

PROPOSITION

n_ fa- ' of a, a'

value

-

4

Let a be an e l e m e n t a

47

of A.

is u n i q u e l y

of the r e d u c t i o n

Then,

there e x i s t s

determined

by a and

a reduction

if _C is c o h e r e n t

the

is unique.

Proof: The p r o o f immediately

of the e x i s t e n c e

by i n d u c t i o n

of a r e d u c t i o n

on rank a.

For the p r o o f of the u n i q u e n e s s preliminary i)

It is c l e a r

involve

The p r o o f

3)

otherwise

the e x p r e s s i o n

no o c c u r r e n c e

If a = a I + a 2 and a I

induction

to state

if Supp a = n, then, of a' and also of Supp a'

of n.

C >a~,

a 2 - - 8 - ~ a ~ are r e d u c t i o n s

a~ ~ n A a ~

~ n

a~ = n A a ~

~ n----~a'

=

a{ ~ n A a l

= n

= a{

of the a b o v e

>a'

= a{ + a~ !

~a'

assertion

we have,

,

a 2

,

can be d o n e v e r y e a s i l y by

on r a n k a.

If a = ala 2 and al---8-~a ~, a 2 - - 8 - y a ~ are r e d u c t i o n s a{ ~ n ^ a ~

The proof

is s i m i l a r

The a b o v e

three

uniqueness

that a

~ n---~a'

to the proof

assertions

of a' by i n d u c t i o n

Suppose I' or ~'

some

= n.

that Supp a = Supp a'; h e n c e

a' = n b e c a u s e

2)

of a' we have

relations:

Supp a = n~--~a'

would

of a can be d o n e

on

• b and a

and that C is c o h e r e n t ;

we have,

= a~a~

of 2).

a l l o w us to p r o v e

immediately

the

lal. ~ c are i n s t a n t i a t i o n s as a p r e l i m i n a r y

of k 2 ~

,

step to end the

m

p r o o f of the p r o p o s i t i o n tive d i a g r a m

of type

we need

to p r o v e

the e x i s t e n c e

of a c o m m u t a -

-

48

-

b

a

d

c

such

that

tiation

any

of

~a~ o u t l i n e d

k

step , by

in b

~ >d

, k' , o r ~' . the

following

and The

c

~ > d proof

diagrams

is a n

is a r o u t i n e (and o t h e r

//a{ + a ~ / a = aI + a2

a{

+ a~

i

a[

identity

÷ a2

a = aI + a2

dI + a2

cI + a2

or

an

induction

analogous

instanon

diagrams).

-

49

-

a1

a = n + aI

ai

n + a~

N o w we can p r o v e a by i n d u c t i o n on

the u n i q u e n e s s

lal: if a----~b

d u c t i o n s of a we c a n c o n s t r u c t

of the v a l u e of r e d u c t i o n

0 ~a'

and a

a commutative

~c

~ >a'

of

are two re-

diagram

h

where b

0 >a',

d---8-~a' and c

taken commutative mutative

following

by the i n d u c t i o n

s e q u e n c e of i d e n t i t i e s , case any reduction

0 >a'

are r e d u c t i o n s ,

the a b o v e r e s u l t and

hypothesis.

is t r i v i a l .

PROPOSITION

5

Then,

a

(3) are c o m 0 >a'

the a b o v e a r g u m e n t d o e s not apply,

Let a and b be e l e m e n t s of A, a is c o h e r e n t .

(2) and

If a r e d u c t i o n

is a s e q u e n c e of i d e n t i t i e s

proposition

(i) has b e e n

~b

!

..........-

~

b

o >!

in this

and the last p a r t of the

an e d g e of T and

there exists a commutative

a

but

is a

suppose

d i a g r a m of type,

-

where

a

~a'•

and b - - ~ b '

an i n s t a n t i a t i o n

50

-

are r e d u c t i o n s

of ~ , ~

, k', p'

and no step in a'--8->b'

or their

is

inverses.

Proof : Remark

t h a t the p r o p o s i t i o n

t i t y or an i n s t a n t i a t i o n proposition way

4 allows

is i m m e d i a t e

of k , p

, ~', p' or t h e i r

us to c h o o s e

for our p u r p o s e s .

when a ~ b

the r e d u c t i o n s

If a' = a, that is,

inverses

in the m o s t

if an i d e n t i t y

tion of a and we are not in the p r e c e d i n g

case,

also a r e d u c t i o n

is i m m e d i a t e .

of b and the p r o p o s i t i o n

For the g e n e r a l

case we need

is an iden-

to p r o v e

that a

or their

then an i d e n t i t y

a preliminary

Then,

0 .~c is a p a t h w i t h no i n s t a n t i a t i o n

inverses

and

we are g o i n g

such that an i d e n t i t y

to p r o v e

the e x i s t e n c e

suitable

is a r e d u c -

*

of ~ , ~

, k', ~'

is not a r e d u c t i o n

of a c o m m u t a t i v e

is

statement:

*

suppose

because

of a.

diagram

of

type

where

a

a

O

> c

a~

O

> c~

vn r Ia . , is a sequence, *

t i o n s of A , ~

, ~' or ~', c

instantiations

of ~ , p

stantiations

of ~ , ~

Observe statement

w i t h at least one e l e m e n t , C >c~

, ~' or ~'

, ~' , ~'

is a s e q u e n c e and in a ll ~' c

or their

that one c o n s e q u e n c e

is that

la~l <

a~ then any v e r t e x

.

their

(because

in it there

inverses).

inverses.

if an i d e n t i t y

n ~c~

v

of the a b o v e is a r e d u c t i o n

has an i d e n t i t y

t i o n on

is no i n s t a n t i a t i o n

This p r e l i m i n a r y

lal f o l l o w i n g

and their a n a l o g o n s :

the m e t h o d

or

' there are no in-

of the c o n d i t i o n s

lal and that

in the p a t h a II

of i d e n t i t i e s

*

tion

of i n s t a n t i a -

*

statement

outlined

of k , ~

as a r e d u c -

*

, ~'

can be p r o v e d

in the f o l l o w i n g

,

p' or

by inducdiagrams

of

-

5 1

-

a = x + y

~

xl+Y

x'

"~ z l + y

a = x + n

~

x'

L

X

a = x +

(y + z)

x'

(y + z)

+

x' (y'

in t h e their

From

this

path

a

in a'

their

inverses.

proposition

are

~

(x'

C >a'

and

b

L

+ y)

x'y'

by

+ z

+ z

,

+ xz

+ x'z'

induction

instantiation

a

/-~

a'

0

is n o

of

C }b'

>-

includes

I

, ~

on

lal

, ~',

reductions

that P'

if

or

there

is a

all

b

b'

of

I

the

, p cases

, i', ~'

or

in w h i c h

the

yet.

not

used

monomorphisms is

"-

instantiation

statement

have

proposition

(x + y)

type

proved

we

to p r o v e

is no a

~

~

immediate

for

L

X v

~ xy

+ z')

there

not

that

distributivity

of

This

was

Note

above

then,

C >b'

+ z)

there

diagram

where

the

is

C >b

inverses,

commutative

it

+ n = b

~

L

a = x(y

+ y = b

that

if

the

hypothesis

and

that

an

for

some

element,

that

immediate

the

arrows

of

consequence

a o f A,

Supp

of

a = n,

-

52

-

t h e n the v a l u e of a n y p a t h f r o m a to b d e p e n d s

~5.

o n l y u p o n a and b.

The c o n c e p t of r a p p e l Let a be an e l e m e n t of A.

A rappel

of a is a p a t h a

O>a'

s u c h that,

i) ii)

Each

s t e p in a

There

law.

speaking,

by a p p l i c a t i o n ,

of 6 or 6# w i t h o r i g i n

a rappel

as m a n y t i m e s

sum of e l e m e n t s

We h a v e to r e m a r k termined

by the origin:

two r a p p e l s w i t h o r i g i n

as p o s s i b l e ,

(XlX 3 + x 2 x 3) +

ii)

(XlX 3 + X l X 4) +

of ~, ~',

their

the rank,

instantiations

that

inverses,

path a

O >a

PROPOSITION

to s t a t i n g

the size and by

to p r o v e by i n d u c t i o n

for the e l e m e n t a. t h a t an i d e n t i t y

if r a n k a = siz a, and t h a t this

that a is a sum of p r o d u c t s

of e l e m e n t s

~ b is n o t an i n s t a n t i a t i o n

of I , P

of X.

6.

Suppose or t h e i r

of a r a p p e l

if and o n l y

the size

of siz a - r a n k a d e c r e a s e s

a of A it is e a s y to p r o v e

is a r a p p e l

is e q u i v a l e n t

of 6 or 6# p r e s e r v e s

is, the v a l u e

on siz a - r a n k a the e x i s t e n c e

and t h a t any

y, and y' p r e s e r v e s

of 6 or 6#: this fact can be u s e d

For any e l e m e n t

We

i n d u c t i o n on siz a-

is a l w a y s n o n - n e g a t i v e

and r a n k and t h a t any i n s t a n t i a t i o n increases

in the e l e m e n t s

is e a s y to handle.

N o t e t h a t this n u m b e r

instantiation

of

(x2x 3 + x2x4).

In t h i s p a r a g r a p h we are g o i n g to use o f t e n r a n k a.

of X.

the e x i s t e n c e

(x I + x2) (x 3 + x 4) e n d i n g

see t h a t this d i f f i c u l t y

to s t a t i n g

is not u n i q u e l y de-

it is e a s y to p r o v e

(XlX 4 + x2x 4) and

in a

of the d i s t r i b u t i v e

is e q u i v a l e n t

that the end of a r a p p e l

in

in a'

t h a t are p r o d u c t of e l e m e n t s

thus,

of 6 or 6#.

of a is a p a t h w i t h o r i g i n

It is e a s y to c h e c k t h a t c o n d i t i o n

t h a t a is the

will

is an i d e n t i t y or an i n s t a n t i a t i o n

is no i n s t a n t i a t i o n

Intuitively obtained

C>a'

that a

i n v e r s e s and t h a t a

T h e n if C is c o h e r e n t

~ c is an i n s t a n t i a t i o n

there exists

a commutative

, l', ~'

of 6 or 6 # .

d i a g r a m of t y p e

-53.©

b

/

~ c such that d ~, ~',

k, ~,

sequences

•

inverses,

of instantiations

is some instantiation is an instantiation

e

t

0

C > e is a sequence their

>

>d

of identities

or instantiations

7,and 7', while b of 6 and 6 #.

of l , ~ ,

or their

C > e, c

Moreover inverses

in d

of

C > d are nv > e there

if and only if a

~b

of the same type.

Proof: The proof can be done by induction by the following i)

on

lal in the form outlined

diagrams.

In the case

a=x+y

X' + y

0

>e+y

x" + y

0

>c

\

we use the induction

hypothesis.

2)

given by

In the situation

+ y

x' + y

a = x + y

~ there are two different

cases.

x

+ y' If x + y

tion of 6 or 6# we can use the construction

% x' + y is an instantiagiven by

-

54

-

x' + y

> x' + y'

O

t

a ~ x + y

x+y' Otherwise,

0

we can take t h e construction

Id.

x' + y' given by

x' + y

O

> x' + y'

x + y'

O

> x + y'

a = x + y

In both constructions 3)

The naturality

we make use of the naturality

of Y allows us to make the construction

x

~+

O

of @. given by

> y i x'

a=x+y

+y 4)

The naturality

of y allows

( ~ +

a=x+

>x' the following

y) + z

O

We omit the analogous The commutativity conditions

construction

> (x' + y) + z

(y+z)

+ (y+z)

5)

+y

is used

cases

O

> x' + (y + z)

for the product.

of the diagrams

of type

in the following

(VIII)

construction

of the coherence

-

55

x[(y + z)w]

-

0

>

a = [x(y + z ) ] / ~ w

tivity of 6)

(VI) and

+ x(zw)

I

(xy + xz)w We omit the analogous

x(yw)

0

>

(xy)w + (xz)w

cases in which we should use the commuta-

(VII).

The commutativity

of the diagrams

of type

(II) is used in the fol-

lowing construction

/•

+ z)x

0

>

yx i zx

+ XZ

~

~

xy + XZ

a =x(y + z)

We omit the analogous tativity of 7)

(I) and

--

S"

cases in which we should use the commu-

(III).

The commutat'ivity of the diagrams of type

(IX) is used in the fol-

lowing construction (x + y) z + (x + y)w

O

>

(xz + yz) + (xw + yw)

a = (x + y)(z + w)

x(z + w) + y(z + w) 8)

The commutativity

"~v > (xz + xw) + (yz + yw)

of the diagrams of type

lowing construction

(IV) is used in the fol-

-

56

-

[(x + y) + z]w --- v

a=

[x+

> (xw + ~)

+ zw

(y + z)Jw

xw +

(y + z)w

~

xw +

(yw + zw)

We omit the a n a l o g o u s cases in w h i c h we should use the c o ~ u t a tivity of 9)

(V).

We use the c o m m u t a t i v i t y

of the d i a g r a m s of type

(XXIII)

in the

following c o n s t r u c t i o n

//•÷y

0

>xiy

0

> ux + uy

a = u(x + y)

ux + uy

We omit the a n a l o g o u s case in w h i c h w e should use the commutativity of 10)

(XXIV).

In the c o n s t r u c t i o n

0

x

>x'

a = ~X

u×'

0

.- >

ux'

We are u s i n g the n a t u r a l i t y of k. We omit the analogous case in w h i c h we should use the n a t u r a l i t y

of~. ii)

We use the n a t u r a l i t y of 6 in the c o n s t r u c t i o n

given by

-

57

-

x ' ( y + z)

©

>

x ' y + x'z

a = x ( y + z)

xy + xz

We o m i t the a n a l o g o u s of 6 and

cases

in w h i c h we s h o u l d use the n a t u r a l i t y

7

S u p p o s e t h a t C is c o h e r e n t , there

are r a p p e l s .

k, p,

is no o c c u r r e n c e

that a

~ > b is a p a t h in w h o s e

of n and t h a t a

Then there exists a commutative

s u c h t h a t a' ~, ~',

> xy + xz

6#.

PROPOSITION

vertices

O

~ v f b' their

a

O

>b

a'

O

>

O>

a' and b

O>

d i a g r a m of t y p e

b'

is a s e q u e n c e of i d e n t i t i e s or i n s t a n t i a t i o n s

inverses,

b'

of

7 and 7'.

Proof: The e x c l u s i o n of n in the v e r t i c e s of a a

O>

b there

is no i n s t a n t i a t i o n

of k , P

and t h e n the f o r m of the p r o p o s i t i o n in w h i c h a their

O>

b implies

, k', p'

y, ¥',

inverses

a l l o w s us to r e d u c e to the c a s e

O > b is an i d e n t i t y or an i n s t a n t i a t i o n

inverses,

or t h e i r

t h a t in

6 or 6#, and this w i l l be d o n e

of ~, ~ ,

~, ~'

,

in t h r e e parts.

The f i r s t p a r t w i l l be p r o v e d by i n d u c t i o n on siz a - r a n k a and

studies

the c a s e

t i o n of ~, ~',

their

= siz b - r a n k b. trivial;

in w h i c h a inverses,

v,

b is an i d e n t i t y or an i n s t a n t i a -

y and y',

in w h i c h case,

siz a - r a n k a

The c a s e siz a - r a n k a = siz b - r a n k b = 0 is

o t h e r w i s e we can use the d i a g r a m

-

a

--~

aI

58

-

b

(P)

• bI

b' 1

(P)

(H) a'

~v

> c '

in w h i c h the

a

% aI

diagrams

hypothesis

with and

proposition path siz

b

6.

a - rank

siz

also

that

of

their

~', The

when

a

~

done

by

siz(au)

where

siz

within rank

b,

O

and

the

have

the

symbol

b,

siz

to be

proof

of

(siz

we

a.

constructed

by

are

of

given the

constructed

the

induction

using

decomposition by

the

rappels,

of

fact

the

that

implies

we

z_ < I

siz

have

to

of

the

a - rank

impose

identities

only

can

is o u t l i n e d

proof

i, ~ ,

6 or

Remark

a)(siz

a - rank

u = 0 if a n d

the

a.

the or

additional

instantiations

y'

a - rank

=

b'

6 is a s s u r e d

a_ - r a n k i

y and

>g' are

(P)

is a s e q u e n c e

a + siz

exception,

been

which

induction

O ~>

possibility

is g o i n g

- rank(au)

a - rank

% bI

(H)

instantiation

on

a - rank

trivial

the

> b'

part

b is a n

induction

siz

< d = siz

inverses,

second

the

b - rank

to do

a'

with

b

in p r o p o s i t i o n

d - rank

condition ~,

> d

> e'

and

symbol

that

a = siz

that

a'

ones

Note O

.% d'

~>

the

the

~ bi

Remark

~ ~

u)

of

the

6#,

that

- rank

proposition

and

for

this

any

will

be

a o f A,

a - rank

u =

u,

if a

suppose in t h e

¢ X. that,

Hence, siz

following

in this

a - rank diagram

a >

case, siz

b

-

-

59

-

b

a

L 0

aI

a !

where

the symbol

(I<)

~b'

(H) in the inside of a diagram means is the reason

part of this proposition proposition

0

c i

~k

!/ d

0

> c

~

tion hypothesis

t

(p)

that the induc-

for the commutativity,

and the induction

(K) the first

hypothesis,

and

(P) the

6.

The third part is going to be the proof of the proposition the case in which a is an immediate

~ b is an instantiation

consequence

bottom path is a sequence value

16.

i) ii)

of k-i or ~-i and this

of the second part and the fact that the

of instantiations

w i t h formal

inverse whose

is an isomorphism.

The concept of normalization Let a be an element of A.

a

for

~v ~

a' satisfying

Any step in a

A normalization

the following

O>

conditions:

a' is an identity

or an instantiation

a' is not the origin of any instantiation Intuitively

application,

to the following:

case

the elements

it is not possible satisfying

of X different

is similar

the condition

from u.

ii)

that are In the

to the concept

but it is only useful when applied

the occurrences

of k and ~.

of

ii).

are ends of rappels because pletely

of instantiations

by

to give a simple characterization

The concept of normalization tion or rappel,

is a path obtained

a is the sum of elements

of elements

of ~ or ~ .

of k or ~.

that if a is the end of a rappel

either u or the product general

a normalization

as many times as possible,

It is easy to prove is equivalent

speaking

of a is a path

to elements

in this case it eliminates

of u in the expression

of reducthat

almost com-

of the elements.

In the

-

general case one typical normalization

situation

-

is the following:

an identity is a

of the element Xl(U + x2) , but Xl(U + x 2)

and an identity malization

60

is not a n o r m a l i z a t i o n

"~ XlU + XlX 2

of XlU + XlX 2 for w h i c h a nor-

is the path

XlU + XlX 2 that in fact eliminates

% x I + XlX 2

all the o c c u r r e n c e s

of u in the e x p r e s s i o n

of

the element. PROPOSITION

8

Suppose that C is coherent and that a is the end of a rappel. Then if a a

O > a' is a normalization,

O > a' are u n i q u e l y d e t e r m i n e d

the element a' and the value of

by a.

Proof : The proof is similar to

(and simpler than)

the proof of proposi-

tion 4. PROPOSITION

9

Let a and b be elements of A that are ends of rappels, a path whose steps are instantiations 7 and y' and a ly.

v r a , b ~

If C is coherent,

such that a'

of ~, ~',

b' n o r m a l i z a t i o n s

k, P,

their

a

v z b

inverses,

of a and b r e s p e c t i v e -

there exists a c o m m u t a t i v e d i a g r a m of type

a

O

>b

a'

v ~

~ b'

O > b' is a sequence of identities and instantiations

~, ~', their inverses,

of

y and 7'.

Proof: It is analogous

to

(and simpler than)

the proof of p r o p o s i t i o n

5. Suppose that a is the end of a rappel and that a - ~ - ~ a' is a normalization.

If a is regular

so is a' and if Adec a =

(ai,'°°,a ~)

-

then if i @ j the set of factors

61

-

of a. is different 1

from the set of

factors

of a. as is an immediate consequence of the definition of 3 regularity, and, moreover, among the factors of any a i there is no repetition

~7.

of elements,

The coherence

as can be also proved almost

theorem

We are going to use the results Theorem

4.2 of

a complete specially

[4], but expressed

illuminating:

remark ~3 of

on coherence

in a more

proof of the equivalence

[4] that holds

immediately.

that this formulation

We omit

difficult

nor

to the same proof given

we are going to give.

is different

in the

language.

that is neither

in fact it reduces

in the formulation

formal

stated

in

We have to

from the one contained

in

[2]. Let A' be the subset of A generated

is the free [+~-algebra algebra

over X - ~n~.

and hence a {+~-algebra,

by all the elements

i x for x,y,z vertices

elements

are in ~' and whose

Theorem

4.2 of

, ¥ , and b.

of _A'"

sum of different

elements

element

that a

tiations

of ~'

of a implies coherence

+ -subalgebra

of the form,

Suppose

~> l

!

~>

steps are elements

c is a path whose

the relation,

result

above

pends upon a and c.

b is a path whose of H ' ~ ,

then the

and C is coherent

v ~ b only depends if and only

some consequences

of A in which there

~,-i or y'

of the edges of

~' ~,-I '7x and x,y,z' x,y,z ,y

that a

of A' is regular

of

for

upon a

if it is the

of X.

We are going to deduce

suppose

of the

the value of the path a

Note that an element

be a regular

The edges of H are a ~ + , . ~ -

[4] states that if a i s regular

, ~'

by X - In}:

and we take as H' the subgraph

whose edges are all the elements H generated --

additively

of that result.

is no occurrence

steps are identities

If Adec a =

(al,''',ar),

Let a

of n and or instan-

the regularity

i ~ j------~a @ a and from this and the

it follows

that the value of a - ~

c only de-

-

Similar c o n s e q u e n c e s PROPOSITION

10

62

-

hold for the product.

(Coherence theorem)

If ~ is coherent and a is a regular element of A, the value of any path a

~>

b depends only upon a and b.

Proof: Let a -~--> a' and b - ~ - > b' be reductions. is p o s s i b l e

0

>b'

,

w h e r e the value of the columns are isomorphisms a and b, in a' ,

,

~*

~f

b' there is no o c c u r r e n c e of i n s t a n t i a t i o n s

, ~

or their inverses,

the p r o p o s i t i o n w h e n in a - ~

hypotheses

on a

and where all the vertices are n or b'.

Hence we are reduced to proving

b there is no i n s t a n t i a t i o n

of k'

hypothesis

~ f b. ~ / b' : the value of it is a monomorphism,

~ f b ', that is, we can

(and will)

that b is the end of a rappel.

By p r o p o s i t i o n

in the

from now on we are going to assume these

hence we are reduced to prove the u n i q u e n e s s b

~'

~ k

Take now a rappel b

~

of

or their inverses and where the symbol n is not involved

e x p r e s s i o n of the vertices:

a

that only depend upon

*

no n is in the v e r t i c e s of a' - ~

k , p

5 it

to find a c o m m u t a t i v e d i a g r a m of type

a',~

~' @'

By p r o p o s i t i o n

7 there is a c o m m u t a t i v e

a

\/

0

of the value of any path assume the a d d i t i o n a l

Let a

O>

a' be a rappel:

d i a g r a m of type

>-b

a o f

where a'

~r

b is a p a t h with no o c c u r r e n c e

6#, and we are reduced

of i n s t a n t i a t i o n s

to prove the u n i q u e n e s s

of 6 or

of the value of a ' ~ A P b ,

that is, we are going to assume that a and b are ends of rappels.

-

Suppose now that a By p r o p o s i t i o n s

w h e r e a'

8 and

~>

63

-

a' and b

O>

b' are n o r m a l i z a t i o n s .

9 there exists a commutative

a

0

>b

a'

O

> b'

b' and the v a l u e s of a

a

and b

O>

d i a g r a m of t y p e

b' d e p e n d o n l y

u p o n a and b, and the fact t h a t the v a l u e s of the c o l u m n s are morphisms

a l l o w s us to r e d u c e our c o n s i d e r a t i o n s

the v a l u e of the p a t h a' cated tion i)

in p r o p o s i t i o n

9.

~v, b', Hence,

that satisfies

we are r e d u c e d

iso-

to the u n i q u e n e s s the c o n d i t i o n s

to p r o v i n g

indi-

the p r o p o s i -

for the f o l l o w i n g c o n d i t i o n s : Every

step in a

their

inverses,

ii) A n y v e r t e x

O>

b is an i d e n t i t y or an i n s t a n t i a t i o n of ~, ~',

7 and Y'

in the p a t h a

vf

b is a sum of e l e m e n t s

is e i t h e r or a p r o d u c t of e l e m e n t s

of X d i f f e r e n t

e a c h of w h i c h

f r o m u.

The n a t u r a l i t y of ® and • i m p l i e s t h a t a n y i n s t a n t i a t i o n -I

of

or Y is c o m m u t a t i v e w i t h any i n s t a n t i a t i o n

of ~',

~'

-I

or y

of ~, ,

and

t h i s p r o v e s the e x i s t e n c e of a c o m m u t a t i v e d i a g r a m of type

-.~

a

>b

C

s u c h t h a t in a ,

O > c every

or y , and e v e r y

step in c

s t a n t i a t i o n of ~, ~-i or 7. of c. i)

For this,

If d

step is an i d e n t i t y or an i n s t a n t i a t i o n ~>

b is an i d e n t i t y or a n y in-

Our n e x t a i m is to p r o v e

the u n i q u e n e s s

n o t e the f o l l o w i n g r e l a t i o n s :

~ e is an i n s t a n t i a t i o n of ~, ~

-i

or y, t h e n A p t d = A p t e.

T h i s can be p r o v e d v e r y e a s i l y by i n d u c t i o n on follows

of

t h a t A p t c = A p t b.

Idl.

F o r m this

it

-

2)

If d

% e is an instantiation

of a rappel, Adec e =

and Adec d =

Adec a =

Ida.

-

of ~' , u.-i or y, and d is the end

(dlod 2,

(d~l,d62,°°°,dGr),

induction on

with

o o o

~¢

,d r )

then,

S r.

This can be proved by

From this it follows that if

(al,a 2 ,°°°~ a r ), then, Adec c =

¢ S r and, 3)

64

as we will see later,

b

(a61, "°',a6r)

determines

If c - ~ - > b is a sequence of instantiations the end of a rappel,

Adec c =

then for i = 1,2,°°',r,

(Cl,''',Cr),

there

identities or i n s t a n t i a t i o n s

-i

Supp a6i = Supp b i.

i ~ j ~Supp

O~

(bl,''',br),

b i whose

steps are

a6i

b i, and

or 7.

But the r e g u l a r i t y

Supp a6i = Supp b i.

u n i q u e l y Adec c and Apt c and by p r o p o s i t i o n

O>

of a imposes that,

a i @ Supp aj, and this proves that6i

termined by the condition,

and y, c is

and Adec b =

From this it follows that for i = 1,2,''',r, hence,

~ uniquely.

of ~, - i

is a p a t h a i of ~, ~

for some

is uniquely de-

Thus b and a d e t e r m i n e

1 the element c is

u n i q u e l y defined. The u n i q u e n e s s

of the value of a

remarks of the b e g i n n i n g

tions of ~, - i

and y.

c has been stated in the

of ~7.

The only thing that remains the value of the path c

~>

O>

to be proved

is the uniqueness

of

d in w h i c h all the steps are instantia-

Suppose that c = c' + c",

then it is very easy

to prove the e x i s t e n c e of a c o m m u t a t i v e d i a g r a m of type, r-% v

C

>d

\d+o/ such that in c

~>

d' + c" all the steps are elements

of type H + ic.

for some step ~ and in d' + c"

~f

and w i t h a trivial

Ic~ we are reduced to the case in w h i c h

induction on

d all the steps are of type Id.+N,

c is the product of elements of X, and the proof in this case is analogous to path a

(and easier than) ~ • c.

the proof of the u n i q u e n e s s

of the value of

--

65

-

REFERENCES [i]

G. M. Kelly,

"On Mac Lane's conditions for coherence of natural

associativities,

commutativities,

etc.", J. Algebra,

4(1964),

397-402. [2]

[3]

G. M. Kelly and S. Mac Lane,

"Coherence in closed categories",

Pure Appl. Algebra,

97-140.

M. Laplaza,

i(1971),

"Coherence for categories with associativity,

tivity and distributivity", [4]

S. Mac Lane,

S. Mac Lane, IV(1970),

49(1963),

commuta-

(to appear).

"Natural associativity and commutativity",

Univ. Studies, [5]

Bull. Amer. Math. Soc.

J.

Rice

28-46.

"Coherence and canonical maps", Symposia Mathematica,

231-241.

MANY-VARIABLE

FUNCTORIAL

G.M.

The U n i v e r s i t y

CALCULUS.I.

Kelly

of New South Wales,

Kensington

2033,

Australia.

Received May 22, 1972

I.

Introduction

I.I

The a u t h o r

structure

carried

categories~

sees a c o h e r e n c e

by a category,

the e x t r a

structure

problem

or more

generally

consisting

axioms.

This v i e w may turn out in the end to be too narrow,

least

as wide

includes

as this.

all c o h e r e n c e

The p r e s e n t problems

In such a s t r u c t u r e among

the data are

variances;

general

functors

as for i n s t a n c e

functor

[ , ]: A °p × A ÷ A.

the data, structure,

by i t e r a t e d

connect

substitution.

The b a s i c

Here

must be a t wide,

a:

functors

and often

structure

a closed

not these basic

of, the basic

variables

a monoidal

such as the a s s o c i a t i v i t y

problem"

but the

natural

of m i x e d

on A i n v o l v e s

structure

substitution

but

transformations

others

means

functors

also i n v o l v e s

(A®B)®C ÷ A®(B@C)

functors

given

a among

in a m o n o i d a l

made

from them

the p r o c e s s

whereby,

from f u n c t o r s T: A x B °p x C ÷ P,

we get the

value

C x E °p ÷ A,

Q: B ° P x E ÷ B, R:

F ÷ C,

functor

T(P,Q,R): whose

P:

at

C × E °p × B ~ E °p × F ÷ P (X,Y,Z,U,V)

and

to the author.

as we speak

®: A 2 ÷ A and I: A 0 + A, w h i l e

equational

view is in fact e x t r e m e l y

known

of many

to v a r i o u s

coherence

of

of v a r i o u s

and n a t u r a l

view of "the most

an e x t r a

by a family

in the g i v i n g

subject

with

functors

definitive

transformations,

as c o n c e r n e d

is T(P(X,Y),

Q(Z,U),

R(V)).

Substitution

-

generalizes functors

composition

natural

Again,

to which

the axioms

axiom for a monoidal

transformations

ing functors

-

of functors,

of one variable.

the pentagonal

67

it reduces

in the case of

for the structure,

structure,

involve

like

not the basic

llke a but others made from them by "substitut-

into them and them into functors";

axiom as given on p. 98 of [7]

involves

thus the pentagonal

the natural

transformations

with components

a(A@B,C,D):

((A@B)eC)®D

+

(A@B)@(C@D)

a(A,B,C)@D:

((AeB)eC)eD

÷

(A®(B@C))@D.

of coherence

problems

and

An abstract tidy calculus

of substitution

suitably

general

Godement

calculus

natural

symmetric

Hence

of a single

do nothing wilder

of natural

functor T(P,Q,R) To(P x Qop x R).

is to give such a

the

we hope in later papers

about

these

and

c: A®B ÷ B®A in a

than permuting

transformation.

is not usually

above

to deal

In the last future plans

seen as a primitive

can be expressed

as an ordinary

In the same way the notion

can be expressed

v(x,y);

but one has a calculus

stitute

a closed

in functorial

and for ordinary

and the

we have not yet overcome.

Substitution

f(z)

the

are all covariant

llke the commutativity

of this paper we say something

difficulties

variable

a

and for

extending

of this paper

the "I" in the title;

therefore,

of many variables

case, when the functors

category,

kinds

needs,

transformations,

The purpose

transformations,

with more general

function

for functors

monoidal

variables.

for functors

of natural

in the simplest

the natural

1.2

kinds

transformations.

calculus

section

theory

circle

calculus

in terms

of ideas within there

is a closed

of a complex

functions

analytic u(x,y)

because

the larger calculus. circle

the

composite

of real functions

of complex

notion;

of ideas

and

they conSo too

centred

on

-

w i t h no p l a c e

substitution,

codomain

is a p r o d u c t

restricted

to s u b s t l t u t i o n - i d e a s

discussion

of c o h e r e n c e

Coherence degree the

levels paper

levels,

there t51

discussion

stitution overcome

volume

say,

following

appropriate refer

of s u b s t i t u t i o n ,

some t e c h n i c a l

calculus

"substitution"

In s a y i n g substitution,

fairly

of the s i m p l e r

substitution-calculus

general

to the

would

last

that we can r e s t r i c t claiming

that has no e x p l i c i t

place

product:

such as the t w i s t i n ~

functor

This must theory makes

or the d i a ~ o n a l

s e e m at first

|9],

designed

explicit

to the For

at these

the f o l l o w i n g

for an a b s t r a c t

levels,

it r e m a i n s

in terms

difficulties

that

of subto be

Finally,

seems

liberally;

cases,

it.

to be true

here

I can only

I suspect

the right

of this p a p e r

for

that

setting.

an We

for a glance

cases.

we are in e f f e c t

= (B,A),

which

still

provide

section

calculus

t(A,B)

moreover

of s u b s t i t u t i o n .

the a s s e r t i o n

but does

they involve.

setting

For some h i g h e r

remain

of g e n e r a l i t y ,

again

true;

as an ideal

slogan;

according

to be said can be e x p r e s s e d

the a n a l o g y

the r e a d e r

at the more

reveals

up a s m o o t h

we i n t e r p r e t

for the

let me n o w q u a l i f y

transformations

is c e r t a i n l y

calculus

but there

level

suffice

is a u s e f u l

in a h i e r a r c h y

problems.

needs

in s e t t i n g

If all our

a calculus

alone

I'm not yet sure,

of the n a t u r a l

of c o h e r e n c e

the h i g h e s t provided

Since

is a s m o o t h

alone,

these

made baldly,

can be a r r a n g e d

true that w h a t e v e r

of d e v e l o p i n g

is that

the a s s e r t i o n

in this

x R whose

problems.

the case?

of g e n e r a l i t y

lower

The point

last a s s e r t i o n ,

problems

like P x QOp

A x B °p x C (or a t e n s o r p r o d u c t ,

are e n r i c h e d ) .

it o v e r s t a t e

-

for f u n c t o r s

categories

This

68

sight

allowance

that we

for f u n c t o r s

- with benefit

- to

can get by w i t h a whose

codomain

is a

t: A × A ÷ A x A given by

functor

unlikely,

to deal w i t h

ourselves

A: A + AxA given by AA = (A,A). since

Lawvere's

notion

of a

an e x t r a

structure

c a r r i e d by a set,

for the f u n c t i o n s

analogous

to the f u n c t o r s

-

t

and A.

Whereas,

structure

however,

69

-

an algebra may well have a law ab = ba, a

carried by a category is unlikely

A@B = B@A;

for this would imply f@g = g@f:

case in any natural example cases of monoldal

categories

is a functorlal equality,

I know of. where

to have a functorlal

law

A@A ÷ A@A, which is not the

C e r t a i n l y there are r e s p e c t a b l e

the a s s o c i a t l v l t y

and also the Isomorphisms

but here we are directly e q u a t i n g two functors

(A@B)@C ÷ A@(B@C) I@A = A, A®I = A;

A 3 ÷ A, and n o t h i n g

llke

t or A is involved.

We do of course have things CAB: A@B + B@A;

this is a natural

w o u l d seem to involve

~.3. natural

isomorphism

i s o m o r p h i s m c: @ ~ @t: A 2 ÷ A, and

the functor t.

we mean by "natural t r a n s f o r m a t i o n

like a natural

We must therefore

explain what

of a general kind".

For T, S: A ~ B we use a double arrow f: T ~ S to denote a transformation

morphisms

f(A):

in the classical

TA ÷ SA s a t i s f y i n g the usual n a t u r a l i t y

Now c o n s i d e r the c o m m u t a t i v i t y monoidal products,

category,

the diagonal

and the e v a l u a t i o n

Each of these is natural the following

sense, namely a family

diagrams

C®D

........

cOD

condition.

CAB: A@B ÷ B @ A in a symmetric

dA: A ÷ A×A in a category with finite

eAB: [A,B]@A ÷ B in a closed category.

in the sense

that,

for f: A ÷ C and g: B ÷ D,

commute:

CAB A@B

of

dA Y-~ B@A

A

=- D®C

C

~- A×A

dC

~-C×C

-

70

-

eAB [ A,B] @A

~B

[ f , 1] e l [ C,B] ®A [ 1,g] @ f ~

[ C , D ] ®C

~D eCD

The first two can if we wish be reduced ations

c: @ ~ @t and d: 1 ~ xA by i n t r o d u c i n g

There Is no similar r e d u c t i o n domain,

natural

the functors

transform-

t and A.

for eAB, where A occurs twice In the

once with each variance.

Such natural Eilenberg-Kelly by T(A,B,C) described where

to classical

[1].

transformations

as e were considered by

W r i t i n g T: A °p x k x A ÷ k for the functor given

-- [A,B]@C and l: A ÷ k for the identity

e as a "natural

transformation

functor,

e : T ÷ 1 of graph re -- E ",

the graph e of e was the i n f o r m a t i o n

as to which arguments

and of 1 were to be set equal in w r i t i n g the components pictorially

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

they

of T

eAB of e,

as

+ +

where the vertical their variances,

columns exhibit

the arguments

and the "linkages"

are to be set equal.

give a composite graph" q~, except contained

of the graph show which arguments

It was shown in [1]

f: T + S and g: S ÷ R of respective natural

of T and of i, wlth

graphs

transformation

in the i n c o m p a t i b l e

that natural

transformations

~ and q could be composed

gf: T ÷ R of the "composite

c a s e s where

the composite

q~

closed loops linking no arguments.

The natural

transformations

of

[1] always had the arguments

to

-

linked

71

-

in pairs , and included as a special case such things as CAB ,

which Im this language would be described as a natural

transformation

c: @ ÷ @ of graph Fc = y given by

In such a case as this, where all the functors are covariant, graph is n e c e s s a r i l y

a bijection

those of the codomain, particular

and is thus identifiable

we can identify

permutation

transformation,

dA: A ÷ AxA, Fd = 6, where

in

the graph y of c with the n o n - l d e n t l t y

but as something autonomous

6 is no longer Just a p e r m u t a t i o n

"natural t r a n s f o r m a t i o n

Clearly the concept

d: 1 ÷ x of graph but the unique

of the codomain to those of the

still further g e n e r a l i z a t i o n

of the

f: T ÷ S of graph ~", both in the

covariant and the m l x e d - v a r l a n c e the total i n f o r m a t i o n

suggests

doing the same with

calling it a natural t r a n s f o r m a t i o n

This in turn suggests

cases,

the graph in general being

about the arguments

set equal;

see §4 below.

of "graph" is what allows us to proceed without

mention of functors

kind of calculus we seek. tions of this p a p e r * a r e those where the functors permutations.

w i t h a graph,

This further suggests

function 2 ÷ 1 from the arguments

explicit

with a permutation;

of r e g a r d i n g e, not as a classical natural

doing the same w i t h c.

notion

of the domain with

of 2.

The n e c e s s i t y

domain.

of the arguments

the

llke t and A, and opens the way to the

We reiterate

that the natural

not those of most general graph, are all covariant

Note that for such graphs

and the graphs

incompatibility

transformabut precisely are only does not

arise.

Graphs were ~*except i n §4

first used in the d i s c u s s i o n

of coherence

-

problems

in [7]; their success

following paper [5] preparation. ing in these

72

-

there suggested

in this volume,

We make

two remarks

the treatment

for which the present

to orient

the reader

in the

paper is a

unused

to think-

terms.

First,

the fact that .the composite

is l, classically

expressed

CBA CAB: A®B ÷ B®A ÷ A®B

by the equation C w

®t

@

is now expressed

more simply by the equation c

¥

@

,-®

with graph

2

~-- 2

®

Secondly, commute"

result

2

the reader should note that the "all diagrams

of Mac Lane [12]

for symmetric

monoldal

does not mean that c = i: A®A ÷ A®A, which is false groups;

the equality

it asserts

is that of natural

not of particular

components

of them.

it is meaningless

to say c = i because

In terms

categories

for abelian transformations,

of 6enerlc

components,

B®A

A@B ~

is not a closed diagram. formations, different

A®B

In terms

it is meaningless

functors.

of classical

because

In our terms,

natural

trans-

c: ® ~ ®t and I: ® ~ ® connect

however,

it is meaningless

because

-

c:

?S

-

@ ÷ @ and i: ® ÷ ®, while they connect the same functors, have

different graphs, with Pc = y and P1 ~ 1. formations

For us, natural trans-

of different graphs cannot be equated;

an "all diagrams

commute" result becomes the assertion that the functor r, sending each natural transformation 1.4

to its graph, is faithful.

We spoke in § 1.1 of a calculus of substitution extending the

Godement calculus. terminology,

The latter calculus is, to use a more recent

the recognition

that the category ~

of categories

(small, or in some universe) has the structure of a 2-category. however,

says less about ~

cartesian closed category,

than the recognition that it is a -×B having the right adJolnt [B,-] where

[B,C] is the functor category. the closed structure; itself,

so that ~

2-category.

The 2-category

structure follows from

for any closed category admits enrichment over

is a ~ - c a t e g o r y ,

which is another name for

In fact vertical composition of natural transformations

is embodied in the category [B,C| functors and horizontal embodied in the M: [B,C] structure.

This,

itself, while both composition of

composition of natural transformations × [A,B]

+ [A,C]

are

arising from the closed

The Godement calculus sees only a small part of the

structure of ~ ,

dealing only with elements of objects of the special

form [B,C]. In the same way, our more general calculus will be expressed in terms of the existence of a closed structure and not even symmetric)

on a certain category;

(no longer cartesian, the analogue of the M

above will then provide our "calculus of substitution",

but it is

the whole closed structure that we need for our abstract discussion of coherence problems. 2. 2.1

The s l n g l e - c a t e g o r y c a s e We begin, purely for pedagogic

simplicity,

with the calculus

-

to d e s c r i b e

needed

all f u n c t o r s

extra

moreover

covariant

structure

of the

f o r m A n ÷ A, w h e r e I with

-

structures

If the

category

V4

c a r r i e d by a single

and all graphs m e r e l y

is to be c a r r i e d

and one m o r p h l s m .

that the right

thing

that a f u n c t o r

A 0 ~ B is Just an object

transformation

between

use this setting

section it down

to c o n s i d e r

two

is r a t h e r

by n the n a t u r a l

Write

~ for the c a t e g o r y

number

numbers,

n or the

An o b j e c t

n = m,

together

that

finite

set

Note

in B. We before

O.

We a m b i g u o u s l y

~1,2,

of T.

There T and

(n,T) ÷ (n,S)

transformation

..., n).

with no m o r p h l s m s

of n as the m o r p h l s m s

is an n E ~ t o g e t h e r

is, u n l e s s

a morphism

with a natural

including

B and C we form a " g e n e r a l i z e d

B n ÷ C; we call n the type

When they are,

A n ÷ B.

with motivation,

w i t h ~ as set of objects,

categories

~B,C).

(m,S) u n l e s s

out h o w e v e r ,

is Just a m o r p h l s m

n ÷ m for n ~ m, and with the p e r m u t a t i o n s

T:

It turns

functors

is to come,

set of n a t u r a l

denote

category"

A 0 = the unit

formally.

is the

Given

functors

of B, and that a n a t u r a l

such f u n c t o r s

to s k e t c h what

(with

permutations).

by A, this w i l l m e a n h a v i n g

A n = A × ... × A, w h i l e

one object

category

n ÷ n.

functor

with a functor

are no m o r p h i s m s S are of the

(n,T)

same type.

is a p e r m u t a t i o n f: T ÷ S of g r a p h

~ of n ~. Such an

f has c o m p o n e n t s

(2.1)

f(A 1 . . . . .

Setting

F(n,T)

exhibiting

An):

= n and

F(~,f)

{B,C) as an object

over P, w i t h a u g m e n t a t i o n The a s s i g n m e n t functor that

~op

T ( A ~ I , ..., A~n)

× ~

= ~ we get a f u n c t o r of the c a t e g o r y

G~/~

.... An). F: {B,C) ÷ ~, of c a t e g o r i e s

F.

B,C ~ - ~ { B , C }

÷ ~/~,

÷ S(A I,

whose

it w i l l have a left adJoint:

is e a s i l y

evident that is,

seen to p r o v i d e

continuity there

a

in C e n s u r e s

is a f u n c t o r

-

75

-

(2.2) and a natural isomorphism (2.3) It is in fact easy to see explicitly what AoB must be. F for all augmentations over ~.

Write

To give an element ¢ of the right side

of (2.3) one must give a functor CA: B n + C for each A 6 A with FA = n, and a natural transformation el: CA ÷ ~A' of graph ~ for each f: A ÷ A' with Ff = ~.

To give CA one must give objects CA(B1,

and morphisms ~A(gl,

..., gn): CA(B1,

..., B n) of C

..., Bn) ÷ CA(B~ . . . . .

B~),

!

where gi: Bi ÷ Bi in B. To give Cf one must give its components, is, morphlsms Cf(B1,

..., Bn): CA(B~I , ..., B~n ) ÷ CA,(B1,

that

..., Bn).

These data are to satisfy the conditions making CA funct0rial, making Cf natural, making ¢(f'f) equal to (¢f')(¢f), and making ¢(1 A) equal to leA. This means that AoB is to have objects of the form A[BI,

..., B n] where A 6 A, B i 6 B, and FA = n; its morphisms are to

be generated by A[gl, f[B1,

.... gn]: A[B1,

..., B n] ÷ A[B~,

..., Bn]: A[B 1 , .... B n] ÷ A'[B1,

..., B~] and

.... Bn] ; these generators

are to satisfy relations corresponding to the conditions at the end of the last paragraph.

One of these, the naturality of Cf, gives the

relation A[ B~I , .... B~n] f[B 1 . . . . .

Bn] m A,[BI,

..., B n]

!

(2.4)

A[ g~l . . . . .

g~n ]

I A'[gl,...,g n] W

A[B'~I,

~--A'[BI,

..., B~n]

fib i, Writing f[gl'

"'''

B'] n

...,

..., gn ] for the diagonal of this, one easily verifies

that this is the most general morphism of AoB, and that the edges of

-

(2.4)

are just the special

usual convention, Next,

regard ~

0 E ~.

Then

A ÷ ~ consisting

(2.2) admits o: ~ / ~

and then augment

an immediate

× ~/~

one has only to define

verifies

AoB by setting

identity

of C ~ / ~

of the constant

ignoring

by giving to functor at

to a functor

Ca~/P

Finally

the augmentation

of B,

..., Bn]) = FB 1 + ... + FB n

definition

o on ~ / ~

J, making

of course.

with the

1Bi.

extension

r(A[BI,

with an appropriate

unsymmetric,

in accordance

÷ ~/~;

AoB as before,

that the bifunctor

a coherent

in which,

as a full subcategory

the augmentation

on objects,

-

A and B i stand for 1 A and

A E ~

(2.5)

cases

V6

on morphisms.

is coherently into a monoidal

one verifies

One easily

associative category

with

- highly

that our original

functor (2.6)

{ , }: ~ o p

itself extends (2.7)

(2.8)

G~/~

(AoB,C)

~ flg~/~(A,{B,C}); category.

This is our goal;

as such is embodied

o {A,B} ÷ {A,C} arising

for later purposes

2-functors,

÷ ~/~,

(2.3) to

as a closed

stitution-calculus ~: {B,C}

x (~/~)

isomorphism

~I~

exhibiting

+ ~/~

to

{ , }: ~ g t / p ) O p

and the natural

useful

x ~

and that

(2.8)

that is, a 2-adJunctlon

in the functor

from this closed

to observe

that

is a 2-natural

(in the older

the sub-

structure.

o and { , } are actually isomorphism

of categories;

sense of ~ - a d j u n c t i o n ,

that of Gray [4]). We now set this down formally.

It is

not

-

2.2

We n e e d some n o t a t i o n

mutations

In permutations.

77

-

for the process

of s u b s t i t u t i n g

per-

For n E ~, deflne a functor Yn: ~ n ÷ ~ by

Yn(ml,

..., m n) = m I + ... + m n,

Yn(~l,

..., Dn ) = H I + ... + nn;

(2.9)

here n i Is a p e r m u t a t i o n

of'ml and n I + ... + n n is the p e r m u t a t i o n

of m I + .,. + m n w h i c h effects

n I on the f i r s t

n 2 on the next block of m2, and so on. define a n a t u r a l

transformation

block

of m I elements,

For a p e r m u t a t i o n

Y~: Yn ÷ Yn of graph

E of n,

E whose

component

(2.10)

Y~(ml,

is the evident

.... mn): mEl + ... + men

permutation

We introduce

that

"permutes

÷ m I + ... + m n

the blocks a c c o r d i n g

the above n o t a t i o n because

b e l o w to refer explicitly,

a l t h o u g h briefly,

w o r k i n g notation,

we drop the Y and set

however,

we have occasion

to Yn and YE" For a

(2.11)

n(ml,

.... m n) = Yn(ml,

..., mn) = m I + ... + mn,

(2.12)

n(~l,

..., n n) = Yn(nl,

..., qn ) = n I + ... + nn,

(2.13)

E(m l,

..., m n) = YE(ml,

..., mn).

The n a t u r a l l t y

of YE Is e x p r e s s e d

by the c o m m u t a t l v l t y E(m I,

n(mEl,

..., m~n)

to E".

of

..., m n) ~

n(m I,

..., m n)

I

(2.14)

n(nEl,

"''' q~n) n(mEl,

I

I n(~l'

...," men)

~ E(m I,

We denote

(2.15)

the diagonal

of this by

E(n 1 . . . . .

nn);

..., m n)

"''' nn)

n(m I ~ ..., m n)

-

so

(2.15)

blocks

is the p e r m u t a t i o n

according

78

-

of m I + ... + m n which permutes

to ~ and at the same time effects

on the a p p r o p r i a t e

block.

case of (2.15) when

Observe

that

(2.12)

time that all our n o t a t i o n name of an object

is consistent

to verify

of space since they are e n t i r e l y

~[nl,

..., n n ] ~ - ~ ( O l ,

to llst them would be a that

~: ~o~ ÷ ~ m a k i n g ~ a o-

i I (the identity

permutation

of 1).

of AoB for A,B E ~ / ~ ;

A m o r p h i s m A[B1, of a m o r p h l s m

A[BI,

all

A[B1,

..., Bnl ~ A'[B~,

where

..., B~I

for

FA = FA' = n,

f: A ÷ A' in A with rf = ~ say, together with

(2.16)

f|gl'

f[gl'

A[I

.... B~1,where

gi: B~_li ÷ B i' in B; this m o r p h i s m "''' gn ] : A[B1, llke

objects

..., Bn| ~ A'[B~,

morphlsms

(It would look more

..., B n] where A E A

Note that this includes

There are no m o r p h l s m s

consists

of the

summed up in the a s s e r t i o n

AoB has objects

with rA = n and B i E B.

m ~ n.

that the

over ~ are denoted by r.

As a category,

~A = 0.

for the last

such p r o p e r t i e s

We give the formal d e f i n i t i o n

augmentations

is the special

with the c o n v e n t i o n

..., n n) is a functor

in ~g~/~ with identity

2.3

(2.13)

We repeat

(2.15) as are n e e d e d in the sequel;

waste

monoid

ni

is also the name of its identity morphlsm.

We leave the reader expression

the p e r m u t a t i o n

is Just the special

~ is r e p l a c e d by i n and that

case of (2.15) when n i is r e p l a c e d by Iml.

the

is denoted by

..., B n] ~ A'[B~ . . . . , B~] .

(2.4) if we wrote

.... gn ] : A[B~l , ... B~n]

-* A'[B~,

..., B~]

!

where

f: A ÷ A' with Ff = ~ and wheregi:

(2.16) with h[kl, Fh = ~, is

..., k n] : A'[B~,

(hf) [PI'

B i ÷ Bi.)

..., B n] -* A"[B~',

The composite

of

..., m"]n.' where

..., pn ] where Pl is the composite

-

(2.17)

B _i _i i

Clearly

AoB

1A[1B1,

...,

1 B ]; note n

-

~ B' ~-li

g-i i

is a category,

79

with

that

~

ki

identities

A[BI,

F1 A = i n since

AoB into

(2.18)

F(A[B1,

..., Bn] ) = FA(FBI,

...,

FBn)

(2.19)

r(f[g

...,

...,

rgn)

in the n o t a t i o n

gn ] ) = r f ( r g l ,

ToS:

AoB

Bn]

=

over ~ by s e t t i n g

= FB 1 + ... + FBn, ,

of §2.2.

The o p e r a t i o n we define,

...,

F is a functor.

We make

l,

a category

" Bi.

for functors

o becomes

a functor

~a~/P

× ~/P

÷ ~/P

when

T: A ÷ A' and S: B ÷ B' over P, the functor

by

÷ A'oB'

ToS

(AtBI,

..., B n])

-- TA[SBI,

..., SB n],

ToS

(f[gl'

"''' gn ]) = Tf[Sgl'

"''' Sgn]"

(2,20)

~/~

is a c t u a l l y

a 2-category;

a 2-cell

a: T ~ T': A ÷ A' over ~, that is,

is a n a t u r a l

transformation

one for w h i c h we have

commutatlvity

in

which

means

that

r(~A)

= IrA.

The

functor

o becomes

a 2-functor

when,

for a: T ~ T' and 8: S ~ S' over ~, we set

(2.21)

mob

(A[BI,

Identifying

..., Bn]) ~

as in

-- eA[SBI, §2.1 w i t h

...,

8B n] .

the full

subcategory

of ~ t / ~

-

of those

consisting

it is i m m e d i a t e o:

~ / ~

from

Another form a category together

with

[B 1 . . . . . |gl"

that

of l o o k i n g

over ~ called

an n-ad

[B1,

• restricts

MB

gi: B~_li

MB is Just

A[BI[Cl,

...,

~ B i.'

satisfies

the p e n t a g o n a l

AoBoC without

[A][BI,

condition

to have

by 1 both

the unique

Define

a new object

different

l[A]

to A and l[f]

that

sending

to f, while

It clearly

of the

in fact

form

normally where

suppress

convenient.

and the unique

morphism

I is also an object constant

to be the category the constant

identity

is defined

we could

We shall

object

of ~ ,

I ÷ ~, n a m e l y two-sided

= Ao(BoC),

associative

I ÷ ~ given by the

J of ~g~/~

augmentation

Then J is a coherent sends

As an object

augmentation

objects

Cm] , and so on.

Denote

with

a second

Note

is immediate.

for coherence;

o as strictly

ZG~/~

(AoB)oC

and its 2 - n a t u r a l i t y

...,

I.

of AoB.

The i s o m o r p h i s m

and treat

category

the fibred

Cml+m2] , ..., Bn[... , Cm]];

the i s o m o r p h i s m

the unit

an n - a d

to

parentheses

..., Bn][CI,

with

the a u g m e n t a t i o n

Cml] , B 2 [ C m l + l , ...,

on morphisms,

is an n

is over ~, MB gets

isomorphism

[C 1 . . . . , C m]

similarly

For B 6 ~ ,

an object

Then AoB is Just

B itself

FB i = m i and m I + ... + m n = m.

define

at 0,

~oB.

is a 2 - n a t u r a l

(A[B I . . . . , B n ] )

("multi-B");

is an ~ t o g e t h e r

over ~ used to give

There

functor

to a 2 - f u n c t o r

at AoB is instructive.

over ~ of A and MB; when

as a category

is the constant

..., Bn] ; a m o r p h i s m

B n] ~ [B~ . . . . , B~]

augmentation

here

augmentation

(2.18)

way

..., gn ] where

product

-

~a~.

÷

× ~

A whose

80

for

I but with functor

of

at 0. a

at 1 E ~.

o; the i s o m o r p h i s m

the i s o m o r p h i s m

All,1 . . . . , 1] to A and f[1,1 . . . . , l] to f.

functor

of

JoA ÷ A

AoJ ÷ A sends

-

we h a v e e x h l b l t e d

Thus

rather monoldal means

symmetric;

sending fails

2-category

to p r e s e r v e

2.4

~t/P

- with

Note that, colimlts

in (2.8)

Observe

"tensor product"

(B1,

we have

to the

a right

formal

o.

the f u n c t o r

has no right

adJoint

adJolnt;

Ao-

yet as

{B,-}.

of {B,C}.

$: n ÷ m induces

here

- or

It is by no

shows,

definition

..., B m ) ~ ( B $ 1

We are c o n c e r n e d

category

AoB = A, the i s o m o r p h i s m

as thls e x a m p l e

that any f u n c t i o n

B$:B m ÷ B n, n a m e l y

as a m o n o i d a l

and t h e r e f o r e

-oB has

We now p r o c e e d

and B id = id.

-

indeed when A E ~

A[ ] to A.

we p r o m i s e d

81

, ..., Bsn);

a functor B $~ = B~B ~

we have

only w i t h the case where m = n

and $ is a p e r m u t a t i o n .

For T,S: transformation formation

B n ÷ C and for a p e r m u t a t i o n

f: T ÷ S of g r a p h

f: TB ~ ~ S, w h i c h

~ means

~ of n, a n a t u r a l

a classical

can be p i c t u r e d

natural

trans-

as a 2-cell

Bn

(2.22) Bn

C S

Equivalently,

(2.23) natural

f is g i v e n by c o m p o n e n t s

f(Bl,

.... Bn):

in the u s u a l

sense

g: S ÷ R is a n a t u r a l composite

gf:

T(B~I , ..., B~n) in each B i.

transformation

T ÷ R of g r a p h

~ fB n

component

(gf)(Bl,

Bn),

If also R: B n ÷ C and if

of graph q, we d e f i n e

q~ to be the c l a s s i c a l

TB n~ = TB~B ~

its

÷ S(B 1 . . . . .

composite

SBq.n====~R; g

..., B n) is t h e r e f o r e

the

the

composite

-

.... ,Bn6 n)

(2.24)T(Bn61

82

-

~ S(Bnl,...,Bnn)

f(Bnl,..-,B~n)

Clearly

this composition

natural

transformation

~ R ( B I .... ,Bn). g(B I, .... B n)

is associative,

and there

I: T + T with identity

is an identity

graph and identity

components.

Given now B,C e ~g~/~ we define is an n E ~ together

with a functor

diagram

the category

T: B n ÷ C making

{B,C}.

An object

commutative

the

T Bn

~- C

(2.25)

v rl

where Yn is as defined

in §2.2.

for n $ m.

(n,T) ÷ (n,S) consists

A morphism

n together

with a natural

that the following (represented

transformation

diagram

by 2-cells)

There are no morphisms

of classical

(n,T)

÷ (m,S)

of a permutation

f: T ÷ S of graph natural

6 of

6, such

transformations

commutes: gn

S

(2.26)

fn

pn

r

v

pn =

I"

y

~ n

Composition described

in

{B,C}

is the composition

in the last paragraph;

when it holds

for the factors.

(2.26) Finally

over ~ by giving it the augmentation

of natural

transformations

for the composite {B,C} is made

r(n,T)

is automatic

into a category

= n, F(6,f)

= 6.

We shall

-

abbreviate

usually

(2.26)

(n,T)

to T and

83

-

(~,f)

to f.

Using the definitions

(2.11)

- (2.13) we may write

in terms of components;

(2.25)

reduces

assertions

(2.26)

(2.27)

P(T(B 1,

..., B n ) )

= PT(rB 1 . . . . .

rBn),

(2.28)

r ( T ( g 1,

..., g n ) )

= rT(rgl,

....

rgn),

(2.29)

r(f(B I,

...,

Bn)) = r f ( r B 1 ,

...,

rBn).

filled,

so that

the functors natural

and trivially

ful-

{B,C} is Just as given in §2.1; It has as objects

T: B n ÷ C of all types n, and as morphisms

{ , } becomes functors

to the third:

these are automatically

transformations

and

to the first two

below and

When B, C E ~

(2.25)

f: T ÷ S of all graphs a functor

(~/~)op

all

a l l the

~.

~ (~!~/~) ÷ ~ a ~ / ~

U: ~ ÷ B and V: C ÷ E over ~, we define

{U,V}:

as the functor over ~ sending T to V T U n and sending

when,

for

{B,C} ÷ {~,E}

f to the 2-cell

Un Dn

~ Bn

(2.30)

Un

~

S

V

in other words,

(2.31) It becomes

({U,V}f)(D I, a 2-functor

if,

B: V ~ V over ~, we define formation whose

T-component

..., D n) = Vf(UDI, for natural {~,8}:

transformations

Dn ~

is the horizontal

composite

V

Bn T ~ C ~ Un

~: U ~ ~ and

{U,V} ~ {~,V} as the natural

Un (2,32)

..., UDn).

E.

trans-

-

Note

that

(2.7)

"op" to m e a n

is the

"reverse

correct 1-cells

official

of

{B,C}

definition

from

happen

that n a t u r a l

have

the same

components.

sense

for all

in the

{I,C}. empty

notation

(n,T)

to T and

For example,

(2.22);

to a b b r e v i a t e (~,f)

It:

B = I both they count

the o b j e c t s

in d e g e n e r a t e

the

cases,

graphs

it

~ and n

In + I n is the i d e n t i t y (~,f)

and

(~,f)

as d i f f e r e n t {0,0)

c o u l d make

morphisms

= ~ where

of

0 is the

category.

We now state closed

2-category;

we leave

Theorem

the m a i n result,

asserting

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

that ~ / P

is a

verification

which

to the reader:

2

There

is a 2 - n a t u r a l

n: ~ / ~

(2.33)

~ i v e n as follows. A ÷ {B,C}

(AoB,C)

isomorphism

of c a t e g o r i e s

= ~g~/~(A,{B,C})

For a f u n c t o r

over P given on o b j e c t s

U: AoB ÷ C over ~, HU is the

((KU)A)(B 1 .....

(2.35)

( ( H U ) A ) ( g I . . . . , gn ) - - U ( A [ g I . . . . , gn ] )

and on m o r p h i s m s

(2.36)

is the n a t u r a l

B n) - - U ( A [ B 1 . . . . , Bn])

by

((EU)f)(B 1 .....

For a n a t u r a l

functor

by

(2.34)

transformation transformation

(HU--)A is the n a t u r a l (B1,

if we take

to f; but

of d i f f e r e n t

{I,I} = ~, and also

In p a r t i c u l a r

2-functor

from

when,

transformations

~, so that w h e n

situation

for this

we have a g r e e d

must be r e m e m b e r e d

might

functor

-

but not 2-cells".

One w o r d of warning: and m o r p h l s m s

84

B n) = U ( f [ B 1 . . . . .

a: U ~ ~ over ~, ~e: ~U ~ ~ over ~ w h o s e

transformation

..., B n ) - C o m p o n e n t

Bn]).

g i v e n by

with

component

identity

(Ha)(A):

graph

(~U)A ÷

and w i t h

-

Bn]):

85

-

(2.37)

m(A[B 1 . . . . .

2.5

From this main result we now extract the generalization

the Godement calculus.

U(A[B I, ..., Bn]) ~ ~(A[B I, ..., Bn]).

The objects and the morphisms of {B,C} replace

the functors and the natural transformations and composition in {B,C} replaces transformations.

of

of the Godement calculus,

"vertical" composition

of natural

By iterating the evaluation

(2.38)

E: { B , C } o B

÷ C

corresponding to the adJunction

{B,C}o

{A,B}oA

(2.33), we get

m {B,C}oB

~

C,

io~ and hence by adJunctlon a functor over (2.39)

~: {B,C} o {A,B} ÷ {A,C].

From the isomorphism J o A = A we also get the functor over (2.40)

n:

J ÷ {A,A}.

These functors ~ and ~ satisfy the associative and identity laws by the general theory of closed categories

[2]

(where they are called M

and J ). of course just sends the unique object i of J to the functor 1 A. morphisms

If we write the images under w of objects and of

as

(2.41)

~(T[SI,

..., Sn]) = T(S I, .... Sn),

(2.42)

"(f[gl'

"''' gn ]) = f(gl'

.... gn )'

then (2.41) is the operation of substitution composition

for functors,

of functors in the ordinary Godement calculus,

is the corresponding generalization

of "horizontal"

generalizing and (2.42)

composition of

-

natural

transformations.

86

-

In detail,

the right side of (2.41)

is the

functor (2.43)

where

Am =

Aml

x

x A

...

mn

~ S I x ... x S n

B x

x B

...

- ~ I b - C

T

FT = n and FS i = ml, with m = m I + ... + mn; and the right side

of (2.42)

is the classical

horizontal

composite

SIA nl x . . . x S A qn (2.44)

A

mI

m x ... x A n

As in the c l a s s i c a l be e x p r e s s e d

B

case,

gives the c o m m u t a t i v e

T(S~I,

....

.

.

x

B

composite

cases when either

~ to (2.4)

(2.42)

can

f or else the

(after r e p l a c i n g

A,B by

diagram f(S I,

(2.45) T(g~l . . . . , g~n )

.

T'

for a p p l y i n g

T(S~I . . . . .

x

S{ x...xS' n the general h o r i z o n t a l

in terms of the special

gi are identities; T,S)

~

TB ~

.... S n) ~--T'(S l,

S~n)

~

..., S n)

T'(gl . . . . , gn )

S~n)

~

T'(S 1 .

. . . .

Sn).

f(s{, ..., s n) In terms of elements,

T(SI,

..., S n) is the functor A m ÷ C

given by

(2.46)

T(S 1 ..... Sn)(AI,... , A m ) = T(SI(A 1 . . . . .

and by a similar transformation

(2.47)

formula

of graph n(nl,

T(gl, . ., .gn)(Al,. . .

and f(Sl,

for morphlsms;

T(gl,

Aml) ..... Sn(...,Am))

..., gn ) is the natural

..., On) with components

, A m ) = T(gI(AI,...,

..., S n) is the natural

transformation

Aml), .... gn(...,Am)); of graph

~ ( m l , . . . , m n)

with components

(2.48)

f(Sl,... , Sn)(A 1 .... ,A m ) = f(Sl(Al, .... Aml) ..... Sn(...,Am)).

-

We

87

-

end this section with the observation that

~: {A,A}o{A,A} + {A,A} and n: I ÷ {A,A} make of the "endomorphism object"

{A,A} a o- monoid.

We shall argue in the following paper [5]

that to give an extra structure on A, of the kind contemplated in §I.I, but with the natural transformations

restricted to those of this paper,

is precisely to give a o- monoid K in ~ / ~ that in fact o- monolds are what coherence

and a monold-map

K ÷ {A,A};

is all about.

The terminal object i~ any monoidal category has a unique monoid structure;

hence ~ is a °- monold in ~ / ~

identified with the endomorphlsm monoid

{I,I}).

~: ~°~ + ~ is the functor sending n[m I . . . . . ~[nl,

..., n n] to ~(ql . . . . .

(in fact ~ can be Its multiplication

mnl to n(m I . . . . , mn) and

qn); its unit q: I + ~ sends 1 E I to

IEP. 3. 3.1

Thgmany-categor[

case

We now indicate the way in which the calculus of §2 must be

generalized to produce a calculus apt for the discussion of extra structures carried, no longer by one category,

but by a family of

categories. An example of such a structure

is a monoldal functor.

Here we

have categories A 1 and A2; functors ®l: A12 ÷ A1 and Il: AlO + A1, with natural transformations functors @ 2 : A 2 2

a I etc., making A 1 a monoidal

÷ A2 and I 2 : A 2 0

category;

÷ A 2 with natural transformations

a 2 etc., making A 2 a monoidal category; a functor ¢: A 1 + A 2, and natural transformations

¢ : CA @2 CB + ¢(A @l B) and ~o: i2 ÷ ¢i 1

satisfying appropriate axioms. has been discussed by Lewis which the identities

The corresponding coherence problem

([I0], in this volume);

the easier case in

Ii, 12 were lacking was discussed earlier by

Epstein [3]. A second example is that of two categories AI, A2, with a

-

structure

monoidal

with

appropriate

88

-

on A 1 given by functors

natural

isomorphisms,

and also

@: A 1 ~ A 2 ÷ A 2 w i t h n a t u r a l

isomorphisms

I~C ÷ C

axioms.

subject

p r o b l e m was identity

to s u i t a b l e

in effect

sidered

a different

variance

functors,

their proofs

problem, arising

apply

variance

The categories example

- Mac

(with I); these only

transformations in this

of c a t e g o r i e s

A (which is u s u a l l y

finite,

(Ak) may be d e n o t e d

by a single

A happier

of it as a c a t e g o r y

way

but

Ak,

to r e g a r d

A over A, t r e a t i n g

is p r o v i d e d

is of course coincide

not

a set of

in our first

k E A, for some i n d e x i n g

set

Such a family call it a

such a p o l y c a t e g o r y

is to think

the set A as a d i s c r e t e

if A: A ÷ A is the a u g m e n t a t i o n

following

discussion

there

of m i x e d

[8].

A, and we shall

category;

A is fixed,

con-

one outside

of |l],

could be a r b i t r a r y ) . letter

really

A 2 = A1 °p - but

although

are those

well

the

but w i t h m i x e d -

are e s s e n t i a l l y

volume

- A 1 and A 2 could p e r f e c t l y

pglycategor~.

(without

problem.

problem,

functors

and

coherence

authors

one c a t e g o r y

of such a s t r u c t u r e

- but a family

§2 of [3]

to the p r e s e n t

Lane p a p e r

carrier

÷ A@(B@C)

from the above by s e t t i n g

in that the

and the n a t u r a l

(A@B)@C

in

of a t h r e e - c a t e g o r y

context

by the Kelly

[ii] with

unchanged

An e x a m p l e our p r e s e n t

Donald

a functor

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

s o l v e d by E p s t e i n

I) and by Mac

®: AI2 ÷ A 1 and I: A10 ÷ A 1

then Ak is A-l(k).

being

a separate

In the

calculus

for

each A.

In the o n e - c a t e g o r y structure

on A was

a

In the m a n y - c a t e g o r y Akl (kl'

x Ak2 k2'

the right

functor

case

x . . .^Akn + A "''"

kn;

things

~)"

case

a functor

An ÷ A ,

it w i l l

and

involved

its

type

be of the

form

in the e x t r a

was

given

, and its type must be s p e c i f i e d

As in the o n e - c a t e g o r y

to c o n s i d e r

are

functors

case,

T: Akl

by

n E N.

by

it turns

x ... x A k

out that ÷ B

n

-

where

A,B are polycategories

form the objects morphisms mutation

89

-

(for the same A); these

of a generalized

are again natural

functor

transformations

satisfied•

f has as before

...• A~n)

naturality

÷ T'(AI•

condition

components

{A,B}.

f whose

of n; but there are no such morphisms

= U'• n = n', and k'~i = ki for i 6 n.

T(A~I•

category

functors

graph

~ is a per-

conditions

are

...• An):

..., A n ) in B , subject

in each Ai; this makes

The

T ÷ T' unless

When these

f(Al•

then

to the usual

sense because

Ai 6 Ak, i

and hence A~i 6 All. of types (kl' above

Then

and graphs;

"''•

{A•B} is a category

~, which

depends

kn; W) and morphisms

conditions

~' •

polycategories ~/~

replacing

Just as ~

Gg~/P;

the

= ki-

the category

over, ~

note that ~ / A

is a full subcategory

~ satisfying

~i

in §2 now carries

replacing

on A• has objects

those permutations

U = ~'• n = n'

Everything

over the category

with the category

of categories•

We proceed

of

and with

is a full subcategory

of Cat/P.

~/A

of ~ / Q ,

to the

details.

3.2

The set A is fixed.

and graphs sense

can be identified

of §2.

For the object

with the object write

(n[kl•

it in this way.

...•

The corresponding with the category (kl• kn]•

Since

~oA × A, using

~) of ~oA × A• and in future

A is discrete

In] , ~) ÷ (n'[l'l,

n = n' and

~ = B'; and then a morphism

(~[l~ . . . . .

l~],

the desired

domain we must have

(3.1)

~ of types o in the

"''' kn; ~) of ~ can be identified

(n[l I . . . . .

~), where

category

we

there are morphisms

...• l'n,] • ~') in ~oA × A only when is necessarily

~ is a permutation

of n.

of the form

For this to have

k'~i = ki for i e n;

so the morphisms

of ~oA × A are Just those of ~.

We shall normally

-

abbreviate

~[~ . . . .

the m o r p h i s m

(~[k~ . . . . .

k~],

90

~) of ~oA

-

k~]

,

of ~oA,

or the m o r p h i s m

x A, to ~, r e c a l l i n g

that

it satisfies

(3.1). The a u g m e n t a t i o n given by functors type

r':

F: A ÷ ~ of a category A ÷ ~oA and

of T and F"T the c o d o m a i n

element, becomes

~oA becomes

type

the category

~, and we r e - f i n d

the

F": A ÷ A. of T.

We call

When

~, A becomes

situation

over ~ is t h e r e f o r e

A has

F'T the domain Just

the unit

of §2 with

one

category

F = r' since

I, F" is

trivial.

In §2 a category category

~ is first

On the other hand,

of ~ / ~ ;

here

~/A

A: A + A with

constant

at 0[ ].

functor

The m u l t i p l i c a t i o n Po(~oA)

as n[Y1 . . . . . Y1 = ml[~l" (3.2)

= (~o~)oA

"'''

~ml|

+ ~oA.

..., n n ] ~ - ~ ( n l , of ~(nl,

permutation

3.3

For c a t e g o r i e s

in §2 as a full sub-

F" = A and

this

of ~ t / ~ F' is the

§2.2 induces functor,

Yi E ~oA;

..., mn)[kl,

On morphisms,

...,

a

on objects,

so if

km I

we also write

Un) ; happily a morphism

....

this

agrees

it as with

the m e a n i n g

of ~oA is i d e n t i f i e d

with

a

(3.1).

A,B over ~ we define

generalizing

the d e f i n i t i o n

in using

same

the

We write

a

over A, namely

subcategory

in

here

then

..., n n) w h e n

satisfying

a full

yn ) where

n(y I . . . . , yn ) ~ n(ml,

~[n l,

regarded

~o~ ÷ ~ d e s c r i b e d

etc.,

m = m I + ... + m n.

was

a category

F: A ÷ ~ where

yn]~-~n(Y1 .....

where

(2.15)

~

becomes

when we i d e n t i f y

functor

of all a category;

F: A ÷ ~ over ~ is in p a r t i c u l a r

F": A ÷ A. category

over

symbol

in §2.3;

o, since

there

a category is no d a n g e r

A is s u p p o s e d

AoB over ~, of c o n f u s i o n

to be known.

-

An

object

91

of AoB is A[BI,

-

..., Bn] where A • A, B i e B, and

where

(3.3)

F'A = n[F"B I,

thus the c o d o m a l n A morphlsm

(3.4)

..., r"Bn.; ]

types of the B i are to match the domain type of A.

is

f[gl'

"''' gn I : AtB 1 . . . . , B n] ~ A'[B~,

' f: A ÷ A' with Ff = ~, and where gi: B~-I i ÷ Bi-

where

there are no such morphisms f: A ÷ A'; and m o r e o v e r defined as in (2.17),

unless

F"A = F"A'

~ is r e s t r i c t e d

AoB is a category;

(3.6)

r,(fIg I, .... gn I ) -- rf(rg I . . . . .

(3.7)

F"(A[BI,

1, oI l,

...,

As in §2.3, identity

3.4

(nIkl,

F'k = l [ k ]

rgn),

(2.21), m a k i n g

0,

...,

o)[

o a

] = 0[ ].

associative

with a two-sided

case J is the category

A, with a u g m e n t a t i o n

and F"k = k.

B,C over ~ we define the category

and similarly

the f o l l o w i n g

F'(T(BI,

..., r'Bn),

as in (2.20) and

o is c o h e r e n t l y

..., kn], W) together

satisfying

(3.8)

0[ l) = n(0,

For c a t e g o r i e s

Set Bk = r"-ik,

it over ~ we set

x Cat/~ ÷ ~_~__~/_~. If B e ~_~__~/A, AoB • C~__~/A; for

J; in the present

F over ~ where

With c o m p o s i t i o n

..., BhI) = F"A.

ToS and a°8 are defined o: C ~ / ~

Of course

then there are no

to augment

r'(A[B I, ..., Bn]) = n ( r ' B I,

2-functor

for

as in (3.1).

(3.5)

n(O[

.... B~]

for C k.

An object

with a functor

analogues

of {B,C} is a type

T: Bkl × ... × Bkn ÷ C

of (2.27) and

..., Bn)) = n(F'B1,

{B,C} over ~.

.... F'B n)

(2.28):

-

r'(T(g I . . . . .

(3.9)

A morphism

92

-

gn )) = n(rg I, ..., rgn).

{B,C} exists only if there is a m o r p h i s m

T ÷ T' in

6: rT ÷ rT'; that is, only if n = n', U = U', and In this case a m o r p h i s m transformation T(B~I,

(3.1) is satisfied.

T ÷ T' is such an ~ together with a natural

f: T + T' of graph

..., B~n) ÷ T'(BI,

~ whose

components

..., B n) satisfy

f(Bl,

the following

..., Bn):

analogue

of

(2.29):

(3.10) where

F'(f(B 1 . . . . , Bn)) F'B i = mi[~ I,

are a u t o m a t i c a l l y augmented

= ~(ml,

..., ~ki]

for some ~j.

satisfied when

over ~ by setting

..., mn) Note that

B,C • ~ / A .

FT = (n[~l,

(3.8) - (3.10)

Finally

{B,C} is

..., ~n ] , ~) and rf = ~.

{ , } is made into a 2-functor by the a p p r o p r i a t e tions of (2.31)

generaliza-

and (2.32), which we leave the r e a d e r to formulate

explicitly. We then have: Theorem

3

T h e o r e m 2 continues

to hold when ~ is r e p l a c e d

throughout

b y_~. Everything modulo sends

some trivial n o t a t i o n a l k 6 J to 1Ak ; (2.43)

(2.41)

- (2.48)

terminal with

object

{A,A}.

(nil I . . . . . (n(ml' ~(nl, 5.5

in §2.5 carries

stand.

and

Now

changes.

QoQ

but otherwise

{A,A} is a o- m o n o i d in ~ / ~ ; is itself a o- monold,

the identifiable

+ Q sends

An], w) [(ml[9 l,

...],

kl ), ...,

.... ran) [91 . . . . ' 9k ]' ~) and ~[nl, .... nn) ; the map J ÷ ~ sends Finally

situation,

The functor n: J ÷ {A,A} now

(2.44) need re-wrltlng,

Q of this category

This map

over to the present

we consider

(toni... , 9k ], An)] .... n n] to

~ to (l[k],

the r e l a t i o n

~).

b e t w e e n the calculi

for

to

-

various

A; to indicate

93

-

the d e p e n d e n c e

on A we write

~A'

°A'

{ ' }A'

a functor

~oA

× A ÷

JA for ~, o, { , }, J . Any map k: A ÷ M of index ~oM × M, that

is,

This has a right

sets

~A ~ ~M; and hence adjoint

and r are d e f i n e d

¢ sending

induces

a 2=functor

~: ~ / ~ A

÷ ~g~/~M"

A: A ÷ ~M to r; ¢A ÷ ~A' where

CA

by the p u l l b a c k

##A

~ A

(3.11)

The functor

functor

~ is a n t l - m o n o i d a l ;

(~g~/~A) °p + ( G ~ / ~ M ) ° P ;

~JA ÷ JM p r o v i d i n g is the i n c l u s i o n coretraction, of ~ / ~ M '

its extra

and p r e s e r v e s The functor

and CJM ÷ JA" Since

4.

of the first

More

general

that

it takes

It suffices

to many

categories

general

natural

+ ~AoM~B

W h e n k: A + M

is w h e n

~g~/~A

and

it is a

as a full s u b c a t e g o r y

J). with evident

maps

¢ preserves

a °M- m o n o i d

on A to an action

¢AoACB

÷ ¢(AoMB)

J (but not

to a °A- monoid,

of the s e c o n d

o). and an

on CA.

transformations

to c o n s i d e r presenting

the o n e - c a t e g o r y

no problems.

transformations

so a functor

it is a m o n o i d a l

~(AoAB)

that

~ embeds

o (but not

is,

are evident.

subset,

¢ is monoidal,

natural

4.1

covariant;

structure

When k is an i n c l u s i o n

¢ is m o n o i d a l ,

action

the maps

of a n o n - e m p t y

it is clear

that

in w h i c h

case,

the e x t e n s i o n

We c o n s i d e r the

functors

first

are

is of the form T: A n ÷ B and its

more

still type

is given

sets w i t h

~ as its

by n E ~. Write

~ for the skeletal

category

of finite

-

of o b j e c t s

set

is the

and w i t h

subcategory

94

functions

of ~ in w h i c h

-

n ÷ m as m o r p h i s m s ;

only

the

isomorphisms

our c a t e g o r y of ~ are

retained.

For natural

¢: n ÷ m in ~ and for T: A n ÷ B, S: A m ÷ B, we define

transformation

transformation f(Al,

f: TA ¢ ~ S, g e n e r a l i z i n g

..., Am):

T(A¢I , ..., Acn)

extends

fairly

functor

category

morphism

A|BI,

immediately {B,C}

gi:

Such n a t u r a l the e x t r a

coproducts. basic

really

to this

~ A'EB~,

structure

Everything

above

we get a new g e n e r a l i z e d

left a d j o i n t

..., B~J

AoB in w h i c h a

is fEgl,

..., gn j where

functors

transformations

is the c o p r o d u c t equational

axioms

w h e n we are

in a d m i t t i n ~

finite

are + : A 2 ÷ A and I: A 0 ÷ A; the

are p: A ÷ A+B,

graphs

and

occur, for instance,

on A c o n s i s t i n g

2 ÷ I, and the unique

the p u r e l y

it has c o m p o n e n t s

..., Am).

case:

transformations

i: I ÷ A, of r e s p e c t i v e function

(2.22);

natural

Bi ÷ B'¢i"

The basic

natural

¢ to be a c l a s s i c a l

÷ S(AI,

over ~, w i t h

..., Bn|

rf = ¢ and where

describing

f: T ÷ S of g r a p h

a

rlT:

1 ÷ 2,

function

I the

~7:1

d: A+A ÷ A,

÷ 2, the u n i q u e

0 ÷ i.

initial

saying

q: B ÷ A+B,

To ensure

object,

we have

that + to impose

that each of the f o l l o w i n g

is the

identity:

A~A+A p

A+B

~

c p+q

given

case;

~A+A

~

q

of the

in p a r t i c u l a r

category

following the

A

d

I

~

I.

iI

paper

~51

also e x t e n d

at once to

free c a t e g o r y - w i t h - f i n i t e - c o p ~ o d u c t s

on a

B is ~oB.

If we want discussing

A

(A+B)+(A+B)~A+B d

The ideas this

A

d

finite

the kind products,

of n a t u r a l we must

transformation

take

those w i t h

needed

graph

in

in ~op;

if

-

T:

95

A n ÷ B and S: A m ÷ B, a n a t u r a l

9: m ÷ n is a c l a s s i c a l there

is a f u n c t o r

morphism

natural

category

-

transformation

transformation

{B,C}

in the l a t t e r now b e i n g

f: T ~ SA ~.

over ~op, w i t h fEgl,

f: T + S of graph Once a g a i n

left a d J o i n t

..., gm j : AtB I,

AoB,

..., BnJ

a

~

!

A'IB~,

.... B~J

following

where

Ff = @ and gi:

paper extend

If we want s h o u l d n e e d more the c o m p o s i t e s to talk about

to this

general

natural

a distributive

The graph,

telling which

the ideas

and c o p r o d u c t s

transformations,

A+A ÷ A ÷ A×A and A×B ÷ A ÷ A+C.

d: A ® ( B e C )

of the

law in a category:

÷ (A@B)@(A@C) arguments

and d-l:

with

together,

components

Similarly

are to be set equal,

llke

if we want

we have n a t u r a l

(A@B)$(A@C)

we

trans-

÷ A@(B$C). now has

to

of two f u n c t i o n s

(4.1)

n

~

k -~l-----m,

and the c o m p o n e n t s

are of the

S(A~I , ..., A~m).

The c a t e g o r y

of o b j e c t s , different

and a m o r p h i s m

diagrams

an a u t o m o r p h i s m of ~ = (4.1) forming

Again

case.

to talk about p r o d u c t s

formations

consist

B~i ~ Bi"

counting

form f(Al,

..., Ak):

~ of types

and graphs

~: n ÷ m is a d i a g r a m as the same g r a p h

(= p e r m u t a t i o n )

of k.

T(A¢I , ..., A¢n)

and n: m ÷ p ÷ t to be the graph

has ~ as its set

in ~ llke

(4.1),

if they d i f f e r

We define

only by

the c o m p o s i t e

~:

÷

in

n ~ q ÷ t got by

the d i a g r a m

t

m

~P

(4.2)

t~.k--...........~q

in w h i c h natural

the square

is a pushout.

transformation

transformation

For T: A n + B and S: A m ÷ B a

f: T ÷ S of graph

f: TA ¢ ~ SA ~, thus:

~ is now a c l a s s i c a l

natural

-

96

-

A n

If g: S ÷ R has graph ~, the composite defined as the composite

gf: T ÷ R of graph n~ is

2-cell An

Ak/ / Aq

v

AP ~

This certainly produces evident continuity However

At /

a functor category

in C ensures

{B,C} over ~.

that a left adJoint

coherence

problems.

a good context

The c o m p l i c a t i o n

is surely

M o r e o v e r its

AoB will exist.

the general m o r p h l s m of AoS is quite complicated,

not yet clear how far this provides

things:

B

and it is

for d i s c u s s i n g

in the nature of

the free c a t e g o r y - w i t h - f i n i t e - p r o d u c t s - a n d - f i n i t e - c o p r o d u c t s

on one generator

Moreover is a natural a natural

is already pretty

complicated.

there is a new aspect

transformation

transformation

of graph

h

~

0¢

respectable

problems;

(4.1),

if f: T ÷ S

and if 8: k + h, there is

of graph

n ---.---.~

namely the classical

that arises here:

composite

m,

e¢ fA e.

This operation

take the "distributive

does occur in

law" situation above,

-

and

write

are

~ and n, then n~ = 1 but

identity with

natural

I:

of s e e k i n g

a calculus

considerations

not h a v l n g

for:

equalizers present

context

categories, either

in the s i m p l e r

that

mention

cases,

doubt

on the

of f u n c t o r s

Ae;

where b o t h

are made

it is p r e s u m a b l y

one f u r t h e r

much e a s i e r by

worth

looking

or o t h e r

we r e p l a c e

and ¢,9 by functors.

¢ or ~ is i, e v e r y t h i n g

generalization

of a c a t e g o r y

products

provided

casts

SA e,

for

case.

to d e m a n d

or c o u n t a b l e

of d and d -1

that of some

This

w i t h no e x p l i c i t

T, that

observe

if we want

of S but i n s t e a d ÷ (A@B)e(A@C).

in the g e n e r a l

Finally,

S ÷ T; if the graphs

and free s t r u c t u r e s

TA e a l o n g s i d e

such a c a l c u l u s

called

right

-

~n ~ I; and dd -I is i n d e e d not the

(A®B)e(A®C)

it is so o b v i o u s l y

coherence

d-l:

transformation

components

wisdom yet

d as d: T ÷ S, w i t h

97

that

limits,

n, m, k in

still

it admit we can do so in the

(4.1)

In the cases

is e v i d e n t l y

like

by small ~ or ~ o p

where

works m u c h as in the p r e s e n t

paper.

4.2

We n o w come

one-category

to the m l x e d - v a r l a n c e

calculus)

a functor

case.

the variances.

AxA°PxA°PxAxA, number

so that

of a r g u m e n t s

considered category

plexity, above.

(5 in the above

w i t h objects

The n a t u r a l

Iwl+Ivl

case). of ~o2,

of +'s and -'s

we might n = Ivl

write

A v for

for the total

In fact a type where

÷ B,

can be

2 is the d i s c r e t e

{+,-}.

for e x a m p l e

corresponding

we take this e a s i e r type of l e n g t h

5[+ -- ++]

transformations

corresponding The ones

With v as above

T: A v ÷ B;and write

as an object

(in the

is of the form T: A x A ° P × A ° P x A × A

and its type is g i v e n by a s t r i n g v = (+ -- ++) indicating

Here

case

first.

may be of v a r i o u s to the c a t e g o r i e s

to P are those For types

got by w r i t i n g

first

levels

of com-

~, ~, ~op,

considered

~ and v, write the s t r i n g

=G

in [i], -~+v

and

for the

~ with all the

-

signs

c h a n g e d a n d then the s t r i n g

biJection columns

between

9B

-

v.

A graph

+'s and the -'s in -~ +v.

it can be w r i t t e n

as an a c t u a l

+

~: ~ ÷ v is t h e n a

If we w r i t e

geometrical

~ and v as

"graph" as in

f i

+

• +

(4.3)

Composition where

of graphs

they meet;

is d e f i n e d by J o i n i n g t h e m at the m i d d l e

the c o m p o s i t e

of 6: ~ ÷ v as in

(4.3)

in

is the f o l l o w i n g

graph

~6 T

+

type

and q: v ÷ T as

-

whose

"linkages"

however,

+

~

~

-

.

~

+

the graphs

the composite

was still defined

closed loops

upon composition,

containing

+

the original

namely loops.

closed

as in

to be incompatible;

as the part of (4.4) We then get a category

A different

loops;

possibility

then we may get more

+

C÷3o

seem to get a category,

graphs without

loops,

as in

'

÷J In this way we would

there,

that we shall call ~ .

graphs

The problem,

+

the closed

is to allow

I

~

~ and ~ were said in [i]

after discarding and graphs

out to taste.

we may get closed

In such cases

of types

-

may now be straightened

is that upon composing

remaining

99

closed

loops

which we shall call P

can then be called slmple

graphs. For a simple formation defined

graph

f: T ÷ S of graph

~: W ÷ ~, the concept ~, where

in [I]; it was moreover

of graph n to get a composite when n and ~ were

to extend;

trans-

T: A ~ ÷ B and S: A ~ ÷ B, was

shown how to compose

natural

transformation

f with g: S ÷ R gf of graph n~

compatible.

As long as the graphs seems

of a natural

are simple,

if A is a category

f: A ÷ A' have graph

over ~

~: ~ ÷ ~ given by

the calculus

of this paper

and B is a category,

let

-

+

i00

-

=

+

+) +)

C+_

Then for B I, ..., BI0 ~ B, a typical morphism of AoB is f[gl'

..., g5 ] : A[B 1 . . . . , B6] -* A'[BT,

..., BI0] where the maps gi in

B go like this: gl ~-

B1

~

B7 B8

g2 B3 g3

gsC

B9

B4 BI0

j That is, g: B i ÷ Bj Joins two arguments linked by ~, and goes from the one which is contravariant in -~ +~ to the one that is covarlant in -~ +v.

It is moreover

clear how to compose such morphisms in AoB as

long as the graphs are compatible. problems:

Moreover substitution presents no

our earlier operation ~(nl,

..., n n) on graphs carries over

well; if n -

+

.....

+

+

+

c; then ~(q , ~) is the result of replacing the linkages in ~ (under a microscope,

as it were) by the graphs q and ~:

-

101

-

__ °'_.

,L'.

+ D

i'+'~•

The problem is that there is Just no {B,C} if we stick to simple graphs, for we cannot compose natural transformations of incompatible graphs; and correlatively that we cannot compose in AoB if the graphs are incompatible.

This suggests that we should allow

,

the more general graphs of ~ .

Then when we try to compose in AoB,

we get something fairly sensible.

If f and k have graphs as in (4.4),

a composite in AoB looks something like

A f~ [B I]

A'[B 2, B 3, B 4, B5, B 6]

A"[ B71 This suggests that a morphism in AoB of the composite graph (4.4) should have a map

hlgl: B 1 ÷ B 7 corresponding to the simple part of

the graph, and something llke g2 B3 (4.6)

~ B4

h31

lh2

B6 ~

B5

g3 corresponding to the closed loop.

There is no way of knowing where

-

to

start

class

the loop

(4.6),

of e n d o m o r p h i s m s

by the r e q u i r e m e n t equivalent.

Then

following.

so define

-

a cycle

in B as an e q u i v a l e n c e

in B, the e q u i v a l e n c e

that, (4.6)

That we are

102

relation

being

generated

for g: B ÷ C and h: C + B, gh and hg are defines

a unique

on the right

track

cycle

in B.

is f u r t h e r

Let f: T ÷ S and g: S ÷ R be n a t u r a l

suggested

by the

transformations

with

i

graphs

as in

(4.4), w h e r e

T:

B ~ ÷ C etc.

Given

B in B and h: C ÷ C in

B, the c o m p o s i t e s

s(~,c,~,c,c)

//~

S(l,h,l,l,l)

~ S(B,C,C,C,C)

S(B,C,~,C,C)S(1,1,h,l,1) S(B,C,C,C,C)

~(B)

R(B)

are all e q u a l general more

define

S(B,C,C,C,C

S(B,C,C,C,C)

S (l,l,l,l,h) ~" S ( B , C , C , C , C

and d e p e n d

result

general

S(B,C, C, C,C)

only on the e q u i v a l e n c e

of this k i n d is e a s i l y

diagrams

a natural

w i t h maps

[h|

of h; the

p r o v e d by c o n s i d e r i n g

C ÷ D and D + C.

transformation

+

class

slightly

Thus we are

led to

T ÷ R of graph

--

+

O as c o n s i s t i n g

of c o m p o n e n t s

with

¢ a cycle

of B.

This

same

t h i n g as a n a t u r a l

f(B,

¢): T(B)

doesn't

÷ R(B),

natural

s e e m a bad idea,

transformation

B(C,C)

in B and

since

÷ C(TB,RB)

it is the in the sense

of [11. Moreover category

such things

to be a s y m m e t r i c

occur

monoidal

in nature. category

Define

a compact

closed

(A, @, I) t o g e t h e r

with a

U

functor

: A °p ÷ A and n a t u r a l

transformations

,

hA:

A @A ÷ I m a k i n g

commutative

the d i a g r a m s

gA:

I ÷ A@A

,

-

g®l .

A ~

103

-

,

,

A @ A

@ A

l@g

,

A

L

A

, ® A @ A

A (This is d e l i b e r a t e l y A , adJoint

A

an iequatlonal

to A in the d e g e n e r a t e

functoriallty

and the n a t u r a l l t y

closed,

with

[A,B]

= A ®B;

exactly

when

the c a n o n i c a l

definition; 2-category

the mere e x i s t e n c e

A, i m p l i e s

of g and h.)

conversely

map k: A®[A,I]

its

Such a c a t e g o r y

a closed

category

÷ [A,A@I]

of

is

is compact

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

I

whereupon [A,A]

A

= [A,I],

= [A,A®I]

vector

spaces

h is the e v a l u a t i o n ,

f o l l o w e d by k -1.

over a field

representations

over

and g is I ÷

Examples

I, or more

I of a group

G.

are the f i n i t e - d l m e n s l o n a l

generally

the f i n l t e - d i m e n s i o n a l

In such a c a t e g o r y

the

composite (4.7)

I

r-A

@ A

~ I

cg is m u l t i p l i c a t i o n incompatibles

(4.8)

by d i m A; we h o w e v e r

~

A

® A

, ~-- B u ®v

cg

which

is m u l t i p l i c a t i o n

cycle

[uv]

by trace

like

defined

we can o r d e r

graph, among

@ B

the c o m p o s i t e

of the

~ I, h

(uv) = trace

but

I don't

the c l o s e d

in the

themselves

~

.

over ~ .

(vu),

depending

loops

on the

to the right

The t r o u b l e

real sense

in a g r a p h

that,

factors

close

It's all very w e l l

see how to make

It then t r a n s p i r e s we must

to be t a n t a l i z i n g l y

on c a t e g o r i e s

k n o w how to define

(4.5),

other.

take

= [vu].

We seem, t h e r e f o r e ,

really

would

cg a n d h to be

I

calculus,

h

is that

to draw p i c t u r e s of the above

so as to tell

ordered

but also w i t h r e s p e c t

the

closed

unless

one f r o m the

to get an o r d e r on the

have

I don't

composite

loops not only

to the n o n - c l o s e d

linkages.

I

-

104

-

yet found a natural way of doing this that permits any decent

haven't

operation

of s u b s t i t u t i o n

of graphs in graphs.

While the search for such a calculus an e m a s c u l a t e d composable

form,

defining Ao8 for categories

occur often;

incompatibilities later paper [6]

it is proved in [7],

do not occur in the problems

in this volume

only contravarlant

functors

in the structure ones.

graphs.

At least

[8], and [i0]

there studied,

that

and in a

explicitly,

arise by p o s i t i n g

The e m a s c u l a t e d

does serve as a stop-gap measure

free closed categories

such that

I show that this is always so when the

for some of the covariant

therefore,

I can only use

over ~

maps in A or in B never have incompatible

such categories

adJoints

continues,

for d e s c r i b i n g

calculus, things

like

and we use it in the following

paper [ 5] •

4.3

If a good ~

S °p*

At the moment

is found,

all will surely be well with ~

I can define an soP*:

•

a graph is not a b i J e c t i o n

~O

of the +'s with the -'s in -~ +v, but a function -'s. sense;

Graphs

closed category,

[14])

assertion

from the +'s to the

can be composed if they are compatible

I think they are always

cartesian ([13],

and

compatible

that,

in the theory of a

and that the "coherence

for these is probably

in a suitable

equivalent

result"

to the much simpler

if A is the free such on one generator,

F: A ÷ S °p* is faithful.

However

of Szabo

I have yet to verify

then this,

and I

=O

know still less about more c o m p l i c a t e d ing paper shows these ideas will supply my lack of wit.

cases like ~

to be worthwhile,

perhaps

If the followsome colleague

-

105

-

REFERENCES

[1]

S. Eilenberg and G.M. Kelly, A generalization of the functorial calculus, J. Algebra 3(1966), 366-375.

[2]

S. Eilenberg and G.M. Kelly, Closed Categories, in: Proc. Conf. on Categorlcal ' Algebrap La Jolla~ 1965 (SprlngerkVerlag, 1966), 421-562.

[ 3]

D.B.A. Epstein, Functors between tensored categories, Invent. Math. 1(1966), 221-228.

[4]

J.W. Gray, The categorical comprehension scheme, Lecture Notes in Mathematics 99(1969), 242-312.

[5]

G.M. Kelly, An abstract approach to coherence. (in this volume).

[6]

G.M. Kelly, A cut-eliminatlon theorem. (in this volume).

[7]

G.M. Kelly and S, Mac Lane, Coherence in closed categories, J. Pure and Applied Algebra 1(1971), 97-140.

[8]

G.M. Kelly and S. Mac Lane, Closed coherence for a natural transformation. (in this volume).

[9]

F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Scl. U.S.A. 50(1963), 869-872.

[10]

G. Lewis, Coherence for a closed functor. (in this volume).

[ 11]

J.L. Mac Donald, Coherence of adJoints, assoclativitles, and identities, Arch. Math. 19(1968), 398-401.

[ 12]

S. Mac Lane, Natural assoclatlvlty and commutativlty, Rice University Studies 49(1963), 28-46.

[ 13]

M.E. Szabo, Proof-theoretlcal investigations in categorlcal algebra (Ph.D. Thesis, McGill Univ., 1970).

[14]

M.E. Szabo, A categorical equivalence of proofs (to appear).

AN A B S T R A C T APPROACH T 0_COHERENCE.

*

G.M. Kelly

The University

of New South Wales,

Kensington

2033, Australia

Received May 22, 1972

i.

Introduction

I.i

We assume

familiarity with the p r e c e d i n g paper [5]

volume,

and we refer the reader again to §l.1 of that paper.

Suppose we have a category an extra structure basic functors,

of the kind considered

there.

A provided with

So we are given

of the form A n ÷ A in the one-category

case, and of the appropriate mlxed-varlance

(or polycategory)

case.

more general

in this

fully-covarlant

form in the m a n y - c a t e g o r y

We pass at once to the w i d e r class of allowable

functors

T: A n ~ A obtained from these and the identity

functor by

iterated

substitution.

transformations

T ÷ S between [ 5] d e s c r i b i n g

We are then given basic natural

certain pairs the arguments

wider class of allowable basic

of these,

to be set equal.

natural

and substitution.

domain and codomain,

certain

obtained

is not closed.

formally-different

from the

1 T and allowing unlimited

com-

If f, g: T ÷ S are two such with the same

it makes no sense to ask whether

they have the same graphs components

We now pass to the

transformations,

ones by adding the identities

position

each such having a graph as in

Ff = Fg; otherwise

f = g unless

the diagram of their

There is now given a set of axioms r e q u i r i n g pairs

f, g: T ÷ S of the same graph to

coincide.

* Preliminary research for this paper was supported by grants from the National Science Foundation and the Louis Block Fund of the University of Chicago.

- 107 1.2

w h i c h other f o r m a l l y - d i f f e r e n t transformations

[14],the

problem

finite

that all diagrams

coincide natural

when

Ff = Fg.

examples

all diagrams

of monoidal

that is, that f, g: T ÷ S always

categories

themselves

My student

commute

because

structures

therefore

in [7], Kelly

with proving

the

do have

a partial

functors

(there

$, ~, $o: A ÷ A'.

with the case of a monoidal

are again non-commuting

result

functor;

of extending connected

result,

the

by a

to deal

except

I, I' and so the corresponding

When the identities diagrams,

ones).

this had been done by

commute"

identities

of the

to a certain

called the proper

the problem

When

- Mac Lane

Here the first step was

[3], with an "all diagrams

$o: I' + $I was missing.

it is Just

examples.

if Ff = Fg and if T, S belong

G. Lewis attacked

A, A' lacked

however,

in the natural

to the case of two closed categories

functor

his

and similar

for closed categories

subset of the allowable

Epstein

u = v: P ÷ Q which would

as closed categories,

form: f, g: T ÷ S coincide

closed

there it was a question

commuting.

coherence

above result

by Mac Lane in

The problem was so posed precisely

not the case that all diagrams

contented

of the axioms.

considered

differently:

set of axioms

commute,

In such structures

studying

such as those

was posed rather

of findin~ a suitable imply

pairs f, g:T--->S of a l l o w a b l e natural

n e c e s s a r i l y coincide as a c o n s e q u e n c e

In the first examples,

that

is that of deciding

The c o h e r e n c e p r o b l e m for the given structure

are allowed

even in this purely

the diagram ~I

----I' ® $I

~

I $°®I

¢I @ I'

~ 185 °

¢I ® ¢I

for, there

covariant

case:

-

108

-

fails to commute

for the forgetful

that once again,

as in [7], the c o n s t a n ~ t u r n

which

destroy

of [7]

commutatlvity.)

could be given:

the form SR. sufficient

Here again,

functor Ab ÷ Set.

Lewis went

conditions

further,

in the first sentence

developed

like that

and gave necessary arbitrary.

of the coherence

of §1.2.

result

if rf = rg and if S is of

for f = g when T,S were

we could call a full solution

(Note

out to be the villains

a partial

f, g: T ÷ S coincide

However

now in the setting

monoidal

The results

problem,

of Lewis,

in this paper,

and

This is what as posed above

re-formulated

will be found in [12]

in

this volume. 1.3

This brings

the general fairly

coherence

loose.

natural

problem.

We spoke

transformations

of the generalized grams

us to the question

commute"

F was

honest

allowable

formall~

natural

{A,A}

model A.

recognized

where

allowable

their realizations

in {A,A}.

the fact that some theories

-®(-®-);

we shall allow

to be distinguished

[7],

functors

overlooked

functors

however,

Then

not writable

for a

in [14], was explicitly

(there

called shapes),

certain

iterates category

in our formal

from fortuitous

Q, T: A n ÷ A

of ~!i were taken to be the

as a strict monoid~l for these

to ~

of ~II is

coincide.

(We have so far ignored require

~Ii = ~!I(A)

f: P ÷ Q, g: T ÷ S could be

"allowable"gf

the objects

different

to coincide,

and the allowable

But the definition

transformations

is

that the restriction

allowable

This difficulty,

functors

said above

of

of [5]; then an "all dia-

model A, fortuitously

to form a freak

in [7],

Even in

{A,A}

faithful.

different

for this particular

general

category

result was the assertion

might,

in

functors

formulation

as if they formed a subcategory

for two formally

composed

Much that we have

of the allowable

functor

of the graph-functor faulty:

of a suitable

and not

for simplicity of the basic

has

(-®-)®- =

theory below;

they are

coincidences.)

the morphlsms

in All were actual

-

transformations

natural

the shapes. positive

are proper";

for a general

model

B.

are the shapes

transformations; coincide

The primary

and its models.

The universal

category over,

of this paper

making

~

above

problem

~

A.

clear the relation

coherence

between

the theory

my debt to Lewis, whose

try for a "full solution"

set me

(we shall now drop this notation) category

~ of types

identity

functor

in the category

consists

and graphs;

1 and admits

C~/~.

is a

moreformal

The basic data

and relations

for K, and in

in calculating

K from its

and relations.

In the first problems commutatlvity

of a diagram

could be "filled little

"functorlality" naturality

and talking

are formal allowable

is to formulate

a system of ~enerators

this view the coherence generators

a formal

it is a o- monold

and axioms provide

of what we

formulation.

since it contains

- these

for all models

K over the appropriate

substitution,

f and g coinciding

talk of models

and whose morphisms

that we should always

1.4

for the

This leads us to a universal

I w i s h to re-iterate

for a suitable

= ~(A)

We cannot even make sense

theory.

in A ~ ( A )

object

looking

and sufficient

then f = g in All if and only if their

in these terms,

insistence

if Ff = Fg and T,S

f ~ g in the A ~

abandoning

to

we seek only a

the formally-corresponding

of the corresponding

realizations

problems

T ÷ S coincide

for f = g, for we may have

whose objects natural

as there,

corresponding

it will no longer do when we seek necessary

are trylng to say here without instead

the functors

where,

of the form "f,g:

for the model A, without in ~ ( B )

-

in {A,A} between

This is satisfactory

result

conditions

109

studied,

in K was established

in" with a trellis

diagrams

conditions

of little

being instances

conditions

as in [14],|3],

like

[13],

by showing

that it

diagrams

known

of the axioms,

or of

(f@l)(l@g)

for the basic natural

= (l®g)(f@l),

to commute

or else of

transformations.

In later

-

studies used,

such as that of [7], where

the methods

reference

commutative "filling

in".

o- monoid

to describe

g in ~ / ~

the particular

from

is that all

of Just such a

structure

to the category

contains

data and axioms;

(or polycategory)

map ¢: K ~ {A,A} of o- monoids. of ¢; same objects

specified actually

~/A

of polycategories).

a 2-monad); A knowledge

free structures; to relate

exists

the history

problem

([9],[10])

of Lawvere

([15],[16],

paper [ 9 ] on free residuated

says something

"any two proofs

ment of the corresponding varlable-twlce

generalized

to Lambek that this result

knowledge

coherence natural

problem

[17]).

would most

That some

for a long time;

The Corollary categories,

reminiscent in terms

transformations

of the

problem as

of T ~ S are equivalent

was highly

to

which

if each

of the state-

of the every-

of [I].

likely be seen,

kind.

Thus we are able

realized

to be as follows.

at most twice",

¢.)

(or on

of a very special

Lemma 5 in Lambek's

appears

of

[11] (for Ko- is

to the free-structure

has been more or less vaguely

like

classes

Ko- on ~

on A is Just KoA.

and Szabo

as I know it seems

the

e: KoA ÷ A,

with it an explicit

the free structure

studied by Lambek

to give a

This shows that the theory

it is a doctrine

of K carries

the coherence

equivalence

(= triple)

in the sense

of course

in fact to give such a

to give an action

for the monad

by K is a doctrine

Just as

if they have the same image under

To give such a ¢ is alternatively

the category

in question,

A is precisely

as K, but morphisms

A as an algebra

necessary

(Our earlier ~!l(A) is roughly

those in K, two being equivalent

exhibiting

all the information

kind of extra structure

much as does the list of basic

variable

conclusion

and relations

in virtue

was

made no explicit

theoretical

of generators

do in fact commute

technique

in".

The

relation

commutativity

An important

of K in terms

diagrams

-

a cut-elimination

used to establish

to "filling

our description

image

ii0

I remarked

upon examination,

- iii to

contain an "all diagrams

paper,

on free biclosed

theorem"

commute"

categories,

for the main result

equivalent";

Szabo,

then Lambek's

cartesian

closed categories.

of Szabo,

drew upon Lambek's

[7],

-structure

as

KoA,

commute"

-

brilliant

to coherence

used the same name in

generalization coherence

and free

of Gentzen's result

and is therefore

false;

as we u n d e r s t o o d

for

commute" result

his

in fact false

clear; that

[17]

give n e c e s s a r y and sufficient

contains

conditions

which I have not yet a t t e m p t e d

it turns out that

"all diagrams

for blclosed categories.

for closed categories

paper

it - nor I think

work, e x p r e s s i n g the free c a t e g o r y - w l t h -

indeed says p r e c i s e l y

The c o r r e s p o n d i n g result

in Szabo's

a modified

result

for equivalence

to re-interpret

thesis is asserted

of proofs

as a direct

to

-

statement

Of course this purely factual c r i t i c i s m of certain of the of Lambek and Szabo is in no way meant

great value

of their work in d e m o n s t r a t i n g

and c u t - e l i m i n a t i o n

of course a category

to detract

from the

the power of p r o o f - t h e o r y

techniques.

If we call a category

A on which K acts a K-category,

of K-categories,

KoA K°~I KoB

~A I~ ~

we get

a m o r p h i s m b e i n g a functor

~: A ÷ B c o m m u t i n g with the actions:

(1.1)

are

Mac Lane and I, independently

finally makes the connexion

"main result"

results

"coherence

free closed categories

Meanwhile

The present

Lambek's

K.

and in his next

but even then we had not any clear notion of the relation

did anyone else.

about

student,later

o b s e r v i n g that an "all diagrams

of Lambek's main result

similarly

result;

Lambek used the name

t h e o r e m to prove a partial

closed categories was false;

coherence

"two proofs with the same generality

his thesis [15], where he d i s c u s s e d

cut-elimination

-

B

-

We

may call these

the structure category

strict morphisms

on the nose.

whose

and an algebra

112

objects

-

of K-categories;

However

K also determines

are functors

a monad

and whose morphlsms

A

~- C

B

~D

for this monad

they preserve

all

on the

are 2-cells

;

is a ~: A + B together with a 2-cell

as in e KoA

-

-

KoB--

~

A

~

B

e

subject

to the appropriate

i~oncept

of a non-strict

a monoidal

functor

We proceed

algebra-axloms.

morphlsm

In this way we get the

~, ~: A ÷ B of K-categories,

is the best-known

example.

now to the details,

but

first the following

comment:

to carry out the above program

in full requires

variable

functorial

for the kinds

transformations first

that occur.

to natural

ing ~oA

calculus

as in [ 5]

We therefore

transformations

x A for the many-category

given in [ 5].

The extension

theory cases

case),

does not occur at some length,

ourselves

at

for which a full calculus in ~ or in ~op is easy;

however,

- and often it can be proved with this

a many-

in ~ (or the correspondis

graphs

cases.

The

for the mixed-varlance

to ~ (and also ~, ~op) provided

therefore,

sad

of natural

so do the mixed-variance

can be made to work in part, corresponding

restrict

with graphs

to graphs

in ~ pose yet unsolved problems;

of which

incompatibility

that it doesn't.

important

extension.

We deal

-

2.

i13

-

Clubs and their algebras

2.1

We put ourselves

in the context

[5 ], so that our category Indexlng-set

A, where

of types and graphs

A is regarded

that we have a closed

structure

category

~/A

the category

of categories). case),

~t/~

We introduce

on ~ / ~ ,

to ~ t / ~

category.

We recall

which has as a full sub-

to one element

and ~ / A

paper

is ~ = ~oA × A for some

as a discrete

of polycategories

When A reduces

reduces

of §3 of the preceding

(= A- indexed

families

(the slngle-category

to ~ .

the word club as a short name for "o- monoid

~g~/~".

A map of clubs is of course

category

over ~ together with functors

such that the following

a monoid map.

in

So a club K is a

p: K o K ÷ K and n: J ÷ K over ~,

d!iagrams commute: poK

KoKoK

KoK

Ko]j

(2.1)

KoK

~K

BoK JoK

KoB ~. KoK

~

KoJ

(2.2)

K--

~K-~,

K

l

Since

J is Just A with the augmentation

Just to give objects appropriate with

~(f[gl'

~(k)

augmentations.

Y_! = 1.

we write

(2.3)

1

T(SI,

To denote

Then

to give 0 is

in K, which w@ denote by 1k, with the When A = {I} this is Just to give i E K

the effect

of ~ on objects

and on morphisms,

.... S n) for ~(T[S 1 . . . . , Sn]) and f(gl'

..., gn]). T(S I,

k~--~-(l[k],k),

..., gn ) for

(2.1) becomes

..., Sn)(R I . . . . , R m) = T(SI(RI,...,Rml), .... Sn( .... Rm))

-

with a corresponding

formula

i14

for morphisms, while

(2.4)

T(~k I . . . . , !k n) = T ,

(2.5)

~(s)

where

= s,

T has domain-type

As always,

-

!~(g) n[kl,

f(~kl,

(2.2) becomes

.... !k n) = f

= g, ..., kn] , S has codomain-type

the name of an object

k, etc.

serves as the name of its identity

morphism. Example {A,A}

2. I

is

For any A 6 ~ / 2 , club,

a

and in particular

as in §§2.5 and 3~4 of [ 5 ] ;

for any A 6 ~ / A ,

it is the endomorphism

club of A. Example

2.2

= {A,A}; structure

Taking A = A in the above example in the case A = {1} it reduces

2.3

natural

numbers

for ~.

Similarly,

Example

2.4

In the case A = {I}, the discrete ~ 0 is a club,

club.

Example

for a general

The operation category

{-,-} of ~ / 2 ! ) .

category.

with n(ml,

Any club K which

also a closed the

The detailed

will again be found in §§2.5 and 3.4 of [5].

Example

discrete

to ~.

we get the club

augmentation

..., m n) = m I + ... + m n as

is a discrete o on ~ / ~

A discrete

In the case A = {1},

category

restricts

(whose internal-hom

to ~ / ~ A '

{-,-}' differs

is the category

consider

numbers

here n+m is the tensor product

however

from

in this

a club K for which

functor

at 1 6 ~.

the

To give n is

The monoid-axioms

categor Y with strictly-assoclatlve

T®S = T(S) and strict ~ of natural

which is

clubs.

to give ~ E K; to give ~ is to give T(S) and f(g).

tensor product

is called a

club is Just a o- monoid

F: K ÷ ~ is the constant

say that K is a strict monoidal

~ of the

A, ~A = ~oA x A is a club.

Thus ~ and ~A are discrete

2.~

category

identity

lo

A typical

n~0 and increasing

and 0 is the identity

~.

example

maps n ÷ m;

-

Example

2.6

115

-

Again with A = {i}, let

be the unit category

with unique

L be any category

object

let I

at all,

l, and let F(I) = i, F(L) = 0.

Then K = 7 + L is a club with I = l, l(T) = T, and T( ) = T for T E L. 2.2

: ~g~/~

For a club K, we have a monad K@-

restricts

to K@-

and §3.3),

: ~/A

+ ~/A.

K@- is actually

and is hence a doctrine

Since

a 2-monad

in the sense

÷ ~/~,

o is a 2-functor

on ~ / A ,

or on ~

of Lawvere

[ll].

this monad on ~g~/A will be called a K-polycategory A = {I}). ~:

A

÷

The K-polycategories

form a category

([5]

§2.3

if A = {i}; An algebra

(K-category

A for if

with morphisms

as in ( i . I ) .

B

A K-polycategory ¢: K ÷ { A , A } ~/~

which

or equivalently

and e in ~ / A

~/A).

is a polycategory with an action

For ¢ to be a club-map

O: K o A + A; here

¢ is in

if A is not in the subcategory

(or itself in ~ / ~

of the usual axioms

A with a club-map

is equivalent

to the satisfaction

by e

for an action:

]~oA

noA ~

KoKoA

KoA

JoA

,

~

KoA

(2.6) KoA

Again we write notation like

~

T(AI,

for morphisms;

(2.3) and

2.2'

symmetric Example monoidal

for e(T[AI,

the axioms

..., An]) with a similar

for an action

examples

are numbered

Let A = {I} and K = ~.

monoidal

2.3'

A

can then be written

(2.5).

The following Example

..., An)

A

category

category.

Then a ~-category

(essentially

Let A = {1} and K = ~.

as in §2.1.

by Mac Lane's

An g-category

A is a strict result

in [14]).

A is a strict

-

Example

2.5'

~-category here

Taking

Example

2.6'

K-category L + A.

A together with

and n,~ come

L(

So the c a t e g o r y

from the u n i q u e

maps

under

is the

K-category

Example

2.2"

The free

strict

symmetric

Example

2.3"

The

free

strict

monoidal

Example

2.5"

The

free

category-with-a-monad

Example

2.6"

objects

I[A]

3.

Clubs

3.1

is A × A(cf. The free

[i0]

To give

is, to give

category

on A is KoA, we have

~g~\L

a

a functor of

(3.1)

category

on A is ~oA.

on A is ~oA.

on A is ~oA;

since

on A is KoA = (I+L)oA with

isomorphic

structure,

to A + L.

of the k i n d we have

or a p o l y c a t e g o r y ,

(B,

category

and r e l a t i o n s

of an e x t r a

by a c a t e g o r y

:

p.95).

and L[ ]; it is clearly

The theory

monoidal

category-under-L

g i v e n by g e n e r a t o r s

to be b o r n e

2.6.

A.

the free

= 1 this

(T, n, ~) on A;

0 + l, 2 ÷ 1 in ~.

) E A for L E L; that

of K - c a t e g o r i e s

to be ~, a

a monad

Since

F(~)

2.5

With A = {1} let K be as in E x a m p l e

A is to give

categories

-

A = {i} and K in E x a m p l e

A is a c a t e g o r y

TA = l(A)

116

is s p e c i f i e d

in mind,

by a t e t r a d

p, D, o)

as follows.

First each

of these

~A = ~oA Thus

there

set B of the names

comes w i t h a type

× A of o b j e c t s

in the t h e o r y

consists

~s the

w E ~, so that

of ~, or e q u a l l y

of m o n o l d a l

of two e l e m e n t s

of the basic

B is a set over the set

a discrete

categories, where

of this m u c h of the theory

A and a f u n c t o r

B + {A,A}

over ~, s e n d i n g

category

over ~.

A = {1} and ~A = ~' B

{@,I} w i t h a u g m e n t a t i o n s

To give a m o d e l

functors;

F® = 2 and rI = 0.

is to give

a polycategory

each B 6 B to a f u n c t o r

-

IB[:

All

... x A l

x

÷ A

of the type

from

generates.

We define

inductively

by:

There

(3.3)

If B 6 B with TI,

is an object

domain-type

=ik(S)

(3.5)

(B{T I,

we can now drop

B 6 B with

This

T ÷ {A,A}

The next

with

that

£'T n)

Of course

B(T1,

..., !l n} 6 T. ..., Tn) by

notation

the unit

functor

B + {A,A}

of clubs;

so that

a model

part

in

{A,A}

Then

(3.4)

in favour

any

ITI for the image

and

(3.5),

of round

over ~ extends

and

brackets. uniquely

of B is Just a T-algebra.

of T E T.

of the tetrad

(3.1)

is a r e l a t i o n

p on T, which

the c o n d i t i o n £T = £S if TpS.

relation

namely (B,p)

...,

,Sml), .... Tn( .... Sm)},

codomain-types.

B{~ll,

the c u r l y - b r a c k e t

It is clear

(3.6)

is an object

n(F'TI,

.,S m) . =. B.{ T I.( S I,

suitable

B{T 1 . . . . , T n} coincides

satisfy

In] , and if

by the ~l"

We identify

must

there

(l[k],l).

by setting

..., Tn})(S I, .

J ÷ T is given

3.2

...,

rll=

= S,

S, S i are to have

Write

domaln-type

it

augmentations

k, w i t h

£'B = n[kl,

T that

£"B.

ToT ÷ Y i n d u c t i v e l y

(3.4)

club

of T and their

F"T i = ki' then

..., T n} in T wlth

and c o d o m a i n - t y p e

discrete

lk in T for each

..., T n 6 T with

B{TI,

to a map

£B.

B to the free

the objects

(3.2)

where

-

n

We now pass

We define

ll7

those

corresponds concerned

of the theory

to the first

wlth

lot of axioms

the functors.

is to be a model

for the structure,

For a model

A of B in which

of the part

-

118

-

ITI = ISI wherever

(3.7)

The relation in §l.1 above. monoidal

It is non-vacuous

categories;

in practice,

however

and we ignored

in the theory

it

of strict

there it is given by

@(®,!)

P @(!, e),

®(I,l)

p i,

A relation or not)

p is often empty

TpS.

p (satisfying

(3.6))

on any discrete

is called a con~ruenc e if it is an equivalence

club T (free relation

such

that !

T(SI,

.... S n) p T'(S{,

Given any p satisfying ing it, which universal clearly

..., S~) whenever

(3.6) there is a smallest

can be given explicitly

algebra.

T/p

The quotient

again a discrete

ToT' and SIPS i. congruence

by the process

~ contain-

familiar

of T by this congruence

club, with its structure

maps

in is

~ and ~ given

via representatives. Returning structed which

in §3.1,

ITI =

to case where a model

ISI whenever

T is the free discrete

of (B,p) TpS;

is a map T ÷ {A,A} of clubs in

for the latter is clearly

of (3.7).

Such a model is therefore

henceforth

write S for the discrete

are what Kelly 3.3

club on B con-

an algebra club

T/p,

- Mac Lane called the shapes

a consequence

for the club

because

T/p.

the objects

We of S

in [7].

The thing given by a set 0 of objects

and a set M of morphlsm~

ii

or arrows

with domain and codomaln

or identities,

has various

cannot be used here, I don't much care

names:

for we have

for the former;

maps M $ 0, but without diagram

"graph" I shall

scheme, already

graph.

composition The latter

in a different

sense;

call it a pre-cate~ory.

Since we can speak of a map of a pre-category

into a category,

it makes

-

perfectly

119

good sense to speak of a p r e - c a t e g o r y

of objects m o r e o v e r A map of pre-clubs restricted

is a discrete

club,

to objects

of the basic natural

(O,M)

I shall call

the part

transformations;

($,9),

together with domain

or D for short,

of (3.1) is a p o l y c a t e g o r y

D ÷ {A,A} of pre-clubs. w l t h an assignment

It is therefore

of a natural

to each d: T ÷ S in ~ with

a pre-category; e x t e n d i n g the

the arrows

A model of

A together with a map

a model A of (B,p) together

transformation

Idl:

ITI ÷

ISI

of graph

Fd = ~.

In the case of a (non-strlct) instance,

over ~, which

F of S, and thus m a k i n g D into a pre-club.

(B,p,~)

a pre-club.

(3.1) is a set R consisting of

further with a map F: D + ~ of pre-categories,

augmentation

If the set 0

is a map of clubs.

and codomaln maps D ~ S m a k i n g provided

over ~.

is of course a map of pre-categories,

The third part of the tetrad the names

-

symmetric

monoidal

category,

for

of D are

r:

®(l,I)

÷

r: 1 ÷ ®(!,I) C: @ ÷ @ ,

all with identity permutation

of 2.

Idl be a natural

graphs except

c whose graph is the non-ldentlty

Note that we have ~ isomorphism;

no way of demanding

a way will be provided

that

in §3.8 below

when we introduce

the second lot a of axioms,

monoidal-category

case will include axioms a~ = l, aa = l, r~ = l,

~r = l, c 2 ~ 1. isomorphisms;

In general,

for example,

of course,

we do not want the

the $ of a monoidal

We shall now construct,

which in the symmetric-

in several

Idl

to be

functor.

stages,

the free club

L on

-

the

pre-club

unique map

120

-

P, so that a map P ÷ {A A} of pre-clubs

L ÷ {A,A} of clubs,

and a model

extends

for (B,p,D)

to a

is Just an

L-algebra. 3.4

For d: T + T' in D with rd = 6, we define an instance

be a formal e x p r e s s i o n of $) of suitable

(3.8)

composed

augmentation

T(S~I,

..., S{n) ÷ T'(S I,

a domain and a codomain

F(d{Sl,

F'S 1 = ml[91,

S i of 9 (that is,

codomain-types:

d{S 1 . . . . , Sn}:

It is a s s i g n e d

of d and n objects

of d to

.... Sn})

..., Sn)-

as in (3.8),

= ~(ml,

and given the

..., mn) where

Fd = ~ and

..., 9ml] , etc.

For the instance

e = d{Sl,

..., S n} of d and for

m = m I + ... + m n objects R i of 9 of suitable

codomaln-types

(3.9) e(R 1 ..... Rm): T(S~I

÷ T ' ( S I ' ' ' S n ) ( R I ' ' ' R m )'

where (3.10)

~ = ~(m I, d{S l, .

..., mn) , by setting .,Sn}(R . .I, .

If we now identify n[kl,

... S~n)(R~I...R~m)

.,R m). =. d{Sl(R . I,

we can h e n c e f o r t h

..., S n) = d{S I,

drop curly b r a c k e t s

We have now e x t e n d e d

,Rml) ..... Sn( .... Rm)}.

d{l~l_ , "''' ~kn }, where

d with its instance

..., k n] = F'T', we find d(Sl,

same objects

we define

in favour of round ones.

9 to a b i g g e r pre-club

as 9, and a d m i t t i n g

..., Sn} , and

an o p e r a t i o n

l~a~ 9, with the

(3.9) which

clearly

satisfies

(3.11)

e(R 1 ... R m ) ( V 1 ... V k) = e ( R l ( V 1 ... Vkl) . . . . .

Rm(...

Vk)),

(3.12) e ( ~ 1 . . . . , ~ n ) = e. Any map 9 ÷ {A,A} of pre-clubs

now has a unique e x t e n s i o n

to

a

P + {A A} of p r e - c l u b s

map I D ~

(3.13) 3.5

121

le(R1, Next,

...,

which satisfies

Rm) l = l e l ( l R l l

for e: S ÷ S' in ~

be a formal e x p r e s s i o n

-

composed

, ....

IRml).

9, we define an e x p a n s i o n of e, an object

F'T = n[kl,

..., kn ] say, and an element

F"S = F"S';

this formal e x p r e s s i o n

of e to

T of D with

i E n for w h i c h ki

is a s s i g n e d

a domain and a codomain,

and is w r i t t e n as

(3.14)

T{!k I, ..., ~ k i _ l , e , ! l i + l ,

...,!kn}:

T(!l I .... ,S .... ' ~kn )

÷ T(~II, it is given the a u g m e n t a t i o n If e is i t s e l f an instance instance

n(l,

..., i, D, i,

of d, then

(3.14)

domaln-type

m[~l,

codomain-type

'''''~in);

..., i) where

Fe = ~.

is called an e x p a n d e d

(3.14)

of e, let P have

"''' Bm]' let J 6 m with ~j = F"T, and let Qk have

~k for k M J.

P(Q1 . . . . ' QJ-I'

Then we define

f' QJ+I'

"''' Qm ): P(QI'

.... R . . . . , Qm )

÷ P(QI' to be the e x p a n s i o n

(3.16)

,S'

of d. Let f: R + R' denote the e x p a n s i o n

(3.15)

"'"

• "" , R' ' "''' Q m )

of e given by

K{IuI_ .... ,iW_J-l'll- l ' ' ' ' ' e ' ' ' ' ' l l n ' l ~ j + l ' ' ' ' ' l U m }

where

(3.17) K = P(QI' " ' ' ' If we now i d e n t i f y

QJ-I' T, QJ+I' " ' ' '

e: S ÷ S', where

i {e}, we find that T(IAI =~ =

.

.

can again drop curly b r a c k e t s

e .

Qm)"

F"S = F"S' = ~, with its e x p a n s i o n

"" ~k ) = T { ~ I . . . . -

n

e ...

in favour of round ones.

~n

},and we

-

122

have now extended ~

We

still with the same objects,

-

~ to a bigger pre-club ~

and now admitting an operation

~ns~ D, (3.15)

which clearly satisfies (3.18) P(RI...Rm)(VI...f...V k)

=

(3.19)

,

=1 (f) = f.

Any map I ~ ~

P(RI(V I. • .Vkl ),...,Ri(Vp...f...Vq),...,Rm(...Vk))

I~

D

'(3.20)

D ÷ {A,A}of pre-clubs

now has a unique extension

to a map

÷ {A,A} of pre-clubs which satisfies

JP(Q1 . . . . .

f' .... Qm )I = JPI(JQII,

We extend to ~

I~

D the operation

"'',

JfJ'

"'"

JQm j)"

(3.9) of §3.4 by

defining

(3.21) T(~kl...e...~kn)(Vl...Vk) which easily (3.22)

T(RI...f...Rm)(VI...Vk)

It is immediate P.

that

We conclude

extension

to a map ~

Write ~

pre-category

~R

~R ln~

~

~

to hold for e E ~

I~

P ÷ {A,A} of pre-clubs which satisfies e,f e ~

I~

D.

D for the free category generated by the

D; a morphism is therefore ~2~

and the identities

With its evident augmentation pre-clubs ~

continue

that any map D ÷ {A,A} of pre-clubs has a unique

T 1 ÷ ... ÷ T n = S of arrows of ~ by concatenation,

..., f(Vp...Vq) . . . . , Rm(...Vk) ).

(3.11) and (3.12)

both (3.13) and (3.20) for 3.6

..., Vk) ,

implies the more general

= T(RI(VI...Vkl),

I~

= T(V 1 ..... e(Vp...Vq),

a string T = T O

D, with n ~ 0; composition

are the strings of length 0.

it is again a pre-club.

D ÷ {A,A} has a unique extension

Any map of

to a functor

is

-

~

~g~

In~t

-

P ÷ {A,A}, which is also a map of p r e - c l u b s to Cg~ Ex~ I n ~

We extend f(Rl,

123

..., Rm) , where

f = fk

D the operation

"'" fl with

( 3 . 9 ) by d e f i n i n g

fi 6 ~

I~

D and

rf i = hi,

to be

f k ( R l . . . R m ) f k _ l ( R n k I ... Rnk m)

where

~ = n k nk_ 1 ... n2.

Similarly equal

we e x t e n d

(3.19),

"'' 1Qj

which

map ~g~ ExR ~ satisfies

in a way

R ÷ {A,A}

(3.13)

quite

and

(3.11),

set M(T,S) equivalence congruence if there

for

from that

if k = 0 we

(3.12),

(3.18),

is a sequence

n >_ 0, in w h i c h

"relation"

of §3.2;

f = f0'

each pair

and

~;

fl'

f,g:

and

D. "congruence"

no c o n f u s i o n

should

~(T,S)

a congruence

composition.

to a

is a functor

M is Just a r e l a t i o n

with

a relation

which

extends

e,f 6 C~t E~R !Bg~

It is called

compatible

~ containing

either

(3.20)

T, S 6 M.

relation

T -- T O + T 1 +

of p r e - c l u b s

~ on a category

for each

of p r e - c l u b s

to use the words

different

A relation

There

result.

on the

if it is an is a smallest

T + S are e q u i v a l e n t

"''' fn = g of m o r p h i s m s

under

T ~ S,

fi-1 and fi are of the form

... ÷ Tk_ 1 v Tk "+ "'" ÷ Tm = S

u~v or v~u.

From a congruence the

P(Q1...f...Qm)

"'" Qm ); again

have

that a map D ÷ {A,A}

We are now g o i n g

where

by setting

"'" fi

We still

set I T ( R I . . . R m) = I.

(3.22).

3.__~7

where

(3.15)

of the P(Q1

... Qm ) = 1.

We conclude unique

... R~m)

If k = 0 we of course

the o p e r a t i o n

to the c o m p o s i t e

set P(Q1

..- fl(R~l

same objects

w on M we get a quotient

as M but with e q u i v a l e n c e

is the c o n g r u e n c e

generated

classes

by the r e l a t i o n

category

M/~ with

as morphisms.

~, we agree

to write

When

-

124

-

MI~ for MIW. Every functor P: M ÷ N such that Pu = Pv whenever u~v factorizes uniquely through M ÷ M/~ to give a functor M/~ + N.

If

P: M + N is givenand if we define ~ by " u~v ~ Pu = Pv", then ~ is already a congruence, the kernel congruence of P, and M/~ ÷ N is faithful. If M is augmented over ~ the relation w is said to be over if ru = rv whenever uwv.

Then M/w has an augmentation over ~, also

called r. Consider now the diagrams (i<J) (3.23 ~ T(SI...Si...Sj...Sn)

T(s]

T(SI...f...Sj...Sn) ~' . S . . . S n) ~T(Sl...oi.. j.

I

s i ..g...s

T(S 1 . .

. S i . ° . S !~ - .

.S n)

'

T(SI...SI...g...S

n)

t m-T(S1.., S i...S~...Sn) T(SI...f...S~...S n)

h(SI...SI...S n) (3.24) T(S~I. • . S _ l l

T(S~l-

.... ~)

T'(SI...Si...S n)

I

T ' ( S 1 . . . k . . . S n)

-k---S~n) I

T(S~I...S'~ - l i "'S~n)~

in ~gt ExR Inst P.

!

~ T ' ( S I . . . S i . . . S n) h(S1...Si...S n)

Because {A,A} is a club, the images in {A,A} of

these diagrams, under such a functor as described in the last paragraph of §3.6, commute. through the quotient of ~

Such a functor, therefore, factorizes ~

~!~

P by the relation which identifies

the two legs of (3.23) and the two legs of (3.24). The same congruence on ~g~ ~ R

!n~

P will clearly result if

we identify the legs of (3.23) and those of (3.24) only in the special

-

Inst P.

case f,g,h,k 6 ~

125

We can go further;

to be in Inst P, for the functors in T or T'.

Since

"functoriality

Write,

relation ~n~

and (3.24)

and "relations

then,

the legs

which

0; write

that

~

identifies F~n~t + ~

f,g,k

them can be absorbed

seem to assert

respectively

the

of h", we call them "relations

of naturality".

for the relation

of (3.23)

we can suppose

"expand"

of T" and the "naturality

of functoriality"

identifies

(3.23)

-

on C ~

~R

~g~

O which

for f,g 6 In~t P, and Ngt for the

the legs of (3.24) for the union

for k ~ Inst P, h ~ ~

of these

two relations.

Define

L = c~t ~ m

(3.25)

Now L admits

~B~

oI(~

the structure

+ ~). of a club.

to see this is to observe

that the operations

ExR Inst P are compatible

with

to [.

Then

diagonal

(3.23)

of (3.23)

extension

gives

an operation

the right vertical and the diagonal The map h[kl, club axioms

an operation

T(kl,

of this diagram

L ÷ {A,A}

at the end of§3.6

(The word

(3.11),

induced

is clearly

and

of ~

Inst P are identified

itself,

(3.24)

"extends"

are of length

are faithfully

..., kn)

a map of clubs;

over The

and by with

still commutes, h(kl,

..., kn).

functorial, (3.19),

C~t ~ p

~B~

and the (3.22). P ÷

and we can assert

to a unique

is legitimate:

of (3.24)

(3.18),

by the functor

in L.)

on ~g~

carry

f ... g ... Sn),

(3.12),

2 in ~

(3.15)

in [ and commute.

the operation

in L; hence

embedded

and

The analogue

extends

the easiest way

and hence

..., k n) is clearly

from

(3.23)

(3.24)

by T'(kl, gives

that a map D ÷ {A,A} of pre-clubs L ÷ {A,A}.

(3.9)

T(S1...

..., kn).

arrow replaced

follow easily

and

can be written

..., k n ] ~ h ( k l ,

The map

{A,A}

and (3.24)

(3.23)

Perhaps

since

map of clubs the legs of

~Z~ $ ~

D, no two elements

ExR ~ B ~

D, and a f o r t i o r i

-

We conclude

that a model

same thing as a polycategory L-algebra

126

-

for the part

A with a club-map

The last part of the tetrad

(3.1)

a model

of (B,p,D,c)

of (B,p,D)

A - for which More

is now a model

Ifl =

Igl whenever

L ÷ {A,A};

is the

that is, an

commonly,

determines

a relation

[e}

~ over ~ on L;

- that is, an L-algebra

~ is given as a relation

if some representatives any relation

is a relation

f~g.

~; it makes no difference

since

on L where

not on L but on ~!~

a relation

[ f] and [g]

~ on the latter

are related

f and g of them are related

under [e]

under G, and since

on L so arises.

For example, monoidal

of (3.1)

or an L-polycategory.

3.8

~x~ Inst

(B,p,D)

category,

the relation

where

e on ~

D is as in §3.3,

~

~Q~

is given

P for a symmetric as follows

(cf.[ 7]

page 98): a(l,_l,®)-a(@,l__,_l) a ®(_l_,a)-a(l=,@,l_)-@(a,1) @(l,r)-a(1,_l,I)

e r(@)

C.C ~ I a-c(®,l).a

e ®(_l,c).a.®(c,l_)

a.aa

1

~.ac

I

r-~c

i

~.ra

1

The last four of these are our way of guaranteeing natural

i sgmorphlsms

in {A,A};

the third already

that

lal,

guarantees

Irl are this

for

Icl. If L any club,

a club-congruence

on L, in the sense of §3.7, which

on L is a congruence

further

~ over

satisfies

f

(3.26)

f~f' and gi~gi imply

If ~ is a club-congruence,

f(gl'

"''' gn ) ~f'(gl'

"''' gn)"

L/~ is again a club, called a quotient

club

-

of i (not to be c o n f u s e d clubs

in §3.2,

-congruence

of

-

the m e a n i n g

to w h i c h we shall

~ of a map

-congruence,

with

127

of " q u o t i e n t "

s e l d o m have

L ÷ M of clubs

and we get c l u b - m a p s

for d i s c r e t e

to refer).

is a u t o m a t i c a l l y

The k e r n e l a club-

L ÷ L/~ ÷ M; we call

L/~ the imase

L ÷ ~.

Any c o n g r u e n c e ~ on L is c o n t a i n e d in a smallest club^ - c o n g r u e n c e ~. If ~ is g e n e r a t e d by a r e l a t i o n a, define a n e w relation

(3.27)

T by

M ( L I . . . f ( N I . . . N k ) . . . L n)

so that

~ consists

congruence

of the

~ generated

T M ( L I . . . f ' ( N I . . . N k ) . . . L n) if faf';

"expanded

instances

by T is c l e a r l y

the

of a".

Then the

club-congruence

# generated

by ~ = w. Returning which

Ill =

now to the club

Igl w h e n e v e r

L of

fog is the

(3.25),

same

an L - a l g e b r a

A for

thing as a K - a l g e b r a ,

where

K is the club K = L/~

(3.28) and

• is g i v e n by

(3.27).

This

Let us now t r a n s l a t e as a r e l a t i o n

over ~ not

(3.27)

to c o n s i s t

on G ~

~

!~

of the e x p a n d e d (3.28), D/(Fun~t ~

~

+ ~

on i but

precisely

on ~ t

instances

of the image

now be w r i t t e n

+ T).

[Y]

as

~R

[a]

L/[~],

(expanded

Our final

result,

then,

a K-algebra

for the

club

of (B,p,D,g)

case where

!~

D.

!~

is.

a is given

Again

define

T by

of a; now T is a r e l a t i o n

of T in L c o n s i s t s

We give a new name the

a model

usual

instances

Then the image

l~a~ D, to suggest

relations.

to the more

of the e x p a n d e d

D.

w h i c h must

then is what

of g.

precisely

The club

is t h e r e f o r e to this

instances

is that a m o d e l

K of

~a~ ~ R

relation

of the) of

I~

• on

imposed

(B,p,D,a)

is

-

128

-

(3.29) We sum up our conclusion Theorem

3.1

The data

POlycategories precisely

(B,p,D,~)

A determine

A.

by p.

instances

of the elements

As a category,

relations

(3.23)

the operations

(3.9) and

(3.15).

Its structure

paragraph

of §1.4;

~ diagram

only if it can be f little diagrams

conclusion"

of K

and instances

as a club is derived

from

are _Fu_nct + _N~t + I_~R is the

we referred

to in the second

in _~_~t ~xR l_nst D commutes

~.ed in in the classical

of these kinds.

i_~s

by the expanded

and to the expanded

The fact that the only relations theoretical

for (B,p,D,e)

to the functoriality

and (3.24),

G.

for

club on B by the congruence

of D, subject

relations

"important

structure

club S of objects

K is generated

of the imposed

3.9

an extra

The discrete

of the free discrete

generated

naturality

describing

a club K such that a model

a K-polycategory

is the quotient

as:

A simple

in K if and

way with a trellis

application

of

of this is the

following. Suppose

we have two theories

(B,D,e)

B' D B, D' D D, and q' D ~; for simplicity Then S' D S, and Cat ~ R clubs

~nst D' D ~

and

(B',D',e')

we take

p,p'

ExR Inst D.

where

to be vacuous.

We get a map of

K ~ K'. Suppose

that for every d E P'-p,

codomain

of d lies in S ' - S .

expanded

instance

either

Then if either

of d, neither

leg of (3.23)

the domain or the

f or g in (3.23) lies in ~

~

is an ~

D;

t

similarly neither

if either h or k in (3.24)

leg lies in ~ Now suppose

~

instance

of d,

~Bg~ D.

further

c' but not by e, neither

is an expanded

that, whenever

u nor v lles in ~

u, v: P + Q are related by ~R

~

P.

It then

-

follows

easily

under F ~ n ~ ' ~

+ ~;

129

-

that if f, g: T + S in ~

+ ~'

+ ~',

in other words,

ExR ! B ~

they are already

congruent

K ÷ K' is faithful.

any d i a g r a m in K can then be deduced

D are congruent under ~ n G ~

+

The commutativlty

of

from the commutatlvity

of its

image in K'

4.

Extensions

4.1

to more ~eneral natural

The e x t e n s i o n

still covariant in ~op

forward,

of the above to the case where the functors

but the natural

(as described although

I haven't

"posessing

If we generalize

of a function

on A.

cannot be handled developed

straight-

finite

on A; an object of ..., A n ] ÷

in the last p a r a g r a p h

of

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

covariant

case with graphs

in these terms until a calculus

in

llke that of [5]

is

for this case.

A more hopeful

natural

transformations

by E i l e n b e r g - K e l l y

case is that of m l x e d - v a r l a n c e of the e v e r y - v a r i a b l e - t w l c e

in [I] .

discussion

of the obstacles

Presumably

if the right

be found as discussed In the meantime

functors

We refer the reader to §4.2 of [5] in the way of a suitable

category

there,

closed loops).

with the one-category

p r e s e n t i n g no d i f f i c u l t i e s

can

above will carry over at once. ~

of simple

We shall deal here for

case,

at all.

for a

calculus.

~* of g r a p h s - w l t h - c l o s e d - l o o p s

everything

and

kind introduced

we must do what we can with the category

(those without

simplicity

in ~ or

~: m ÷ n and maps A~i ÷ B i in A.

describe

The more general

4.2

graphs

strictly-associative

a bit further as suggested

§4.1 of [5], we can similarly

are

~op itself is the

..., A n ] where A i ~ A, and a m o r p h l s m n[Al,

..., B m] consists

-equalizers

have graphs

checked the details.

and ~°PoA is the free such category

this is n[Al, m[Bl,

transformations

in §4.1 of [51) would seem to be quite

club for the extra structure products",

transformation~

the e x t e n s i o n

We assume

to p o l y c a t e g o r i e s

familiarity both with

-

[

i]

and with

§4.2 of [ 5 ]; ~

We call a category morphisms

is d e f i n e d

K over ~

We write

~!m/~

between

2-category.

~

It contains

as a u g m e n t a t i o n

We i n d i c a t e d operation

morphism

...,

A[BI,

f: A ÷ A' with

in §4.2

(-Ul,

gl"

of [5]

Bn] , where

..., B n] ÷ A'[Bn+I, Ff = ~, where

-u2,

codomain

under

indices

...,

~ such On+l,

A

[BI~

g]

g

g: A'

{ A"

I S 4,

[B 9 '

B5,

!0

~::

,

B 7, B 8]

r~n.].l

that

Objects

an

are

u i = ±; a

is f[gl'

and where

Because

B2,B3I

is

an a s s o c i a t i v e

o n] with

that p has

as we go:

A 6 ~

(Recall

if B 6 ~ .

Bn+m]

we wind

when

o i is + or -.)

we may as well

no problems;

categories

it is a

at 0[ ].

..., On+m).

in n + m.

Fg are

of simple

properly

in ~ ...,

composable

Ff and

how to define

2k = n+m,

..., -On,

presents

each

FA = n[ol,

"''' gk are written,

composition composing

for any

subcategory,

functor

with values

p, q 6 n + m are mates

of their

if,

more

as a full

..., e n ] where

AoB in ~ / ~ ,

like A[BI,

which

simple

them;

the constant

of P*=o is n[el,

sign in

in the latter.

for the category

over P* and all functors =o

object

-

f: T + S and g: S ÷ R of K, the graphs

compatible.

given

130

gi:

..., gk ] where

Bp ÷ Bq; here

the - sign and q the + As for the order take

it to be the order

of the

happily

in

simplicity,

around

the graphs

-

We

augment

obvious J for

AoB over P* when ~0'

process

of r e p l a c i n g

have is a f u n c t o r

have no way of c o m p o s i n g

-

B has a n o n - t r i v i a l linkages

o is the unit c a t e g o r y

we don't

131

by entire

augmentation, graphs;

I with a u g m e n t a t i o n

category

natural

{B,C} right

the i d e n t i t y

I[+].

The one thing

adJoint

transformations

by the

to AoB,

for we

of i n c o m p a t i b l e

graphs.

We can still We can also define club-map case,

Now s u p p o s e describe

an e x t r a

set ~o2

of o b j e c t s

etc.,

carrying

we are given data

structure

there

that

~ require

theory

of a c o m p a c t

o t h e r hand,

category, author

us to compose closed

will

contravariance

arises

through

good cases

over

§3.6

where

- §3.8,

it may h a p p e n

(with a p o l o g i e s

of D

[8],

the

is the

of [5].

On the

be c o m p a t i b l e ;

this

and [I~ for the theories enriched

a later p a p e r

adJoints are

the a n a l o g u e

of

over a c l o s e d

show it to be the case w h e n e v e r

all c o m p o s a b l e s

that

graphs;

even though

in §4.2

in

to Dr. Johnson)

an example

D may always

categories;

obtaining

the

suppose

D with i n c o m p a t i b l e

as given

positing

to

respecting

We must

to §3.6,

transformations

of c l o s e d

volume

carry

~ns~

(3.1)

of M ~ {A,A} by MoA ÷ A

no i n c o m p a t i b l e s ;

in ~

of n a t u r a l

and of maps

a relation

This may h a p p e n

category

in this

In these

say

to be the case in [7],

categories,

by a

We then meet no p r o b l e m s

W h e n we come

and we m e r e l y

composables

has been s h o w n closed

over ~ .

is not c l u b b a b l e .

axioms

as in

3.2 u n c h a n g e d .

f: T ~ S, g: S + R in ExR ~ D ~

(B,o,P,a)

it is true,

Now B is a set over the

the r e p l a c e m e n t

§§3.1 and

§3.3 - §3.5.

if so we are stymied,

as a o- monold.

to this in the c o v a r i a n t

A.

of ~:, and 0 is a g a i n

that it is a u g m e n t e d

are

not,

(B,o,D,o)

on a c a t e g o r y

Always modulo

over

is e q u i v a l e n t

namely

KoA ÷ A.

we can take over

of course

K in ~!~/~:,

or K - c a t e g o r y ;

but by what

an a c t i o n

augmentation.

a club

a K-algebra,

K ÷ {A,A},

namely

define

[6]

the only

to c o v a r i a n t

compatible, of T h e o r e m

by the

functors.

we may 3.1;

such a

-

case

is said to be clubbable.

132

-

Of course

To extend to the m a n y - c a t e g o r y as category on A.

case is easy:

of types and graphs by the appropriate

A type is now

(nI±kl,

over.

just replace

~,

~

which depends

..., ±kn I , ~) and a graph is as before

except that it must mate elements

4._~3

§3.9 too carries

corresponding

Let us return to the unclubbable

case.

to the same k.

To say that we are

stymied does not mean that we are out of the game - only that we cannot describe

a model of

We successfully

carried over

objects,

(B,p,D,o)

as an algebra for some club K.

§§3.1-3.2

to get the discrete

and further - taking O to be augmented

§§3.3-3.5

to get the pre-club ~ R

can still form the free category it over P*. ~O'

!Bg~ D. ~

it is Just not a simple

~

over P* - carried over =O

When we come to §3.6, we ~g~

O on this,

category

f, g, k E ~ D ~

in the sense of §4.1

the analogues D and h E ~

of (3.23) and

~Ss~ D, and form the quotient

again it would have made no difference

f, g, k E ~

~Sg~ D.

g on ~g~ ~ R um...

excellent

Finally, ~B~

coming to §3.8, suppose that the

and the vj compatible of T = ~

in (3.27)

we cannot

case be the free model on A.

T h e o r e m 4.1 (B,p,O,~)

on.the

category

A.

to make

K as in (3.29).

But in that case the free

and even in the unclubbable

The . category K ~iven by

continues

for this

form KoA, which would in the

model KoA on A is equal to K as a category, augmentation;

written as

(see [7), Lemma 5.3,

and we can form the category

Since K is not simple, clubbable

L

... v2v I with ui, vj E E~R !Bg~ 0, have

Then the d e f i n i t i o n sense,

category

if we had taken

D only relates pairs u,v which,

u2u I and v n

the u i compatible concept).

We

(3.24) in §3,7 for

as in (3.25);

morphlsms

and augment "

can still write

relation

club S of

apart

from its

case we still have this:

(3.29)

is the free model of

-

Proof

It is a model:

133

-

first of all a model

T E S we get a functor

using

(3.23),

of (B,p,P), components obvious

Idl(S1,

instance

of (3.24). u(SI,

~B~

Finally

P,

Next it is a model

transformation

Idl with

..., Sk); it is natural

by the

it is a model of (B,p,D,G),

..., S n) T v(S1,

..., Sn) by (3.27),

whence

Ivl. If A is a n ~ model,

objects

Kk E A.

of models

= Kk.

of (B,p),

directly

from

~

R, extends

I~

a functor

Certainly

since

(3.2) and

K since it respects

(3.3)). ~

~ ,

~,

The value ~

without

construction;

and ~ .

of

in practice,

having proved

we cannot

on A as KoA, which

clubbable

K whether

it is a club

at the

in the unclubbable

§3.6-stage

on A, much as we constructed

way to express

it is this:

K.

(B,p,D,~)

to B the objects

of the

in this volume.

case get the free model

we can still construct

the free model

we may be

by a cut-elimlnation

this in the paper [61

1, FA), to get B'; by setting

through

on A is of great interest

perhaps

it directly

form from

(or

factorizes

even when K i_~s a club,

is now undefined,

by adding

such map ¢: S + A

0

the free model

we do precisely

Although

(B',p',D',~')

a family

of ¢ is then forced on

D, and finally

able to prove this only a ppsteriorl, argument,

is a unique

to be able to talk about

or not, not so much because but because

there

(B,p) alone is certainly

to ~

It is valuable

in itself,

A ÷ A is in effect

We are to show that there is a unique map 4: K + A of

models with ¢(ik)

A~(0[

for gi E E ~

gi by composition.

..., S k) = d(S1,

extension

for

ITI(g I, ..., gn ) = T(g l, .... gn);

for d E p we get a natural

since u G v implies lul =

in the first

and for arbitrary

since

since

ITI: K ~ ÷ K by setting

ITI(S I . . . . , Sn) = T(S I . . . . , S n) and the latter is defined

of (B,p),

directly

In fact the simplest a n e w theory

A of A, with augmentation

p' = p; by adding

to D the

-

134

-

of A to get D'; and by adding to a the relations h = fg

morphisms

and k = IA, where h is the actual composite

fg in A and k is id A in A,

to get a'

Form the K' for this theory,

subcategory

of K' determined by the objects of domain-type

is then easy to see that K[A]

and let K[A]

is the free

(B,p,D,a)

be the full 0[ 1.

- model

It

on A; its

set of objects is of course So ~ A . The monad K[ ] is Just not determined by knowledge of K as a category over ~ ;

we lost too much information by throwing away the

closed loops when we composed incompatibles

in ~ .

If we take the

example of compact closed categories given in §4.2 of [5], then the morphism

(4.7) there in the corresponding g has empty graph in P* =O

fails to inform us that there will be morphisms K[A] for u,v 6 A. would help;

and ~

llke (4.8) there in

Once again it is clear that a suitable ~* if found

(4.7) would then have a graph consisting of a single

closed loop, KoA would now make sense, and would contain a morphlsm (4.8) depending on a cycle in A.

The conviction is strengthened that

K[A] is of the form KoA for a monold K in ~ / ~ * but not-yet-found, 4.__~4

~*.

The ideas in §4.2 above surely extend to the mixed-varlance

cases corresponding to graphs in Z or in ~ o p --

n[Gl,

..., Gn ] to m[~n+l,

function, (-~l'

for some suitable,

compatibility

~

O

..., an+m] , where ~i is + or -, is now a

and not a biJection,

"''' -Gn' an+l'

A graph in S °p* from

--

from the +'s to the -'s in

..., an+m).

There is a simple concept of

for such graphs - namely the absence of closed paths in

the composite graph - and there is an evident way of Composing compatible graphs.

I don't see how to make S °p* a category:

one can

O

no longer define a composite of incompatibles Just by "discarding the closed loops"; but in the clubbable don't occur,

case, where incompatibles

§4.2 surely carries over.

Just

The theory of cartesian

closed categories has natural transformations

of this kind.

It will

-

135

c e r t a i n l y be c l u b b a b l e by S z a b o ' s for this theory. "Generality

I rather

Theorem"

other words

5.

that

Coherence

5.1

"all d i a g r a m s

(5.1)

problem

i m a g e s u n d e r the

@: ~g~ ~ R

in the s t r o n g e s t determination

that

for

club-structure,

=

the c l u b b a b l e

A map

A in C ~

it c o n s i s t s

sense,

D have

+ ~

+ ~)

= K;

in the c o m p l e t e

and r e l a t i o n s

and w i t h its

(B,p,D,e);

we then

KoA on any A, i n c l u d i n g

which talked entirely a canonical

a: K ÷ L of clubs has and m f a c t o r i z e s

while

ourselves

its

entirely

to

T ÷ S is to be a m o r p h i s m

it

~ which

is a

im a; the c l u b map K/w ÷ L is of clubs.

are to be t h o s e

of K, but

that

K/~

a morphism

Its c l u b - s t r u c t u r e

into c l u b - m a p s

K ÷ K/w = im a ÷ w i m a ÷ L;

Recall

a n e w club w i m a c a l l e d the

aT ÷ mS in L.

and a: K ÷ L now f a c t o r i z e s

of m o d e l s ,

of c l u b - m a p s .

a kernel-congruence

Introduce

of a; its o b j e c t s

by

as K + K/w ÷ L; as we said in §3.8

of m, w r i t t e n

as K.

in terms

factorization

K ÷ K/w is a q u o t i e n t map

has the same o b j e c t s

(5.2)

~B~

this to the f o r m u l a t i o n u s e d for i n s t a n c e

we call K/~ the i m a 6 e

wide image

~R

Inst D / ( F u n ~

~R

(We are r e s t r i c t i n g

to i n t r o d u c e

club-congruence,

faithful,

in the w e a k e s t

case.)

Lane in [ 7 ] ,

is c o n v e n i e n t

in

functor

f r o m the g e n e r a t o r s

To r e l a t e Kelly-Mac

in [15]

O

K, w i t h its a u g m e n t a t i o n ,

as a K - a l g e b r a .

for CCM"

Lane

(B,p,D,e),

of course k n o w the free K - p o l y c a t e g o r y structure

the

F: K + S °p* is f a i t h f u l ;

diagrams

however,

of the club

Theorem

([151,[16])

commute".

Inst D ÷ ~

sense,

theorem

Szabo calls

"Coherence

- comparison with Kelly-Mac

in d e t e r m i n i n g w h i c h

commutative

that what

in [l~ and the

The c o h e r e n c e

consists

cut-elimination

suspect

will turn out to be the a s s e r t i o n

-

is o b v i o u s ,

-

the first first)

is a quotient

is the identity

identity"

map,

-

the second is faithful

on objects;

on morphisms.

with their images

136

(like the

the third is faithful

We may identify

in wim a.

and

Clearly

the morphisms

and "the

of K/~ = Im a

the factorization

(5.2) is

functorlal. In the covarlant factorlzation

case,

let A be a K-algebra,

and apply

(5.2) to ¢ and to F: K ÷ ~ in the commutative

the

diagram

~(A,A} (5.3)

to get a diagram we shall write ¢I

¢2

(5.4) ~ Note further

as

r2

that if m: K ÷ L is a club-map,

via KoL ÷ LoL ÷ L; so all the objects K-algebras,

while

L becomes

a K-algebra

in (5.4) are not only clubs but

all the maps are not only club-maps

but maps

of

K-algebras. Passing now to the mlxed-variance cannot

copy all of this,

corresponding ~upposing

object

in closed since ~

to ~ is not a club.

for incompatible

of A and agree loops equal

case, we

since we have no {A,A} and since the ~

A to be non-empty)

composition

(but clubbable)

We can however

by adopting natural

to compose

some harmless

transformations:

by setting

to this object.

is at any rate a category;

write

N(A) definition

pick some

all arguments

We can write

we compose

of

occurring

the bottom

by ignoring

line

closed

-

loops.

We

can therefore

137

-

write

¢1

¢2

~N(A)

- F2

(5.5)

~

~

G

G

Here ~!! N(A) and ~ii G are Blearly clubs, namely quotient clubs of K.

The other categories

N(A) and G, while not clubs since not

simple, are at any rate K-algebras;

and all the maps are K-algebra

maps. ~ii N(A) is what, in §1.3 of the Introduction, we called ~(A),

and K is what we called the "universal ~!i".

We have changed

notation now to allow comparison with the formulation of Kelly-Mac Lane in §2 of [7]. G is what they called G, the "category of shapes and graphs"; ~(A) is what they called ~ = N(~), the "category of shapes and natural transformations"

(they had y for 4).

that is, closed categories

in the case of [7].

are what they called the subcategorles N(A) and in G respectively; sub-K-categories

These were K-algebras, ~i! N(A) and ~ii G

of allowable morphisms in

they were defined there as the smallest

containing all the objects, which clearly agrees with

their present definition. The results of [7] were finding the maps T ÷ S in ~I~ compatible,

G; (ii) that any two such maps were

so that the case is indeed a clubbable one;

was surjective, finally

(i) that there was an algorithm for

trivial for us because

(iii) that F 2

F 3 is a quotient map; and

(iv) that F2, when restricted to the full subcategory

A!~ N(A) determined by a certain subset ~ S, was faithful.

of

S of its set of objects

-

our terms,

In

when restricted functor have

~

-

the last is equivalent

to the full subcategory

FB, or equally

the functor

to the assertion

of K determined

r: K ÷ ~ ,

that,

by P ~

is faithful.

S, the

For we

:

Lemma 5.1 U

IS8

If u,v: T + S i_~n K and if $1 u = $1 v for all models

A then

Vl

Proof

Take the model A to be K itself,

map ~: KoK ~ K. precisely itself;

u(R1,

The

(R1,

between

..., Rn) - component

..., Rn) ; the

(~,

so $1 u = $1 v implies The club ~ K and ~

seen as a K-algebra of $1 u

=

..., ~) - component

N(A) is a quotient

club of K intermediate

to All G when A is the unit category

always

From our present

5.2

objects

of interest

The complete

solution

of all the determination (B,p).

This presents

otherwise

u

u = v° D

A ~ K; it reduces

the proper

lul is then

is therefore

G; we have Just seen that it is equal

a model).

via the

point

to K when 7 (which is

of view it is a red herring;

are K, and its image ~

of the coherence

problem requires

of the set g of objects

of course

no difficulty

leads to a w o r d - p r o b l e m

G under

first

of K from the data

if p is vacuous,

which may be anything

but

from trivial

to insoluble.

The next step might perhaps

as in |7]

Then it remains image

F-l~;

composition;

often be in practice

via an algorithm,

to determine

law of substitution

f(gl'

In one case these

of the image ~

for each ~: T ~ S in ~

to know K as a category and finally

the determination, G of F: K ÷ ~ . G the inverse

we must then determine

its law of

to know it as a club we must determine

its

"''' gn )" last two steps

are unnecessary,

namely when

-

F is faithful. finished stitute

In this

139

-

"all diagrams

commute"

case,

the Job is

when we know ~!~ G, for we know how to compose graphs,

The typical symmetric

and this forces

example

monoldal

composition

and substitution

of this case is Mac Lane's categories;

there

result

the category

strict

symmetric

5.3

between

monoidal

on a category

(B,p,D,~)

axioms

form p and e; typically

(5.6)

for is ~, and

of categories;

we have of course

that all interesting

or a polycategory Certainly

For

K = ~. extra

are given by data and

there may be axioms

not of the

we may have a theory

(B,p,D,~;~),

a model

of which is a model of (B,p,~,G)

not wrltable theory

categories

as in §3.

in K.

K and ~ is that K has more objects.

We do not mean to suggest

structures

[14]

of graphs

All G is all of ~, so that F: K + ~ is an equivalence the only difference

and to sub-

in the forms

in nature,

p and ~.

however,

satisfying

further

axioms

I have not yet come across

for which the coherence

problem

T

such a

differs

from that of (B,p,D,G).

One example enriched

is the problem

over a closed

this volume.

This

category,

of a natural

studied

is a 3-category

by Kelly-Mac

problem;

T iS the assertion

for the part

(B,p,D,~)

already

(where K i is the part of K with codomain-type coherence

problem

Again,

in

(5.6)

But the club K

has K 1 and K 2 discrete i), and there is no

other than that for all K-algebras.

consider

as given in Chapter inessentially,

is of the form

that A 1 and A 2 are discrete.

of the theory

Lane [8]

and if we write

A1, A2, A 3 for the A, A', Y of [8], the theory where

transformation

the theory

I §2 of [21.

to bring

of "closed First

categories

let us change

it closer to our present

without

@",

the theory

form, by dropping

-

what was

there

able by I.

called

consists

li = l, and f i n a l l y [2]

is e s s e n t i a l l y

(5.7)

together with

of the axioms

The map f ~-~

JI = iI"

K: A(A,B)

[l,f]JA,

In the c o h e r e n c e

(8,p,R,a)

want.

that

A of

(B,p,O,a),

assertions

÷ A(I,[A

in those

to ~ o p ,

aspects

related

topos.

It is clear

Free m o d e l s

closed

that

that

case,

closed

categories

the theory

K would

(5.7).

finite

limits

extension

from

(B,p,D,a).

of e l e m e n t a r y

It

topoi, and

then i t s e l f be an e l e m e n t a r y

K w o u l d be e n o r m o u s l y

We r e f e r p r i m a r i l y

to L a m b e k

[9]

and to Szabo

all c o r r e s p o n d

for

could also be so e x p r e s s e d ;

and Szabo

these

are

for this

with

of a s u i t a b l e

to L a m b e k

categories;

that there

problem

satisfies

the a p p r o p r i a t e

to Coherence,

categories,

gives what we

I doubt

to the c o h e r e n c e

in terms

club

but

(5.7)

problem

for c l o s e d - c a t e g o r i e s - w i t h o u t - @

K itself

after

to prove

to the c o h e r e n c e

- relation

|10] on b i c l o s e d

the e x t r a a x i o m ~:

we may want

if K is the club

to know w h e t h e r

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

6.1

Moreover,

suspect

whether

6.

we can make

can be e x p r e s s e d ,

w o u l d be i n t e r e s t i n g

as in

given by

the s o l u t i o n

of c a r t e s i a n

or r a t h e r

of i; a

with ir = 1 and

satisfying

for such c a t e g o r i e s ,

(5.7).

The t h e o r y

to ~ o p

B]),

by the s o l u t i o n

I strongly

and c o l i m i t s

together

(B,p,O,e)

l o o k e d h a r d at this

without

D consists

I to act as the i n v e r s e

may only give us Ku = Kv; but then

are not i m p l i e d

(B,p,~,a)

p is vacuous;

the

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

problem

I haven't

any c o h e r e n c e

[ , ] (replacing

Then a c l o s e d - c a t e g o r y - w i t h o u t - @

that u = v: A ÷ B in A, w h i l e for

types;

CC1 - CC4 of [2],

a model

is in any case r e p r e s e n t -

of two e l e m e n t s

and I, of a p p r o p r i a t e

of L, J, i as there,

-

the f u n c t o r V, w h i c h

Then B c o n s i s t s

( , ) of [2])

140

complicated.

on r e s i d u a t e d ([I~

categories

and Ch.I of [15])

to graphs

in ~ ;

and

on

I have yet to

-

in these

examine

which involves

terms

graphs

Szabo's

vacuous

A, but w i t h o u t When it comes

cases,

mention

(B,p,D,~)

and c o n s t r u c t of clubs,

result,

Dealing with

But then to d i s t i n g u i s h

different

"generalizations",

a coherence

result

them roughly

6.2

the graph,

we w o u l d

KoA r a t h e r

essentially

by

(3.3),

place

they have

to r e p l a c e

the c o n c e p t

where

of of

the v a r i a b l e s

to speak of these

as a d m i t t i n g

c: A@B ÷ B@A and l: A®B ÷ A@B;

as "u = v if Fu = Fv" b e c o m e s

as we c o n s t r u c t e d

but w i t h

the objects

then set up a formal

T,S E R.

are in e f f e c t

for

of KoA

T(=S b e c a u s e

p is vacuous)

of A as s t a r t i n g - p o i n t

They give axioms the maps

system

P whose

and rules

formulae

of i n f e r e n c e ;

f: A ÷ B of A, the i n s t a n c e s

d(T1,

the d E 0 w i t h T i E R, and i: T ÷ T for T E R; the rules are in effect,

in

B(T l, (appropriately

are T ÷ S the axioms ..., T n) of of i n f e r e n c e

for B E B,

T1 ÷ S1

(6.1)

to e x p r e s s

than K, they have

c, l: A®A ÷ A@A,

state

of graphs.

of our Ii.

They where

they have

KoA on

as in c: A@B ÷ B@A i n s t e a d

serves

namely

mention

the set R = So Q~ A of o b j e c t s

inductively, and

the free m o d e l

"u = v if u and v are e q u i - g e n e r a l " .

They c o n s t r u c t

(3.2)

directly

therefore,

f r o m A in t h e i r m o r p h i s m s ,

no l o n g e r i n d i c a t e

categories,

as in §3, w i t h p in fact

and i n d e e d w i t h o u t

c: @ ÷ @, and this to some e x t e n t graph.

closed

O

to a c o h e r e n c e

it in o t h e r terms. variables

w o r k on c a r t e s i a n

start w i t h a theory

in their

-

in S °p*. =

They

141

altered

T2 ÷ ..., T n)

S2 ÷

• ..

B(S 1 . . . . , S n)

when B is of m i x e d

T ÷ R

R ÷ S T÷S

Tn ÷ Sn

variance),

and the c u t - r u l e

-

(I

say

"in e f f e c t "

because

142

-

they may give an a d J u n c t i o n

A@B ÷ C

directly

by

A ÷ [B,C] and

A + [B,C] rather

than,

it comes P(T,S)

by its unit

to the same thing.)

equivalence

relation

6(T,S)

objects. that

as we would,

of all proofs

P(T,S)/

A@B ÷ C

and form the q u o t i e n t

"imposed"

composition

a category,

relations

that

the same

with

~ are verified;

cut.

category

clubbable

case;

in f o r m i n g D - so that

effective

proof-theory borrowed

would

What

to assert

is that

[4],

that

the

composition

to assert

that

that

in A, and relation

this yields

of §4.3 in the

non-

they a l l o w the cut a l r e a d y

B has to be more on ~

more

~

then,

9.

there

is

of T + S, for in

"complicated"

cut-eliminatlon

a trick

~

~

complicated

of i n f e r e n c e ,

the p r o o f s

than T or S.

technique

that K e l l y - M a c

the a c c o m p a n y i n g

and Szabo do,

and that

that

be in the case A = I our ~ R

way of e n u m e r a t i n g

however,

to assert

IT; it has

cut is one of the r u l e s

the

KoA on A.

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

+ ~W~ that we impose

due to G e n t z e n

Lambek

it has

is their c o m p o s i t e

that

relation

this by e x t e n d i n g

- without,

it has

really

the t e r m R may be a r b i t r a r i l y overcome

an

and show

free m o d e l

or even as our K[A]

otherwise

So long as the no o b v i o u s

this;

identities

the m a i n d i f f e r e n c e

+ ~

is the

It is not h a r d to see d i r e c t l y

their e q u i v a l e n c e

than the F~ng~

They

with

as our KoA,

P - which

Just

the cut,

the d ~ D are natural,

1 T is 1 A if T 6 A; and finally

compatible

(6.1)

and i n d e e d

in P/8 of f,g 6 A

the c o m p o s i t e

in P/6 by u s i n g

to a c h i e v e

functorial,

they d e s c r i b e

P/B is a p r e c a t e g o r y w i t h R as

Thus

via the cut is a s s o c i a t i v e

that

on P(T,S),

of 9; but

for T, S E R the set

~(T,S)

~ is c h o s e n

the B E B are

consider as trees;

They d e f i n e

The r e l a t i o n

They now

as e l e m e n t s

of T ÷ S, w r i t t e n

= P/~(T,S).

it is a c t u a l l y

and counit

Lane g r a t e f u l l y

proof-theoretic

is to set up,

in

still

language.

with T ÷ S

-

as formulae,

-

a new formal system Q, whose

and whose rules

of inference

inference

in Q, however,

are parts

of the terms

rule of inference effective

143

Assigning

are theorems

are derived rules in P.

are such that the terms

in the thesis

of the set Q(T,S)

they get a function

Q(T,S)

~ P(T,S)

the cut is not a

cases this allows

of proofs

of Q a particular

of

in the hypotheses

an

of T + S in Q.

to each axiom of Q a particular

and to each rule of inference

of P,

The rules

- in particular

in Q - and in favourable

enumeration

axioms

proof of it in P,

derivation

called expansion;

of it in P,

and they prove

that the composite (6.2)

~(T,S)

is surJective. the kernel

of (KoA)(T,S)

that everything

Q(A) etc., relation

the definition

= P(T,S)/~(T,S);

now only awaits

e(T,S)

on Q(A)(T,S)

called equi-generality. this:

An

f,g 6 Q(A)(T,S)

Except

in [17],

are equi-general

is a g' ~ Q(A')(T',S')

Q(a) g' = g.

Because

not necessary

to take all A' here,

of the nature

n~ of the "arrow" map 0 ÷ 1).

number of n's,

a new

~: A' ~ A, and for all T', S' 6 Q(A') with

f' = f if and only if there

non-identlty

in

in A, so that

Q(~) T' = T and Q(a) S' = S, there is an f' E Q(A')(T',S')

coproducts

is

an effective

above is functorial

Lambek and Szabo define

is essentially

if, for all functors

Q(m)

which

of y(T,S).

Observing

equivalence

relation

KoA = P/8 is also given by ~/y.

free model

determination

we can write

for the equivalence

they now have Q(T,S)/y(T,S)

the desired

determination

~ P(T,S)/~(T,S)

Writing y(T,S)

of (6.2),

other words effective

+ P(T,S)

so that

of the deductive

with

with

system Q, it is

but only those which are finite

category

~ (two objects

0, I and one

In fact given T,S we need only try a finite

e(T,S)

It is easy to see that

is a finitely-decidable f ~ g(y)

implies

relation.

f ~ g(e).

The

- 144

-

is shown by Lambek and Szabo (see e.g. Lemma on p.l17 of [I0])

converse

to be equivalent

to the following, which we state for simplicity in

our own language, using the fact that Q/y = KoA: there is at most one map P[A1, Ai E

..., Ap] * Q[Ap+I,

n~, if AI,

= y

..., Ap+q] i n K ° n~, where P, Q E K ~ d

..., Ap+q are all different.

and KoA is calculated effectively as Q/e. That 7 = E is asserted in [9]

(Coherence theorem for BMM, p.l16); later corrected in [17].

Ff = Fg, and if hl, by an appropriate

also in [15], but in this case

is faithful;

choice of distinct Ai E

..., Ap+q].

form above, it in

for if f, g: P + Q with

..., h n are the non-identity maps of

..., h n] and g[hl,

Q[Ap+I,

(Proposition 5) and in [i0]

In view of its equivalent

fact says exactly that F: K ÷ ~

f[hl,

When this is the case,

n~, we get

n~ the two maps

..., h n] from P[A1,

..., Ap] to

Thus in [10] and [15]this result is false, since

F is certainly not faithful for the theories of biclosed or of closed categories.

In [17] Szabo has

modified

[15] by replacing E by a

finer relation e', which in that context he still calls "equl-generality",

and

by

asserting

that

7 =

¢'.

Another result given in each of the above references,

under

various names, is that the canonical functor A ÷ KoA is full and faithful. l[f].

In our language,

this functor sends A to I[A] and f to

It is certainly faithful since l[f] # fig] for f # g.

It is

full precisely when there is no non-ldentity map h: I ~ i in K. ~h is necessarily endomorphisms

Since

i, this is so when r, restricted to the

of ~ in K, is faithful;

it is certainly so for closed

categories by the results of [ 7 ], since i is a proper shape. 7. 7.1

Non-strict mor~hlsms of K-categories We Just amplify a little here the indications

given in §1.4;

we hope at another time to study some of the problems suggested by

-

145

-

the concept. Write ~

for the 2-category whose objects are functors

~: A ÷ A', whose morphisms

$ ÷ ~ are 2-cells in ~

of the form

e

A

~B

A'

~--~B' e,

with the obvious composition,

and whose 2-cells(e,eJe) ~ (X,X;8) are

pairs of natural transformations commutative

y: 0 ~ X and 7': e. ~ x' rendering

the following cylindrical diagram of 2-cells: X A

B @

A'

:t

There is an obvious 2-functor .: ~ / ~ object

(K,$) to Ko$:

KoA

÷

x ~

KoA', sending the m o r p h l s m

(poe, poe,, po~),and sending the 2-cell

÷ ~

sending the

(p,(e,e,,a))

to

(~,(~,7')) to (eoy, ~oy').

This 2-functor satisfies the axioms for an action of ~ / ~

on ~gB, so

that we have Lm(K.$) = (LoK),$, etc. Now let K be a club in ~ / ~ . 2-monad) on ~ ;

Then K=- is a monad

(indeed a

an algebra for this monad is a functor $: A ÷ A'

together with an action K-$ ÷ $; that is, a 2-cell

-

146

-

KoA

~

KoA '

~

A

A

e ,

satisfying the axioms for an action. be equivalent

to the following:

into K-categories,

These axioms are easily seen to

e and e' are actions making A and A'

while the components

¢(T[AI,

..., An]) satisfy

(besides naturallty in T and the A i)

(7.1)

¢(T(S1...Sn)[A1...A m ] )

= ~(T[Sl(A l...Aml) ..... Sn(...A m ).T(~(SI[A l...Aml]), .... ~(Sn[...Am])) (7.2)

~(l[A]) = I. We call an algebra for K~- a non-strict map of K-categories;

it consists of K-categorles satisfying and relations

A and A', a functor ¢: A ÷ A'and a natural

(7.1) and (7.2). (B,p,~,~)

it is clear from (7.1) and (7.2) that ~ is

determined by the ¢(B[AI,

..., An]) for B 6 B, and further that these

can be chosen independently

if p is vacuous.

(7.1) to assert the naturality !~g~ 9.

It suffices moreover by

of ¢ in T only for maps T + T' in

Thus we get, for instance, when K is the club for symmetric

monoidal categories, ¢(@[A,B])

When K is given in terms of generators

the usual concept of a symmetric monoldal functor;

is a natural transformation

¢: CA@'¢B + ¢(A@B) and ~(I[ ])

is ¢o: I' ÷ ¢I; asserting the naturallty gives the usual three axioms.

of ~ for instances

of a,r,c

-

147

-

REFERENCE S

[ 1]

S.Eilenberg and G.M. Kelly, A generalization of the functorial calculus, J. Algebra 3(1966), 366-375.

[2]

S.Eilenberg and G.M. Kelly, Closed Categories, in: Proc. Conf. on Cate6orical A16ebra~ La Jolla~ 1965 (Springer-Verlag, 1966), 421-562.

[ 3]

D.B.A. Epstein, Functors between tensored categories, Invent. Math. 1(1966), 221-228.

[4]

G.Gentzen, Untersuchungen Gber das logische Schliessen I, II, Math. Z. 39(1934-1935), 176-210 and 405-431.

[5]

G.M. Kelly, Many-variable functorial calculus I. (in this volume).

[6]

G.M. Kelly, A cut-ellminatlon theorem.

[7]

G.M. Kelly and S. Mac Lane, Coherence in closed categories, J. Pure and Applied Algebra 1(1971), 97-140.

[8]

G.M. Kelly and S. Mac Lane, Closed coherence for a natural transformation. (in this volume).

[9]

J. Lambek, Deductive systems and categories I. Syntactic calculus and residuated categories, Math. Systems Theory 2(1968), 287-318.

[ I0]

J. Lambek, Deductive systems and categories II. Standard constructions and closed categories, Lecture Notes in Mathematics 86(1969), 76-122.

[ ll]

F.W. Lawvere, Ordinal sums and equational doctrines, Lecture Notes in Mathematics 80(1969), 141-155.

[ 12]

G. Lewis, Coherence for a closed functor. (in this volume).

[13]

J.L. Mac Donald, Coherence of adJoints, associatlvitles, and identities, Arch. Math. 19(1968), 398-401.

[14]

S. Mac Lane, Natural associativity and commutativity, Rice University Studies 49(1963), 28-46.

[15]

M.E. Szabo, Proof-theoretical investigations in categorical algebra (Ph.D. Thesis, McGill Univ., 1970).

[16]

M.E. Szabo, A categorical equivalence of proofs (to appear).

[ 17]

M.E. Szabo, The logic of closed categories (to appear).

(in this volume).

COHERENCE

FOR A CLOSED FUNCTOR

Geoffrey

The University

Lewis

of N e w South Wales,

Kensington

2033, Australia.

Received M a y 22, 1972

§0 I n t r o d u c t i o n

In the first p u b l i s h e d work on coherence, coherence

in tensored

and without

categories

symmetry.

functor between

with and without

In [2], Epstein

two symmetric monoidal

Kelly and Mac Lane in [5] examined

We consider the structure

coherence

identities,

showed coherence categories

coherence

and with

for a monoidal

without

in a closed

identities. category.

consistin~ of two closed

categories and a closed functor between them. is to prove partial

[6], Mac Lane showed

results

The object of this p a p e r

as was done in [5]

for a closed

category.

As t h e r ~ the purely symmetric monoidal elimination partial

covarlant

categories

and a monoidal

theorem is proved,

coherence

results.

case is treated functor.

However the situation

commute";

abelian groups

situation

is worse

as in [4].

when

proof The

of the club.

than in [5] because

it is no longer the case that

for example,

to sets,

p r o o f of

At the same time the c u t - e l l m i n a t i o n

result will then be a partial d e t e r m i n a t i o n

diagrams

two

Then a cut-

which leads to an inductive

will also yield a proof that the theory is clubbable

the purely monoidal

first:

~ is the forgetful

the following diagram:

even in "all

functor from

-

149

-

¢I

I' @' ¢I

¢I @' I'

¢ o @ ~

~l@, ¢I

fails

to commute.

In this p u r e l y

case we c o m p l e t e l y sider,

part

The part

of E w i t h

object

of H* is

of E with

permutations,

monoidal

map,

f is

discrete

B is

equivalent

and a is a (p,u) p',

(~, h,e);

constitute

We thus have

u', where

to the c a t e g o r y

r and A are J o i n t l y

faithful,

There

The

H*.

An

20. x: n + p

are m o r p h i s m s

only when n = n',

u = u'

The p e r m u t a t i o n s

Ff in the sense

r: H* ÷ ~ = ~oA × A.

is not

integers

and x'~ = hx.

the g r a p h

skeletal

to ~.

~: n ÷ n and 8: u ÷ u are

of f, that we S h a l l

~ is the

we con-

with two o b j e c t s

equivalent

- shuffle.

x', a')

A: H* ÷ 8, w h e r e

category

of K e l l y

[3];

h

call Af. We also have a n e w f u n c t o r of finite

faithful,

i.e.

sets.

While

F

f = g iff Ff = rg

Af = Ag.

When we come a graph

it take

to the club N for the

function

its value

w h i c h bears

the

F, but we depart

in a m o d i f i c a t i o n

same r e l a t i o n

A: H* ÷ ~ to A: N + D. are J o i n t l y club

category

type A is

h: p + p' is a function,

and 8 t o g e t h e r

have

Like all the clubs

(n, p, u, x, e); where n, p, u are

is a new c h a r a c t e r i s t i c

and

E.

codomain

type

(n, p, u, x, ~) to ~ ' ,

Such a m o r p h i s m

¢I

the club

over A~the

codomain

is an i n c r e a s i n g from

describe

E is a c a t e g o r y

A and B.

@'

¢o

N.

We c o n t e n t

faithful proper

faithful:

functor,

§4. 2-3 of [4]

as G does

to ~, and we e x t e n d

An object

a complete

with proving

to the full

by l e t t i n g

We define

we do not a t t e m p t

ourselves

we still

G of ~ .

In the case of N, it is not

when r e s t r i c t e d

objects.

to ~

from

closed

that

subcategory

is c a l l e d p r o p e r

a category

D

true that

r and A

description

of the

r and A are j o i n t l y of N d e f i n e d

by its

if in its f o r m a t i o n

no

-

[T,S]

is i n v o l v e d ,

or T c o n t a i n s IT,S]

where

either

150

-

T is n o n - c o n s t a n t

¢ and S does not.

(It is the

and S is constant,

same t h i n g to say that

is n e v e r used w i t h FT # 0 and FS = 0, or AT # 0 and AS = 0).

For s i m p l i c i t y categories identity

(i.e.

monoidal

strict

and i d e n t i t y

and to f a c i l i t a t e

symmetric

monoidal

Isomorphisms

this we first

are

consider

strict

categories.

§l D e t e r m i n a t i o n monoidal

first w i t h

all a s s o c i a t i v i t y

morphisms),

non-symmetric

we deal

of the club

in the strict n o n - s y m m e t r i c

case

We seek the club

F, in the sense

of E4|

for the s y s t e m g i v e n

by:

Two strict n o n - s y m m e t r i c V ~ {A, ®, I, a, ~, r} and monoidal

functor

FA and

2, or, as we shall identified

V' = {A', ®',

¢ = {¢,¢,¢o}:

We write

monoidal

categories I', a', ~',

r'},

and a

of F of c o d o m a i n

types

Type A o b j e c t s

can be

V ÷ V', as in |lJ.

FB for the p a r t s

say, of types A and B.

w i t h the n o n - n e g a t i v e

integers.

Type B o b j e c t s

i and

are of the

form:

C I @' C 2 ®' where

C i is e i t h e r

l' or Cn i.

Type A m o r p h i s m s by e x p a n d e d

instances

are

|NB:

In e x p a n d e d

If m -- 0, the object

l: n ÷ n.

is w r i t t e n

Type B m o r p h i s m s

of

¢: Cn @' S m ÷ ~O:

... 8' C m

¢(n + m),

and

I' ÷ ¢0.

instances

of ¢o

I' is u n d e r s t o o d ,

1 ®' ~o ®, i: __I' @' ¢n ~ l' @' ¢0 @' Cn|

e.g

I'.

are g e n e r a t e d

- 151

The generators

satisfy

functorial

-

relations:

I ®' f @' I > D 1 @' E 2 @' D 3 ®' D 4 8' D 5

D 1 @' D 2 @' D 3 @' D 4 @' D 5 l

i ®' g @' i

I

J

~i

4/

®' g @' I

%

D 1 ~' D 2 @' D 3 ®' E 4 @' D 5

> D I ®' E 2 @' D 3 @' E 4 @'

D5;

i 8' f @' I naturality

relations

(vacuous

in this

club);

and the given relations:

D @' Cn @' Cm @' Cp @' D ®' ¢(n+m)

®' Cp @' E

D ®' Cn ®' ¢(m+p)

D @' ¢(n+m+p)

i~'

,o~ ~1//"

®'E

|

D 8' ~0 @' ~n @' ~ E

®' E

._. ' ~°e'

I i

~ D

i

@' ¢n @' ¢0 ®' E

D @' ~n @' E We shall describe f

n or n

~

p

a

a category

H.

The objects

of H are either

~ u, where n,p,u are non-negatlve

f:[n]

+ [p]

and a: [p]

÷ [u]

empty

set, and [n] is the set

we shall omit the square

are increasing {1, 2,

brackets

integers,

functions.

[0]

..., n} for n > 0.

in this

context.

and

is the

Henceforth

The morphisms

of

H are i: n ÷ n and

~- p 'l h

n

~. q

rl g where h is increasing, defined

"~u -I1

(sometimes

(1,h,l))

> u b

such that the diagram

commutes.

in the obvious way so that (I, k, I).

written

(i, h, i) = (I, kh, i)

Composition

is

-

F is isomorphis

152

-

Theorem

i.I:

to H

Proof:

Let B be the category with the type B objects of F as objects,

and such that B is the free category on generators of ¢ and ¢o. congruence

expanded instances

Denote the relations on these generators in F by p; the Then FB = B/[p].

thus generated by [p].

We want to show

that B/[p] = HB, where H B is the subcategory of H without the objects n, and the morphisms B' = B

I: n ÷ n.

We shall exhibit H B as B'/[p']

where

and p' = p. Consider the object D of B:

Cn I @' ... @' Cnrl @' -l' @' Cnrl +

i

@' ... @' ¢nr2 @ !' @ "'"

... ® i' @' Cnru_l +I ®' ... @' Cnru P 7 n i. Define f: n ÷ p by f(i) = j if i=i n I + ... + nj_ 1 < i < n I + ...+nj. Define a: p ÷ u by a(i) = J if Let p = r u and n =

rj_ I < i < rj.

Then there is a biJection between the objects of B

and H B given by the correspondence

between D and n ~ p ~ u.

Let ~i: P ÷ p-I be the increasing surJection which takes the value i twice.

Let 6i:p ÷ p+l be the increasing injection which fails

to take the value i. following morphisms

Denote by i

> P

11

b(~i ~ u

~o i

n

respectively,

the

in HB:

f n

f,b and ~ , b '

> p-i ~±f

Ii > b

f n ------~> p and

II~

u

b6 i > u

~±

n

~I

~ p+l ~if

~ u b

Let B' have the same objects as HB, and be the free category on generators i f,b and ~if,b"

i Identify D ®' Cn i @' Cni+ I @' E with ~f,b

~

I

@'

~

@'

i

D @' ¢(n i + ni+ l) @' E

-

153

-

D ~' E

and

with ~i f,b"

I

i @' ¢o @, 1

D @' ¢0 @' E This is a bljection between the generators

of B and of B'.

The

isomorphism between B and B' sends p to the isomorphic relations B'.

The problem reduces to showing B'/[p']

is that the relations

p' in

= H B. The point of the proof

p' have the same form as the relations

x below.

Define a functor K: B' ÷ H B by the biJections Just described on the objects and generators. n >_ 0, and morphlsms

Let A be the category with objects

increasing maps. f

F(n

Define a functor F: H B + _~ by

a > p

~u) = p,

f

a

g

b

It Is well known that _A has generators and relations

and

i . p ÷ p - I and ~ip p + p + 1 Cp.

x:

i i = Bp_l~ p

i J+l ap+ 1 6p

J > i

i+l ~p+l ~

J < i

1

J = i

i = -1 Gp

i J+l Op-1 ~p

J ~ i

i = 6~÷i ~p

i B~-I ~p+l

J m I

Let C be the category with objects n ~ 0, and morphisms generated by ~pi and Bp. i

freely

Let L: C ÷ A be the functor which is the

identity on objects and generators.

There is a functor G:

B'

÷

which is the identity on objects and such that G ( ~ , b) = G~ and G(~,b)z = 61" P

We have FK = LG.

C

-

We

onto,

154

k n o w that ker L = [T].

-

We are r e q u i r e d

that K is

and that k e r K = [p'].

Choose

a morphism,

~, of HB: f

bh

n

~p

n

> q

~u

i

)u.

hf Since h E 8, h

may be w r i t t e n hI

P = Po .......... > for g e n e r a t o r s

h i of ~.

-

h

h2 Pl

m

> P2

"'"

But ~ is the bh m ~ p

-

hlf

>Pm=q

composite:

... h I = bh >

bh m • . .

n -

(*)

b

as

f n

u

h2

\

~" Pl

~ u

~1

... 11 n -

n I~

h2hl f

Pm-1

hm

~ hm

" ' h.l f

h3 "

~ u

bh m >

Suppose

u

b

~I

>Pm

So K is c l e a r l y

FKf -- FKg.

~h2 bhm > P2 --

hm_ 1 ... h l f >

n

~ u

onto.

f = g [p'].

T h e n Gf -- Gg [T]. So LGf = LGg,

But F is faithful, so Kf = Kg.

If Kf = Kg,

then FKf = FKg,

But by the d e c o m p o s i t i o n

§2 D e t e r m i n a t i o n _

case

to prove

in

(*),

Thus

ker K D [ p ' ] .

so LGf = LGg,

f = g [p'].

of the club in the .,

thus

and Gf = Gg [~].

Thus ker K = [p'].

strict

symmetric

QED.

monoldal

-

We seek to describe strict symmetric monoldal the full subcategory

1 5 5

-

the club F* for a monoldal

categories.

of F* determined

We first describe

objects are Cn I @' Cn 2 ®' ... ®' Cnp.

(For convenience we shall omlt

are ~: n ÷ n where ~ 6 S(n), the permutation

Type A morphlsms

group on n elements.

Type B morphisnmof

involve i'" ~

F whose domain and range do not ^

(2)

Cn~l @ ... @ Cn~p

(3)

¢~: Cn + Cn, where

The relations

> Cn I @ ... ® Cnp, where ~ 6 S(p); ~ e S(n);

are: Functorlal: l@f@l > A®B'

A@B@C@D@E l®g®l

@C®D®E

~

l@g@l

1

A @ B @ C @ D' @ E

(a)

) A @ B' @ C @ D' @ E

l@f@l

Natural: 1~¢~¢n

~1

>

A @ Cn @ Cm ® B

A @ ¢n ® ¢m @ B 1®0®1

A ® ¢(n÷m)

> A @ ¢(n+m) @ B

® B

1 ® ¢(2(~,n)) (b)

Type

are generated by:

(1)

(2)

i'

integers n, and type B

the prime and write thls as ¢n I @ Cn 2 @ ... @ Cnp).

(I)

F', which Is

by the objects not involving

Type A objects of F' are non-negatlve

B morphlsms

functor between

@ 1

~(IAI,'--,IAp) A~I @ ... @ A~p

.) A 1 @ ... ® Ap fl ® """ @ fp

~ f~l ® "'" @ f~p

B~I ® ... ® B~p

^

~( IB 1 ,.. "'iBp )

>

B1 @ ... @ Bp

-

(3) (a)

156

-

Given relations:

If h, k are composable in F, then the composite in F' is the

composite in F; similarly for identities. ^^

(b)

=

(o)

¢(~n)

~n

= ¢~.¢n ^

I @ T @ l ->

A @ Cn @ Cm @ B

(d)

A @ ¢(n+m) ® B

A @ Cm@¢n @ B

> A @ ¢(n+m) @ B

~" 1 @ ¢(T(n,m)) @ 1

where T is the non-ldentlty F~ is isomorphic n # m, ~ (n,n) = S(n)).

element of S(2).

to the category ~. (Objects n, ~ (n,m) = ~ if We shall show that F~ is isomorphic to the

category H' The objects of H' are increasing maps f: n ÷ p.

A morphism of

H' is a commutative diagram f >p

n

>q

n

g

where ~ 6 S(n), and h is any function. (~,h).

We sometimes write this as

The composite with (~', h') is (~'~, h'h). We shall now study the morphisms

f: n + p, and a permutation n (in the notation of [3]

n(l an_ll,

Given an increasing

n: P ÷ P we write as n f the permutation of

§2):

..., i an_Ip): P(al,

where a i = f-li.

of H'.

..., ap) ÷ p(a~]_ll, .... a11_ip )

In (i) h may be written as kn where n E S(p) and

-

k E _~ (p,q).

157

-

Then (1) may be written as : f n

-> p

nr.(nf) -i n

)

p

(2) kaf.

(nf) -t

n

>q

~(nf)-i I

11 krlf~ -1

= g ~q

n

This is a composite of three morphlsms of Hi and is uniquely determined by n and k.

We examine what happens when h may also be written as

k' n' for n' 6 S(p), k' E ~ (p,q). Let q* = n' n -I.

Thus k = k'n*. Suppose i < J. If kl < kJ,

then k'q*i < k'n*J, so n*i < n*J because k' is increasing.

If kl = kJ,

then k'n*i = k'n*J, so n*i may be either less than or greater than q*J.

It follows that q* = lq(nl,

..., qq): p ÷ p where qi 6 S(k-l(i)).

For any I E p, k'i = (k'(n*)-l(n*))i = k(n*i) = ki.

Thus k = k'.

We define @ in H: : nl

÷ Pl ) @ ( f 2 : n 2

÷ P2 ) = 2(fl'f2):

fl

2(nl'n2) + 2(PI'P2)

f2 2!nl,n2)

= ],2(~i,~] 2 ) 1

~ql

/

2

gl

~q2 / g2

2(nl,n2)

2(f ,f ) I 2~ 2(Pl,P2)

]12 (hl,h2) ....~ 2(ql,q 2) 2(gl,g 2 )

® is strictly associative. By the decomposition

(2) we note that there are three types of

generators of morphlsms of H':

- 158

f type (C):

n

-

> p

(3) n

nf.

(nf)-I) p f

type

(D):

n .

~ p

(4)

> q

n hf f

type

n

(E):

>p

(5) n -

>p f

where ~ may be written as A

I(~i , ..., ~p): P(al, where a i = f-l(i), Clearly (l,hl).

..., ap) ÷ p(a I, .°., ap)

and ~i E S(ai).

(nlg,nl).(~f,n)

(l,h) = (l,hlh);

= (~inf,nl n) where g = nf(~f)-l;

and (~', I).(~,i)

= (~'~,I).

We now prove: Lemma 2.1:

l_~f s is of type (w), and 8 is of type

exists m' of type (*), and 8' of type

(%), then there

(%), such that s8 = 8's', where

6

Note [3]

(I)

m is

C,

(2)

w is C, % is E

(3)

" is

D,

% is

% is

D

E

We extend the notations

i(~I,

..., ~p) and n(l, i, ..., I) of

§2 in an obvious way to the case where

longer merely permutations.

~,n are functions

and no

4~

r~ v

II

v

v

x~

i.-i

o21

i

t__l

v

Vr

v

v

v

0:1

f

v

F..a

i-a

%

I...1

i..a

i-i

,.0

..I

II

~=~

--I

I-i °

II

I-I

I

i

I 0 SO v

II

I

C"

,M v

v

II

0

,-I

. •

•

I-I

~

v

V

: :

V

g

">,.~

r~

v

r~

H

v

¢XZ

II a~

v

-

161

-

P(fl, "''' fp) P(nl,

...,

) p(l, ..., i)

np)

11(~ 1, ...

~p) P(fl' "''' fp)

~ I > p(l, ..., i)

P(nl~ ..., npl

(q(Pl .... ,pq))(l, ..., i) h = h(l, ..., I) =

l

(l(h I .... ,hq))(l,...,l) (l(h I ..... hq))(f I, .... fp) $ ) (q(l .... , I))(i, ..., i) (q(Pl,...,Pq))(nl,.-.,n p) l(f I, ..., fp)

P(nl, ..., np)

)

1

p(l, ..., i)

(q(pl,...,pq))(l,...,l) (l(hl,...,hq))(f I .... fp)

(q(pl,...,pq))(nl,...,n p)

i (l(hl'''hq))(l''*l)

> (q(l, ..., i))(I, ...,I)

q(Pl(nll, ..., nlp I) .....,pq(nql,...,nqpq))

~

l(l(nll, .... nlPl),...,(l(nql,...,nqpq))

1

q(pl(nll,. • .,nlp I ), .... pq(nql,.. .,nqpq) ) I= (l(hl'''''hq))(fl'''''fP) (q(pl,...,pq))(nl,...,np) >(q(l, ..., I))(I, .... I) = 8'a'; where nil , ..., nlPl, ..., nql , ..., nqpq Is a relabelllng of nl, ..., np, and slmllarly for n.

-

We n o w d e f i n e M(¢n)

M(h)

-

M:

F~ ÷

H':

where

h E

F

a functor

n ÷ I;

=

M(A@B)

162

= M(A)

--

n

@ M(B); f

> p

n

>

q

Hf f corresponds

)

to n

n

p

~% i

in H;

>I

)q

Hf = p C n ~ l , ....

M(~)

~(1 P(nl, M(¢~)

n

> i

MN = i, N M = I.

; and

of the m o r p h l s m s

)

the r e l a t i o n s

of F~.

a f u n c t o r N: H' ÷ F~.

We shall

@ ... @ ¢(f-lp);

® ...

) = the m o r p h i s m

(2), and

see later that

__ ¢(f-I~i ) ® ... ® ¢(f-1~p)

¢(f-11)

(i),

Thus M is a functor.

Let

n ÷ p) = ¢(f-ll)

(4)

>p

= n

be seen that M p r e s e r v e s

We define

N(

>P

= Mf @ Mg.

on the g e n e r a t o r s

N((3)

V n

I)~

..., np) > 1

It can e a s i l y

N(f:

.....

-- n

M(f@g)

n~p)

® ¢(f-lp); in F c o r r e s p o n d i n g

to

(3)

- 163

n

>p

ii n

>

-

>I

I q

1

- - ~

in H; and

i

hf

N((5)

) = $a I @ ... @ Cap

I

¢~1 ® "'" ® @~p

Sa I @ ... @ Cap Since any morphism N(

(I)

(I) of H' can be written in the form (2), we define

) = N(~(nf)-l,l).N(1,k).

N(nf,n).

It is necessary

to show

that N is well defined. We suppose that h may be written kn'.

We have seen that

n m = n,n -I can be written as lq(n I, ..., nq): q(k-ll,

..., k-lq)--*q(k-ll,

Denote n'f(n'f) -I by E-

Then

..., k-lq).

g N/

) p

n

II -.

\

>!

(where d is the map which makes the diagram commute)

d

¢(g-lnml)

~

... ® ¢(g-lnmp)

n" ~ n I ® ... e ¢(~-11) ® ... ® ¢(g-lp)

nq (where k = N(l,k))

¢(d-11)

® ...

® ¢(d-lq)

-

¢(g-ln*l) ® . . .

O

¢(d- I)

¢(d-lq)

...

®

®

¢(nl(l, $(d-11)

-

¢(g'n*p)

...,

...

O

164

I))

® ...

® ¢(nq(l,

...,

I))

¢(d-lg)

®

(by relations (I) and (3d))

N/

n

,

>p

I~

kg

~k

n

>q

n

> q

Thus N({(n'f) -I, i).

N(l,k).

N(n,f,n ,)

: N(~(n'f) -I, I).

N(1,k).

N((n*)g,n*).

= N(~(n'f) -I, i).

N((n*) g, I).

= N(~(nf)-ll).

N(l,k).

N(l,k).

N(nf,n) N(nf,n)

N(nf,n)

So N is well defined. To show N is a functor it now suffices to show that N preserves composition.

We use the following lemmas.

Lemma 2.2.

If m and a' are composable ~enerators of tFpe (*)

of the morphisms of H', then m'a is of tFpe (*), and N(m'a) = N(m') N(m), where * is (1) C, (2) D, (3) E. Proof:

(i)

Let g = ~f(~f)-l.

Then

g N

f

g

n >p

N

f

~f

~ n

>

-- N

(n~)

n~ n

>p

- 165

by relation

-

(3b). hf

f = N n

khf by relation

hf

khf

(3a).

(3)

f

f = N

n

>

n

>

f by relation

(3c).

Lemma 2.3: Proof

>

f QED.

In Lemma 2.1, N(a)N(8)

(I)

By relations

(2)

By relation

(3)

By relations

= N(a')N(8')

(3a) and (3d). (2b). (1), (2a) and (3a).

QED.

Let a = e 6 7 and a' = e' 6' y' be two composable morphisms H'; where y and

y' are of type (C), 6 a n d 6' are of type (D), and e

and e' are of type

(E).

Then

~' y' E 6 y

=

E'

=

c' 6' c I Yl

=

E' ~2 61 Yl 6 V

by Lemma 2.1(3)

=

¢' s2 61 62 V2 y

by Lemma 2.1 (I)

by Lemma 2.1(2)

6 V

where Y1 and Y2 are of type are of type

of

(E).

(C); 61 and 6 2 are of type (D), c I and E 2

But

N(e') N(a) =

N(c'

=

N(¢')

6' y') N(~')

N(c

~ y)

N(y')

N(~)

N(6)

N(y)

- 166 =

N(E') N(~2) N(BI) N(62) N(Y2) N(y)

=

N(c'~ 2) N(~IB 2) N(Y2y)

=

N(~'~)

by Lemma 2.3

by Lemma 2.2

It is easy

We have thus shown that N: H' ÷ F~ is a functor. to check that NM = IF~ and MN = IH,. Theorem 2.4.

Thus we have proved:

H' is isomorphic, to F~.

We now consider the whole club, F*, i.e. we now admit the type B object i'.

Clearly

F~ is isomorphic to ~.

The objects of F~ are C l @ ... @ Cm where C i is either i' or Cn i.

(For convenience we write i' as i).

morphisms

The generators

of the

of F~ are:

(I) Expanded instances

of morphlsms

of F~; and

^

(2) ~: C~l ® ... ® C~m ÷ C 1 ® ... ® Cm where ~

S(m).

Relations are: (I) Functorial,

as relation

(2) Naturality,as

relation

(1) for F'; (2b) for F'; and

(3) Given: (a)

If h I = l@kl@l:

C@DI@E ÷ C@D2®E and h 2 = l@k2@l:

are expanded instances A@B3@C;

and

(b)

= ~n.

~n

C@D2@E + C@D3@E

of kl, k 2 E F~, then h2h I = l@k2kl@l:

Let H* be the following category.

A@BI@C

An object of H*,

(f: n + p,u,m) consists of an object f: n ~ p of H', a non-negatlve

-

integer

u, and a (p,u)

mi<mJ.

If p~i<J

(g: n ÷ q,u,B) morphlsm is

then mi~mJ).

is

(~,h,e),

of H', and

-

m.

(i.e.

A morphlsm

where

e E S(u) .

(~,h):

The

a E S(p+u). of H* from

If i<J~p

then

(f: n ÷ p,u,m)

to

(f: n ÷ p) ÷ (g: n ÷ q) is a

composite

of

(~',h',e')

and

(~,h,e)

e,e).

(~'~, h'h,

T he0rem Proof.

- shuffle

16?

2.5~F~

We shall

is i s o m o r p h i c

define

functors

to H u.

M*:

F~ ÷ H * and N*:

H* ÷ F~, such

that M*N * = IH, , and N m M ~ = IF~. Let M* a p p l i e d

to

Cnlp 1 @ -i@ n21 @ .. • @ Cn2p 2 @ i® -.. @ ~@ Cnu+l, I ...

¢nll @ • -- @

... @

be

(f: n ÷ p, u, ~), where

M(¢nll

¢nu+l,Pu+l f: n ÷ p is

@ ... ® Cnlp I @ ... @ Cnu+l, 1 @ ... @ Cnu+l,Pu+l)

Thus n ~

u+l Pi Z ~ i~l J=l nlj

and

P

~

u+l Z i= 1

of I; and ~ is the p e r m u t a t i o n

Pi

;

u is the n u m b e r

of o c c u r r e n c e s

sending

(I,2,...,Pl,PI+I,...,PI+P2,PI+P2+I,

..., PI+P2

+ ... + P u + l , P + l , p + 2 , . . .

... p+u) to (1,2,

..., p l , P + l , P l + l , . . . , p l + P 2 , P + 2 , P l + P 2 + l ,

..., p + U , P l + P 2 + . . . + P u + l ,

"''' Pl + P2 + "'" + Pu+I ) Let Suppose M(k)

M*

l@k®l: C ~ D @ C 2 ÷ C ~ D ' @ C 2 be an e x p a n d e d (C i) = (fi: ni ÷ Pi'

= (~,h).

Then M * ( l @ k ® l )

ui'

instance

ml ) for i = 1,2;

=(~',h',l)

where

of k e F~.

and suppose

(~',h')

is:

-

168

(fl: nl ÷ Pl ) @ M(D)

-

® (f2:n2

l@(~,h)

÷ P2 )

@ i

(fl: nl ÷ Pl ) @ M(D')

® (f2:n2

÷ P2 )

^

Suppose

~: C ÷ D where

MiD = (g: n ÷ p, u,

8).

a

2(p,u)

The p e r m u t a t i o n :

~

= p+u + p+u--> p+u

may be w r i t t e n

M*C = (f: n ÷ p, u, m) and

8 -I ) p+u = 2(p,u)

as 2(~I,~ 2) for ~I e S(p),

I

~I'

It can be v e r i f i e d (3b),

We now define define

this

positions

to be N(f:

m(p+l),

N*:

from

the r e l a t i o n s

(1),

(2),

F~ to H*.

H* ÷ F~.

We require

n ÷ p) with

l' i n s e r t e d

m(p+2),

be:

u

that M m p r e s e r v e s

so is a functor

Let Mm(~)

2

>p

(3a) and

~2 e S(u).

... m(p+u).

N*(f:

n ÷ p, u, a). We

u times

It is easily

at the

seen that MWN w = 1

and NmM * = 1 on objects. ^

There

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

~ in F~:

^

~: N*(f: where

n ÷ p, u, a) ÷ N(f:

!u is i@i@

... @i, where

1 is m e n t i o n e d

(f: n ÷ p, u, a) ~ (g: n ÷ q, u, composite:

n + p) @ lU

8) define

u times.

N*(~,h,0)

If (~,h,e):

to be the

-

169

-

N*(f: n ÷ p, u, m)

L ~

N(f: n ÷ p) @ lU N(~,h)

@ ;

N(g: n ÷ q) @ ! u

N*(g: It can be verified §3 Determination We shall treat follows

a similar

non-strlct

that M'N* = I and N'M* = I.

of the club in the non-strlct

the symmetric

argument.

and S are type A objects.

(I) a:

monoldal

The non-symmetrlc

case

case

to the club for the strict

case.

for the club E are I, i, and T@S where The type B objects

are I', ~', T@'T'

T

and

T and T' are type B, and S is type A.

The type A morphisms following

case.

QED.

We shall show that the club in the

case is equivalent

The type A objects

¢S, where

n ÷ q, u, B)

of E are generated

by expansions

of the

instances:

(T@S)@R ÷ T@(S@R)

and a -1, T@(S®R)

(2) £: I@T ÷ T, r: T@I ÷ T, ~-l:

~ (T@S)@R;

T ~ I@T, r-l:

T ÷ T®I;

(3) c: T®S ~ S®T for type A objects

T,S,R.

The type B morphisms (i) a', a '-I. (2) ~', r'

~,-I

,-I.

of E are generated

by expansions

of:

-

(3)

c';

(4)

Cf:

(5)

¢, ¢o.

170

-

CT ÷ ¢S, where f: T ÷ S is a type A generator;

Type A morphlsms

satisfy the following relations:

(1)

Functoriality

of @;

(2)

(a)

Naturallty

of a;

(b)

Naturallty

of r (naturallty

follow from relations

(3)

of r-l,l,l-l,a -1 will

(2a),

(2b),

(c)

Naturality

(a)

aa -I = l,a-la = I ~

(b)

The diagrams MC1 - 7 of |11 ; and

(e)

(3a) and (3c));

of c;

c

I®T

-I = I,~-I~

-- l, rr -I = l,r-lr = i;

> T@I

T

The type B morphisms (i)

(2)

(a)

Functoriality

of ®';

(b)

Functoriality

of ¢, i.e. ¢gf = Cg. Cf;

(a)

Naturality r,-l,

(3)

satisfy the following relations:

of a', r', c' (and consequently

~', a' -I

~,-I);

(b)

Naturality

(a)

a' a '-l = l, ..., as in (3a) for type A morphlsms;

(b)

As (3b) for type A morphisms;

(c)

As (3c) for type A morphisms;

(d)

The diagrams

Theprem B.l and faithful•

We describe l.e. E(T,S)

of ¢;

MF2, MF3 and ME4 of [ i].

a surSectiv ~ functor J: E + F* which is full = F*(JT,JS).

Thus [ is equivalent

to F*.

-

Proof:

171

For each object T of E, we describe

isomorphism U(T): T ÷ K(T) in E. ¢o.

-

~(T) will not involve c, c', ¢ or

Let K 0 = I, K I = I, K n = Kn_l® ~.

K n' = K'n_l @'I'._

Define

an object K(T) and an

Let K8 = I', K~ = ~',

en,m: Kn@K m ÷ Kn+ m by:

en, 0 = r: Kn@I ÷ Kn; en, 1 = i: Kn® ~ ÷ Kn+l; 8n, m is the composite: a -I

en,m_l®l

Kn@Km--------~ (Kn@Km_l)@ ~

> Kn+m_l@~ = Kn+ m

Define e'n,m.- -nK'@'K'-m÷ K'n+m analogously. Isomorphlsms

not involving

8n, m and e'n,m are

c,c',¢ and ¢o.

Let K(I) = I, K(~) = i, and n(I) = l, n(1) = l. If K(T) = K m and K(S) = Kn, then let K(TgS) T@S

= Km+ n and

nT@nS ~a~ ...)... KmgK n

(T@S) be the composite:

Km+n.

Let K(I') = I', K(l') = ~', and hi' = i, n !' = i. KT = K!(A.,m ± ..., A m ) and KS = K~(BI, K~+ n (AI,

..., Am, BI,

If

..., Bn) , let K(T®S) be

..., B n) and n(T®'S)

be the composite:

T@'S

~T ®' nS ' A l" Km(

.

" .,

Am) @'Kn(BI"

.

" ., B n)

8~,n(A I, ..., A m , B 1 . . . . , B n) K'_ m~n (A_, ±

"''" Am, B1,

"''' B n ) "

If R is a type A object, and KR = Kp, then let K(¢R) be ~'{¢Kp} = ¢Kp and n(¢R) be ¢(~R):

CR ÷ ¢Kp.

Define L as a partial objects of F* given by:

function from the objects of E to the

-

172

-

L(Kn) = n; L(I') = ~; L(!') = !'; L(¢K n) = Cn; L(K~(A 1 . . . . , An) ) = LAI@' LA i is defined. certainly

J is

When T is an object of E, define JT as L(KT).

surJective

on objects,

and L is InJectlve.

Let f: T ÷ S be a morphism in E. fn fn-1

... ®'LA n when

We may write f as

"'" fl where each fi is either an expansion of an instance

a, a -I , r,

r-l, '£, £-i, a' , a,-l, r' , r , - I

£',

or

£ t-1

of

; or is of the

form: (l@gi)@l:

(P@R)@Q ÷ (P@R')@Q,

(l@'gi)@'l:

or

(P@'R)@'Q ÷ (P@'R')@'Q

where gl is an instance

of c, f', ¢, ¢o or cf.

J(f) = J(fn ) J(fn_l)...

J(fl).

J(fl ) = 1.

Let J((1@gi)@l)

Define

If F i is of the first type, let

be

l@Jg i @ l: JP @ JR ® JQ ÷ JP @ JR' @ JQ and similarly

for J((l@'gi)®'l);

is c: Rl@R 2 ~ R2@RI,

where Jgi is defined below.

and J(Rj) = rj, let Jgl be ~(l,l):

2(r2,rl) , where T is the non-identity c': R3@'R 4 ÷ R4@'R3,

let Jgi be T(1,1):

gi is Ch: CR 5 ÷ ¢R6, let Jgi be ¢(Jh): or $o, let Jgi be ¢ or ¢ o relations

element of S(2).

If gi

2(rl,r 2) + If gi is

2(JR3,JR 4) ÷ 2(JR4,JR3). ¢(JR 5) ÷ ¢(JR6).

respectively.

If

If gi is

Since J preserves

the

on E, J is a functor.

Suppose f: LS

÷ LS' is an instance

J'f: S + S' as a morphlsm of E. composite of instances

in F,.

We define

If f is 6: n ÷ n, let J*f be a

of a, a -1, and c with graph ~ (unique by [6]).

jwf is defined similarly

if f is ~, or Cf'.

If f is ~ or ¢o, then

-

173

-

so is J*f. Suppose f is l@'f'@'l. of an instance

f' in E.

LT@'LS@'LR ÷ LT@'LS'@'LR,

(The @ case is analogous).

an expansion

Let jmf be the

composite: K((T@'S)@'H)

~

(~,s)e,~))

-I

(T®'S)@'R

l

(l@'Jmf')@'l

(T@'S')@'R

~

n(CTe'S')e'R)

K((T@'S')@'R) Suppose

g = l@'gl@'l:

(TI@'SI)@'R I ÷ ( T l @ ' S 2 ) @ ' ~ i s

a morphism of Fu,

such that KT 1 = T, KR 1 = R, KS 1 = S, KS 2 = S', and Jgl = f' "

Then the

composite

K((T@'S)e'R)

~ l

(n((TI@'SI)@'RI)) -I

(TI@'SI)@'H I (l@'gl)@'l

(TI@'S2)@'R I In((TI@'S2)@'R I) K((T®'S')®'R) equals jif, by the naturallty and the coherence

of a,~, r, a', £', r' and their inverses,

of symmetric monoidal

categories. m

If f: T ÷ S in E is written fn "'" f2fl where fi is an expansion

of an instance,

relations

in F* are preserved by J*, so J*;F* ÷ E is a functor.

We now describe

then let J*f be J*fn''"

J*f2"J*fl"

The

the blJection between E(T,S) and F*(JT,JS).

- 174 There

-

is a correspondence between f: T ÷ S and Jr: JT ÷ JS, and

between g: JT ÷ JS in Fi and the composite : nT T

J*g ) KT

(~S) -I ) KS

) S.

Thus E(T,S) is isomorphic to FI(JT,JS).

QED.

§4 Coherence for a closed functor We describe N, the free model on A = {A,B} for a closed functor between two categories. The objects of N of type A are i, I, T®S and [T,S] where T and S are objects of type A.

The objects of type B are ~',I',T@'S, [T,S]'

and CR, where T and S are of type B, and R is of type A. Morphisms of type A are generated by expanded instances of a, a -1, b, b -1, c, d, e; where b: T@I ÷ T, d: T ÷ [S,T@S] and e: [T,S]@ T ÷ S.

Type B morphisms are generated by expanded instances

of a', a '-l, b', b '-l, c', d', e', #f, ~ and $o; where f Is a type A generator. Type A morphism satisfy the following relations: (i) Functoriallty of @; and [,]: [ f,l]

[T',S]

A [T,S]

[l,g]~

I[ l~g]

IT',S']

.% [T,S'] [ f, I]

(2) Naturallty of a, a -1, b, b -1, c; and d and e:

-

175

-

d T

) [ S,T@S]

f

[ S,T'®S'] [g, i] T'

~ [S',T'@S']

l

d [ T,S] @T

) S

{ f, 1] / ~I / [T',S ®T]

g

[l,g~ f ~ " - ~ [T' S']®T'

~ S' e

(3) diagrams

aa -1 = l, a-la = I, bb -1 = l, b-lb = l, and the

CI-C6 of [5]. Type B morphisms (i) Functorlality

satisfy the following relations: of ®'

[

]'

and ¢:

Sf ¢T

~ ¢S

@(gf) (2) Naturality

CR

of a',a '-I, b',b'

(3)(a) The relations (b) The diagrams (c) The relations

-1,

~ c', d', e' and ¢;

that a' and b' are isomorphisms; C1-C6 of [5] for a' . . . . .

e';

that ¢ is a closed functor,

i.e.

diagrams MF2, MF3 and MF4 of [1]. We now describe a precategory morphisms

graphs,

and a precategory

morphisms

R-graphs.

G with objects G-sets and

R with objects R-sets and

G-graphs may be of type A or type B.

and the codomaln of a graph of type A(respectlvely

The domain

type B)are also of

-

176

-

type A (respectively B), A G-set is a llst (possibly empty)

composed from the four

elements +A, -A, +B, -B; together with another e l e m e n ~ A or ~ which specifies which type the G-set is. or -B in the llst.

G-sets of type A do not have +B

Examples of G-sets are:

{~; A} {+A, -A, -A; A} {+A, -A, -A; B} {÷A, -B, -B, +B, -A, +B; B} We sometimes write the G-set as {L;A} or {L;B}.

A D-set is a list

(possibly empty) composed from the two elements + and -. are lists, define preceding

L 2.

L 1 v L2 as the llst comprising

Let - L be the llst obtained from

sign of each of the elements of L.

{L1;A}@ { / 2 ; A }

If L 1 and L2

L 1 and L2, with L 1 by changing the

Define @ on G-sets and D-sets by:

= {L 1 v L2;A}

{L1;B}@ { L 2 ; B } = {L 1 v L2;B} K 1 ® K 2 = K 1 v K2, where K1, K2 are D - s e t s . Define [,] on G-sets and D-sets by:

[{L1;A},

{L2;A}]

|{L1;B},

{L2;B ]

[K1, K2} = (-K1) If of ~.

~ is a G-set,

Define ~ ,

,~, ~

set of + elements of W.

= {(-L1)vL2;A

}

{(-L1)vL2;B}

v K2

let

~

be the set o f +A elements o f the l i s t

similarly.

If ~ Is a D-set,

Similarly define ~-.

consists of:

(i)

a blJectlon

from

°

to

o

let ~+ be the

A graph f: W ÷ v

-

(2) A P-graph

a biJection

P-graphs.

-

from .~ U ~B to PB u ~ .

f: p + 9 consists

Suppose

177

of a function

from p

+

f: ~ ÷ ~ and g: 9 ÷ ~ are either

Then g and f are

u 9 both

said to be i n c o m p a t i b l e

-

+ to p- u ~ . graphs

or both

if there

is a

subset 91' 92' of the elements (i = l, 2, to 91"

of the

..., n);

Otherwise

compatible for P.

list

Consider

(i)

~21to

~21+l

1

f maps

~2i-i

(i = l,

to 92i

..., n-l),

and 92n

we say that

we define the

9i e L(9)

of 9, such that

and g maps

g and f are compatible. If g and f are + + + gf: ~ ÷ ~. Suppose pOePA u PB for G, or pOep sequence:

PO' 91' where

n_>

"''' 92n

(the llst

92'

"''' ~r'

of 9),

a

and a E p- u ~+

f maps

PO to 91"

and ~i to vi+ I when

(2)

g maps

9i to 9i+i w h e n

(3)

If r is odd,

such

that:

i is even

and

2 ~ i ~ r-l;

If r is even,

a similar

"o"

for W O e

~'

Then

8ep-

consists

i ~ i ~ r-l;

~r to m.

~r to m.

sequence

"",

u ~+.

and w 0 and 8 set up a f u n c t i o n function

is odd and

m e w + and g maps

~ e ~- and f maps

We define

i

B

The

'

correspondences

between

from p+ u ~" to p- u ~+.

of two b i J e c t i o n s

between

PA u ~A

PO and s,

In G this + and PA u ~A'

-

and ~B u ~B and WB u ~B"

178

-

Thus we have defined

gf: ~ ÷ w for graphs

and P-graphs. Theorem

4.1

Suppose

or all P-graphs. compatible Proof:

f: ~ + ~, g: 9 ÷ ~ and h: w ÷ p

Then g and f are compatible,

Iff h and g are compatible,

Suppose

and h and gf are

and hg and f are compatibl 9.

g and f are incompatible.

Vl'

Then there exist

"''' 92n

n a 1

in L(9), such that f maps ~21-I to ~21 for i = i, 92n to ~l' and 921 to 92i+i compatible

are all ~raphs

for i = I,

..., n-1.

..., n; and g maps But if h and g are

then hg maps 92n to 91' and 921 to 92i+1"

Thus hg and f

are incompatible. Suppose

g and f are compatible,

and h and gf are incompatible.

Then there exist

"''' W2n

~i"

in L(~),

~

1

such that gf maps ~2i-1 to ~2i' and h maps ~2n to ~l' and

~2i to ~2i+l" (possibly

n

Since gf maps ~2i-1 to ~21' there exists

empty)

91,I'

9i,2"

"''' 9i,r i

in L(~) such that g maps ~i,J+l when J is even;

(where r i is even)

~2i-i to 9i,i'

and f maps 9i,J

~i,r i to ~21' and ~i,J to 9i,J+l when J is odd.

r i = 0 for all i, then h and g are incompatible. always

a sequence

0.

If

r i is not

Consider:

91,i'91,2 ' "''' Vl,r I' 92,1' When j is e v e ~ hg maps 91,J consecutive

Suppose

to

"''' 92,r 2"

to 9i,J+l"

terms in the sequence

"''' Vn,l'

"''' 9n,r n"

If 9i,rl and 9J,l are

(including

the last and first terms),

-

179

-

then hg maps v i,r i to vj, I. Thus hg and f are incompatible. Theorem

4.2

Suppose

or all S-graphs. are compatible. Proof:

f: ~ ÷ v, g: v ÷ ~ and h: w ÷ p are all 5raphs

Suppose

that g a n d f are compatible,

+ ~0 e U .

Suppose

If (h(gf))~ 0 -- ~, then there is a sequence

"''' Ws' e'

where ~i e £(~) and ~ e ~- u ~+; such that gf maps and ~s to ~ if s is even,

a if s is odd.

(I)

and h and gf

T h e n h(gf) = (hg)f.

~0' Wl'

~2i+l'

QED

and h maps

We have the f o l l o w i n g

vO,l,

..., v O,rO

where f maps ~0 to v0,1,

W0 to Wl' ~2i to

~21-1 to ~2i' and ~s to

sequences

in £(v).

for r 0 odd,

and v0, i to v0,1+ 1 when i is even;

and g maps

V0,r0 to Wl" and v0, i to v0,i+ 1 when i is odd. (2) where

Vi,l,

for r 0 even,

f maps vi, j to vl,j+ 1 when J is odd, and g maps w2i to Vi,l,

Vl,ri to w2i+l' (3)

and vl, j to vi,j+ 1 when J is even.

If s is even:

Vs,l, where

..., v i,ri

f maps VS,rs

w s to Vs,1,

..., VS,rs

for r s odd,

to ~, and Vs, i to Vs,i+ I for i odd; and g maps

and Vs, i to Vs,i+ 1 for i even.

Consider

the sequence

~O,~O,l,.--,VO,ro,Vl,l,...,~l,rl,...,Vt,l,...,Vt,rt,Vs,l,''',~S,rs where

t -

s-2 if s is even, 2

maps ~0 to 90,i;

and t = s-__._lif s is odd. 2

90,i to VO,i+ 1 when i is even;

91,j

,e

We see that f to vl,j+ 1 when

i = i,

..., t and j Is odd; 9s,i to ~s,i+l when i is even;

to a.

Also hg maps VO, i to 90,I+i when i is odd; 91,J

and Vs,rs

to ~i,j+l

-

when i = i, 9i,ri

..., t and J is even;

to uJ,1 when 9i,rl

sequence;

and 9t

,r t

Suppose

9s,i

to e if s is odd.

to U s , i + l

p-"

Thus

We define

f@g:

~ to 8 if e i t h e r

Thus h(gf)

f maps

a to 8 or f maps

G* and D*.

of G and 9 r e s p e c t i v e l y .

G(~,~)u{*}

= 9(~,~)u{*}.

b o t h in G* or

(i)

both in D*,

the c o m p o s i t e

Similarly

QED.

b o t h graphs

or b o t h

÷ [~,Pl.

m to 8.

Let

Let [f,gl

a to 8.

and 9' are the objects

either

in the

= (hg)f.

~ to 8 or g maps

the c a t e g o r i e s

and D*(~,~)

terms

~x~ + ~@p and [f,gl : [u,~]

g maps

Let us define

w h e n i is odd;

~ = ((hg)f)~0"

f: ~ ÷ ~ and g: ~ ÷ p are e i t h e r

f®g map ~ to 8 if e i t h e r map

-

and ~J,1 are c o n s e c u t i v e

(h(gf))p 0 = ((hg)f)p 0 for P O ~

9-graphs.

180

The o b j e c t s

of G*

Let G*(~,~)

=

If f: ~ ~ ~ and g: ~ ~ ~ are

then their c o m p o s i t e

in G or 9, r e s p e c t i v e l y ,

is:

if g and f are

compatible;

(2)

*, if g and f are i n c o m p a t i b l e ;

(3)

*, if e i t h e r

By T h e o r e m s Define

4.1 and

or

g or f is *.

4.2 c o m p o s i t i o n

in G* and D* is a s s o c i a t i v e .

@ and [ , ] in G* and 9" as in G and 9, except

and if,g] identity

= *, if e i t h e r graphs

Define

f or g is *.

and D - g r a p h s

y(~,~):

Since

G* and 9" are

there

that

are obvious

categories.

~v~ + uv~ by l e t t i n g

y(W,~)(x)

equal:

+ + x c ~ + c ( ~ p ) +, when x c p c(Wv~) •

Define

x E ~ + c ( ~ v p ) +, when

x c ~+c(pv~)+;

x a ~-c(~vv)-,

when

x ~ ~-c(~v~)-',

x E ~-c(Wvu)-,

when

x e ~-c(~v~)-

6(~,~):

~ ÷ (-~)vwvv

= ~ by l e t t i n g

f@g = *

and

6(~,u)(x)

equal:

-

+ + x e ~ c w , when x e W-,

when

181

-

+ x e ~ ;

x e ~-cw-;

-x e -x

Define

e(~,v):

( - ~ ) + c w +, w h e n x e ~ - c ~ - ; + + e ~ c~ when x e (-~)-c~-.

(-~)vvv~

+ v as

and

( 6 ( v , ~ ) ) -I

I

We

define

r:

a functor

N ÷ G*:

=

(g;A}:

r z , = (~;B};

r~ =

{A;A};

r~,

ri

r(T@S)

= rT@rs;

r([T,s]) If

(L;A~

T@S

then

+ S@T)

= FT@FS;

r([T,s],) r(¢S)

ra = Fa -1 = Fb = Fb -1 r(c:

{BIB};

r(T@'S)

= [rT,rs];

rS =

=

=

= [rT,rs];

{L;B}~

= i;

= y(L(T),

L(S));

r d = 6, re = e; r(f@g)

= rf@rg;

r([f,g])

= [rf,rg];

ra,

=

ra '-I

rc,

=

y,

r(f@'g)

=

rd'

rb' =

=

6,

rb '-I

re'

=

=

i;

e;

= rf@rg;

r([f,g],)

= [rf,rg];

r(¢f) = rf; r(~)

= l, r(¢ °) = z; a n d

r(gf) We

AT = AI'

=

rg.

define

{~} =

rf. a functor

for

At'

=

all {~};

A(T@'S)

= AT@AS:

Af = i:

{g} ÷

{~}

A:

N ÷ ~*:

objects

o f NA;

ACES)

{+};

=

A([T,S]') for

all

= [AT, AS]; morphisms

o f NA;

-

182

Aa'

= Aa '-I = Ab' = Ab '-I = I;

Ac'

= y, Ad' = 6, Ae'

A(f@'g) A(¢f) A(¢)

= Af@Ag;

= I:{+}

= ¢;

A([f,g] ') = [Af,ag];

+ {+};

is the R-graph:

{+, +} ÷ {+};

A(¢ °) is the R - g r a p h : { g } A(gf)

= Ag.

-

÷ {+};

and

Af.

It can e a s i l y

be v e r i f i e d

that

F and A ~ r e s e r v e

the r e l a t i o n s

on ~4.

Let the c e n t r a l a, a -1, b, b -1 and c. generated

morphism

of N A be those

Let the c e n t r a l

by a', a '-l, b',

b '-l,

generated

morphlsms

c' and Cf w h e r e

by

of N B be those f is c e n t r a l

of type

A.

If f: A®B ÷ C is a m o r p h i s m d A

If g: D ÷ E,

let

F

(usually

> [ B,C]

written

l@g

~f and

Let of m o r p h i s m s

e

for type

the type A c o n s t r u c t i b l e of N A w h i c h

satisfy

central

~ F

the f o l l o w i n g

Every

morphlsm

CA2:

If f: T + S is in the class then vfu:

B.

morphisms

CAI:

CA3: f@g:

) be the c o m p o s i t e

~ [E,F] @E

in the same way

v: S ÷ S' are c e n t r a l

let ~f be the c o m p o s i t e

[ 1,f] ) [ B,A@B]

[ E,F] @D

Define

of NA,

be the s m a l l e s t conditions:

is in the class.

and if u: T' ÷ T and

T' ÷ S' is in the

class.

If f: T ÷ S and g: P ÷ Q are in the class

T®P ÷ S®Q.

so is

class

-

183

-

CA4:

If f: A®B + C is in the class so is ~f: A ÷ [B,C]

CA5:

If f: T ÷ S and g: P@Q ÷ R are in the class

g(®l):

([S,P]®T)®Q

+ R.

Let the type B constructible morphisms

of N B which

CBI:

so is

satisfy

morphisms

the following

be the smallest

class of

conditions:

CAI

CB2:CA2 CB3:CA3

with ®' replacing

CB4:CA4

with ®' and [,]' replacing

CB5:CA5

with ®' replacing

CB6:

If f: Kn(SI,

is a type A constructlble

® @ and [,]

®

..., Sn) ÷ T (notation morphism,

then the composite

Cn K~(¢S I,

.... ¢S n)

is a type B constructlble

as in §3)

Cf ~ ¢(Kn(S I .... ,Sn))

morphlsm.

We define

> CT

T 0 to be ¢o; ¢i to be l;

and if n > I, ~n to be the composite:

K~_I(¢S 1 . . . .

~

,

¢(Kn_I(S I,

¢n_l@'l

. . . ,

Sn_l))

®'¢S n

1;

¢(Kn(S I, An object

CSn_ I) @'¢S n

.... Sn)).

T of N is constant

An object

T of N is integral

An object

T of ~ is C-free

if it does not involve

if it does not involve

if it does not involve

f: T ÷ S of N is said to be trivial

¢.

i or i'.

[ , ] or [ , ]' A morphism

if T and S are constant

¢-free

integral.

Lemma

4.3:

A trivial

con structlble

morphlsm

in N is central.

and

-

Proof:

As Lemma

For each constructlble

h: T + S i~nN, at least one

is true:

(I)

h is central

(2)

h is of the form x

f@g

T

where

-

6.1 of [5].

Pr0P0sltio n 4.4: of the following

184

> A@B

y ) C@D

) S

f and g are non-trlvlal.

(3)

h is of the form ~f

Y

T

(4)

~ [B,C]

) S

h is of the form @l

X

T (5)

where

~n

both.

f and g are constructlble, For convenience

Similarly

morphisms

h of the forms

we are using @ to represent

(2),

(3),

< > and of type

(4),

etc.

(i) Axiom CB6 is clearly (2) If B is a constant

generated

rS;

® or @' or

For brevity,

(5) are said to be of type ®,

¢ respectively.

The proof is the same as Proposition

need to show that there

Y ~¢B

and x and y are central.

for [ , ], a,b,c,d,e,I,l,

of type ~, of type Proof:

Cf )¢(Kn(A1,...,An))

...)...K~ ( CA 1 , .... ¢An )

Notation:

> S

h is of the form

X

T

g ) C@D

~ ([B,CI@A)@D

satisfied

integral

6.2 of [5] except

that

and

C-free object,

is a central m o r p h i s m

by Just ® and I (or Just ®' and I').

u(B):

we

B + I.

But B is

If B = I, let u = I.

-

If

B = C®D,

185

-

let u(B) be the c o m p o s i t e :

u(C)® u(D)

b

C@D

~ I@I

~I.

QED.

For the p u r p o s e s object

T a non-negative

following

inductive

of our i n d u c t i v e integer

p r o o f we i n t r o d u c e

r(T~ called

its rank,

defined

for each by the

rules:

r(1)

=

0

r(_l) = 1 r(T@S)

= r(T)

r([T,S]) r(¢T)

Note

+ r(S)

= r(T)

+ r(S)

+ 1

= rT + 1

that rT = 0 if and only if T is a c o n s t a n t

If f: T ÷ S, we say the rank of f, r(f),

Lemma

4.5:

Proof:

If T ÷ S is c e n t r a l

Use Lemma

Proposition

4.6:

integral

is r(T)

then r(T)

C-free

object.

+ r(S).

= r(S).

6.3 of [5].

(The C u t r E l i m i n a t l o n

Theorem)

h: T ÷ S and k: S®U ÷ V o_~f N are c o n s t r u c t i b l e

If the m o r p h i s m s then

so is the c o m p o s i t e

morphism h@l T®U

Moreover~

the g r a p h s / k

Ak and A(h@l)

Proof:

shall proof.

omit

and

) V .

r(h@l)

are c o m p a t i b l e ,

a n d the P - ~ r a p h s

are c o m p a t i b l e .

The g r a p h s

considering

k ~ S®U

and D - g r a p h s

the same a r g u m e n t s any f u r t h e r

are

shown

to be c o m p a t i b l e

as used in each

reference

to the graphs

case

below.

or D - g r a p h s

by Thus we in the

-

We

those

use the same

of the

double

IS6

-

induction

form ~, IT, S] and

S.

as in [5]•

We need

only

Prime

objects

consider

the

are

following

cases: Case

7 h i s of type

the same way as Case Case central

Case

|A,B]@U

h@l

There

¢.

Since

is c o n s i d e r e d

C

I'

•

" o~

¢Cn)

¢, k is of type

this < >.

there

case

case

case does not exist. We have

g(@l) ~([B,C]@D)@E

I: CA is a s s o c i a t e d

is c o n s i d e r e d

in the

> V.

via x w i t h

..

¢An),

.,

IK~(~ n, i, K m ( ¢ ( K n ( A l,

•

~¢

B 2,

l,

...,

m e ~

...,

¢.

factor

6 Subcase a prime

as Case

of type

¢B 2,

..•, An)), m(¢f,

K~(¢B l,

via x w i t h

in the same way

10: h and k are both

' K 'n(¢A l, Km(

a prime

same way as Case

2: ¢A is a s s o c i a t e d

is c o n s i d e r e d

Case

2.

factor

6 Subcase

We have

of D.

of E.

3.

k(h@l)

=

CB m)

I)

¢B 2,

...,

CB m)

..., i) ¢B m)

!~m ¢ (Km(B I

..., Bm))

,c w h e r e m , 1. trivially

if

in

is no

are two subcases:

Subcase This

7, k is of type

x ~ CA@U

Subcase This

This

+ Kn(¢ !

9 h i s o f type

T@U

®.

5.

8 h is o f type

morphlsm:

¢, k is of type

But; ~m. K m ( * f ,

1,

....

m -- 1, a n d by n a t u r a l i t y

1) -- dO(Km(f, 1 . . . . . of ¢ if

m ~ 1.

1 ) ) . ~ m, Also

-

g. Km(f,

i,

...,

187

I) is constructlble

Cm.Km(¢n,

l .....

l) = Cm+n-1

¢(g. Km(f,

I,

1))¢m+n_l,

....

All morphlsms

Proof:

As the proof of Theorem

and

Thus k(h@l)

so is of type ¢.

4.7:

=

QED.

of N are constructlble. 6.5 of [5]

and ¢o are constructlble. (i) Cf, by letting

by induction,

by §3.

Theorem

Cf,

-

except we need to show

In CB6 we obtain:

n = 1

(2) [, by letting n = 2 and f = I (3) ¢o, by letting n = 0 and f = 1. QED.

Corollary

F and A take their values

4.8:

In particular, Proof:

objects

N is a club.

Immediate

by induction

The prime

objects

A (respectively

C-free

on rank,

object

and CR.

B) may be written

object

of type A (respectively

S is said to be reduced

B).

if S is either

formed by ~ and @ (respectively

Every

as S(R1,

of type A (respectively

said to be reduced

in view of Proposition

of type A are i and [T,S].

of type B are ~', [T,SI'

is a prime

in G and D respectively.

The prime T of type

..., R n) where each R i B), and S is an integral An integral

I (respectively

~' and @').

if S is reduced,

object

4.4.

T = S(R1,

C-free

object

I') or just ..., R n) is

and if Pi is reduced

whenever

Ri = CPi" Lemma 4.9: a central Proof: Lemma

Given any object isomorphism

Use the result 4.10:

T, we can find a reduced

object

T' and

z: T ÷ T' in N. of Lemma

In proposltion

7.1 of [5].

4.4 we can suppose

that the objects

-

188

-

A,B,C,D

i_n_n(2); A and D i_~n (4), an__~dAi, B in (5) are reduced.

Proof:

As Lemma

7.2 of [5].

We now define proper T and S are proper, proper,

then IT,S]

is not constant, proper,

is proper

on @', [,]',

S is constant

and T is not C-free.

that every

constant

and T

If S is

C-free

then T and S are proper;

T is proper

object

that T®S

and that if ¢S is proper

if and only if each of its prime

for convenience

etc.

Thus @ stands

Suppose h: T ÷ S is central

Use Lemma

we have been omitting for @ and/or in N.

the

@'

Then if either T or S

7.3 of [5].

Let h: T ÷ S i_nn N.

(i) S is constant~ (2) S i_~s ¢-free~

If T is proper and:

then T is constant;

then T i s C-free.

The proof of (i) is the same as that of Proposition Except

we need consider

CB is constant Thus

If T and S are

so is the other.

Lemma 4.12:

[5].

Observe

is proper

- whence

Note that again

is proper

Proof:

S is C-free

unless

if and only if T and S are proper;

Lemma 4.11:

Proof:

or unless

are proper,

If

is proper.

Remark: primes

and [T,S]'

that if [T,S]

then S is proper factors

I, I', 1 and i' are proper.

then T®S and T@'S are proper.

then ¢S is proper.

is proper;

objects.

¢(Kn(AI,

is constant.

because .... An))

S is.

h = Cf'~n:

Kn(¢AI'

Since T is proper,

is proper.

By induction

7.4 of

"''' CAn) ÷ CB. But each A i is proper. ¢(Kn(AI,

.... An))

Thus T is constant.

(2) is proved

in the same way.

We now show how to eliminate

QED.

constant

C-free prime

factors

-

from an object

Lemma

4.13:

189

-

T.

Given

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

an object

T, we can find an object

f: T ÷ S in N such

S w i t h rS ~ rT,

that:

(1) S is r e d u c e d (2) S has no c o n s t a n t being precisely

those

(3) I f

isomorphism

Proof:

7.5 of

4.14:

are proper.

factors,

are n o n - c o n s t a n t

its prime

or not

with

S-free

object

factors

C-free.

Fx = Ff and

R, and a c e n t r a l

Ax = Af.

[5].

Let h: P@Q ÷ M®N be a m o r p h i s m

Suppose

q E G(FQ,FN),

prime

so is S.

is a c o n s t a n t

x: T ÷ S@R,

As L e m m a

Proposition

of T w h i c h

T is properp

(4) There

C-free

Fh = ~@n and Ah = ~'@q'

[' e D ( A P , A M )

P,Q,M,N

for ~ e G ( F P , F M ) ,

and ~' E D(AQ,AN).

p: P ÷ M and q: Q + N in N such

of N, w h e r e

that h = p@q,

Then there

exist

Fp = ~, Fq = q, Ap = ~'

and Aq = ~'

Proof:

We use i n d u c t i o n

of P , Q , M , N constant

to be r e d u c e d

Suppose

of P®Q is R(S1,

P®Q = R(SI,

fi:

and have p r i m e

h is central.

l, and p and q,

7.6 of [5].

where

By L e m m a

4.13 we may s u p p o s e

factors

which

each

are not b o t h

and C-free.

Suppose equal

on r(h).

gl'

...,

...,

If h is of type A, then

satisfying

the p r o p o s i t i o n ,

h is of type B. Sn).

Sn)

the prime

Then h may be w r i t t e n

f = l(fl'''''fn~

g2 are g e n e r a t e d

Suppose

by a',

R(S~,..

exist

But

p = gl f' and q = g2 f''.

factorisation

S' M' gl®g2~ ., n) = @N ~ M@N

a '-I, b', b '-I and

f may be w r i t t e n

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

as:

Si ÷ SI is Ck i if S i = CT i where k i is c e n t r a l

is 1 o t h e r w i s e .

~' and q'

as f'@f":

c'; and

of type A, and fi

P@Q + M'@N'.

Let

-

Suppose Proposition of Fh.

Similarly D-graphs

h is of type @.

7.6 of [5].

Define

Define

a D-graph

define ~':

graphs

AY ÷ AY',

p':

T':

If h is of type 7.6 of [5]

-

Form X,Y,U,V,X',Y', a graph

U' and V' as in

p: FX + FX' as the r e s t r i c t i o n

AX ÷ AX' as the r e s t r i c t i o n

o: FY ÷ FY',

case is as in P r o p o s i t i o n

Proposition

190

T: FU ÷ FU',

AU ÷ AU',

K':

of Ah.

K: FV ÷ FV',

AV + AV'.

The rest

and

of this

7.6 of [5].

~ or of type except

< >, we use the same m e t h o d

we n e e d

to c o n s i d e r

graphs

as in

and P - g r a p h s

as above.

If h is of type x P@Q

) K'(¢AI'n

We may

suppose

f o r m ¢T,

¢~ let h be the c o m p o s i t e :

Cn ..., ¢ A n ) ~

Cf ¢(Kn(AI,

M = ¢B and N = I.

so A(P@Q)

= {+, +,

E a c h prime

..., +}.

-empty,

and AQ- is empty.

element

of AQ + to AQ- u AN + = g.

Q = I.

We can t h e r e f o r e

..., An )) factor

Suppose

By the form a b o v e

Y ~ CB

) M@N.

of P@Q is of the

Q ~ I.

Then

AQ + is n o n -

of h, Ah maps e a c h

We thus have a c o n t r a d i c t i o n

so

let q be I and p be the c o m p o s i t e : b -I

P

h ) P@I

b )M@I

~M.

QED. Proposition are p r o p e r

4.15: objects.

and Ay E AB-.

Proof: of

Let f: A@B ÷ C be a m o r p h i s m Suppose

for each

Then B is c o n s t a n t

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

in N, where A , B , C

x e FB +, Y e AB +, that

Fx e FB-

and C-free.

4.12 a n d 4.14 in the p r o o f

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

7.7

[ 5] •

Proposition proper that

4.!6:

objects

Let h:

([Q,M]@P)@

in N, w i t h [Q,M]

rh is of the form ~(<~>

N ÷ S be a m o r p h i s m

not a constant

@ i) for 6 r a p h s

C-free

between

object.

~: FP ÷ FQ,

Suppose

-

n:

F(M@N)

D-graphs exist that

+ FS. ~':

AP + AQ,

objects ~,~'

Suppose

F,G,E,H

that

O':

191

Ah is of the

A(M@N)

÷ AS.

and a c e n t r a l

can be w r i t t e n

-

in the

Suppose

morphlsm

that

there

do not

x: P + ( [ F , G ] S E ) @ H

such

p(<~> ® 1) r(([F,G]®E)eH)

2 FQ,

AX

~'(<~'>

® i)

) A(([F,G]@E)®H)

AP

respectively

® i) for

forms

rx FP

morphisms

form ~ ' ( < ~ ' >

for graphs

p, e

P

and P - ~ r a p h s

AQ

p', a'

Then there

p: P ÷ Q, q: M®N ÷ S in N, such that h = q(

exist

@ 1), where

Fp = ~, rq = n, Ap = ~', Aq = ~' Proof:

We p r o c e e d

as in P r o p o s i t i o n

consider

both graphs

type

¢.

But h cannot be of type

with

any prime

Proposltion proper

factor

4.17:

objects

and P - g r a p h s ,

and we need

¢ since

in the d o m a i n

Let h: K~(¢A1,

in N.

7.8 of [5]

[Q,M]

for c e n t r a l

Proof:

...,

¢C n)

h cannot

be of types

¢.

QED

between

Cf ¢ ( K n ( C 1 . . . . , Cn))----~¢B

w or < >. If h is of type

h is central.

Suppose

h is of type y

....

suppose

CA n)

Then n = I.

Thus h = Cf w h e r e

g®k ~ C@D

a prime

f is

form.

®, so h may be written: z ~ E@F

E to be CB, and F to be I.

via y with

¢, h

form.

So h = ¢f. ~i.i is in the d e s i r e d

associated

of type

...,¢ An) ÷ ¢B be a m o r p h i s m

Cn

in the d e s i r e d

Suppose

We may

be a s s o c i a t e d

x.

is c e r t a i n l y

K~(¢AI,

cannot

h is of

Then h may be written:

¢An)----~ Kn(¢C l,

Obviously

central.

that we n e e d

the case w h e r e

of a m o r p h i s m

x

Kn(¢A 1 .....

except

factor

of D.

> CB. Suppose Thus

for some

i, ¢A i is

AD + is not empty.

By

-

form z(g@k)y

the

so no factor

k is trivial,

so h cannot

4.18:

in N, such

Let h,h':

that

-

of h, AD + is m a p p e d

contradiction

Theorem

ig2

to AD-

CA i is a s s o c i a t e d be of type @.

We use i n d u c t i o n

suppose

that none

with

T ÷ S be two m o r p h i s m s

on r(h)

of the prime

This

D, thus

a

D = I and

QED.

Fh = Fh' and Ah = Ah'

Proof:

u AF + = g.

between

proper

objects

Then h = h'

= r(h').

factors

By Lemma

4.13,

we may

of T or of S is constant

C-free. Suppose

h and h' are both

central,

h may be w r i t t e n

f = l(fl, .... fn )

g

T = P(R 1 . . . . , R n) where

P is an integral

factors

of T; g is g e n e r a t e d

by a ' , a ' - l , b ' , b ' - l , c R i is

CM i in w h i c h

h' is of the same equal same

object;

...,

by a , a - l , b , b - l , c

case

)S

R n are the prime if h is of type A, and

fl is Ck i for central it is s u f f i c i e n t

where

is so by a p p l y i n g

k i of type

to show that

Fx = Fy, Ax = Ay,

form as f, and y is of type

one of the other

R1,

"''' Qn )

if h is of type B; and fi: Ri + Qi is 1 unless

form,

the identity,

But this

and where

B and g e n e r a t e d

Theorem

3.1.

A.

x,y:

Since

M + N

x is of the

by a ' , a ' - l , b ' , b ' - l , c '

We may now suppose

that h is

types.

If h is of type of

P(QI' C-free

as

~ or of type

@ we p r o c e e d

as in T h e o r e m

2.4

[ 5] •

Suppose

h is of type

x T.

¢.

Then h is t

~ ) K~(¢AI,

By p r o p o s i t i o n

....

CAn)

4.1~ y - l h ' x - I

Cf

¢(in(Al,

.... An))

= y-lhx-l:

K~(¢AI,

y ) CB

...,

CAn)

~ S ÷ CB may be

written z K~(¢A1,

...,

¢An)

>K~(¢C1,

...,

¢Cn)

¢n)¢(Kn(C 1 ..... On) )

Cg ) ¢B

-

193

-

-

¢(gw)

= K~($A I, .... S A n ) $ n ) $ ( K n ( A l for a central ~ b y

, ..., An))

Theorem 3.1.

Since Fh = Fh', F(y-lh'x -I) =

r(y-lhx-l),

i.e. r(¢f.~n) = r(¢(gw).$n).

F(~ n) = i.

Thus rf = r(gw).

Af = A(gw).

...~. SB,

ThUS r(~f) =

r(¢(~))

since

Since f and gw are type A morphisms,

By induction f = gw, hence h = h'.

Suppose h is of type < >, say h is x T

®l ,)

g

([B,C] @A)@D

) C®D

~S.

It may happen that Ff and Af are of the forms:

p(
Fy rA

> r ( ( [ F , G ] ®E)®H) Ay

and

p'(®l) ) A(([ F,G] ®E)@H)

AA

> FB,

........

>

for a central m o r p h l s m y: A ÷ ([F,G]@E)@H),

A B, graphs p,a, D-graphs p',a'.

Using the naturallty of < > (which follows from the naturallty of e), it follows that Fh = T(<e>@l)~, D-graphs

T',~'.

Ah = T'(@I)~'

for graphs T,~,

But ~ = rz and ~' = Az where z is the central

morphlsm: x T

l@y ) ([B,C]@A)@D

) [B,C] @(([F,G]eE)@H)

ac ([ F,G] @E)@(H@[ B,C] ). Possibly c and a' are of the forms:

ru rE...

~(<~>®i) ~ r(([x,Yl®Z)®W

AU AE-

> rF, K'(<X'>®I)

> A(([X,Y]@Z)®W)

> AF

for a central m o r p h i s m u: E ÷ ([X,Y]@Z)®W, K',X'

graphs K,X, and D-graphs

But E has strictly fewer prime factors than A, since [F,G] is

a prime factor of A but not of E; Z has strictly fewer prime factors than E; and so on.

Thus this process terminates,

and ultimately we

-

for Fh a n d

have e x p r e s s i o n s

.

,(<6>®1) , r((IQ.~®P)®N)

Av AT

n'(<~'>®l)

Moreover [Q,M]

h ' v -1 we conclude

is not constant

AS

It follows

C-free

since T has no constant

and h ' v -I = q'(@l)

for

M@N ÷ S, w i t h Fp = Fp', Fq = Fq', Ap = Ap', from the inductive hypothesis

q = q', so that h = h'.

This completes

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

F r o m P r o p o s i t i o n 4.16 applied to h v -I and

that hv -I = q(

@l)

P ÷ Q and q,q':

Aq = Aq'.

"

and ~,~' are as in the Statement

C-free prime factors.

p,p':

~ rs.

) A(([Q,M]@P)@N)

where v is central, 4.16.

-

Ah of the form!

rv rT

194

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

4.18.

that p = p' and

-

195

-

REFERENCES

[ 11

S. Eilenberg and G.M. Kelly, Closed categories, in: Proc. Conf. on Categorical Algebra, La Jolla, 1965 (Springer-Verlag, 1966) pp 421-562.

12]

D.B.A. Epstein, Functors between tensored categories, Invent. Math. I (1966) 221-228.

[ 3]

G.M. Kelly, Many-variable functorlal calculus. I. (in this volume).

[~l

G.M. Kelly, An abstract approach to coherence. (in this volume).

[51

G.M. Kelly and S. Mac Lane, Coherence in closed categories, Journal of Pure and Applied Algebra I (1971) 97-14o.

[6]

S. Mac Lane, Natural associativity and commutativlty, Rice University Studies 49 (1963) 28-46.

A CUT-ELIMINATION

THEOREM

G.M. Kelly

University

of New South Wales,

Kensington

2033,

Australia.

Received May 22, 1972

i.

IntroductiOn We assume

volume, Let

familiarity

with the earlier paper [2]

and now refer p a r t i c u l a r l y

(B,p,D,G)

category

to §4.2 and §4.3 of that paper.

be as there the theory of an extra structure

or on a A-indexed polycategory,

with natural

in this

transformations

One problem, have incompatible

as we saw,

graphs;

in the m i x e d - v a r l a n c e

of the e v e r y - v a r i a b l e - t w l c e

K be the free model on A as c o n s t r u c t e d

on a

kind;

case and let

there.

is that composable

morphisms

of K may

then K is not a club, and the free model on

A cannot be w r i t t e n as KoA.

Even when K is a club,

easy to prove this fact directly

it may not be

from the c o n s t r u c t i o n

of K given

there.

Another p r o b l e m comes composite

in K, the object

from the fact that, if R ÷ S ÷ T is a

S may have more variables

(that is, higher

type) than either R or T; this forms a barrier to certain kinds of inductive

argument,

which,

absent in the purely

like the i n c o m p a t i b i l i t y

covariant

Both of these problems Kelly - Mac Lane results,

([3]

problem,

is

case.

were overcome

for p a r t i c u l a r

theories

and [4]) and in Lewis [6] by c u t - e l l m l n a t l o n

inspired by that of Lambek [5]

and hence ultimately

by the

in

- 197 work

of Gentzen

can be built composition this

(the analogue

as an i n d u c t i v e

-elimination theory

of this p a p e r

with

the

to some

a purely club

IP°l:

corresponding

IP°I(AI , A 2,

admits

a right

object

in the P.

x ... x A k ~ m

+ A~o

the

corresponds

...,

o an i a 6 n °. that,

then

when

we suppose

(n°[l~,

A of G such

to pO,

K.

× A; that we

choose o InO],

Then

o

a )

a model

if

the p a r t i a l

..., Aio_l , -, Aia+l , ..., Ano):

functor

Akiao ÷ A~o

adJoint.

For s i m p l i c i t y theorem

All

for each

type

as

cut-

Namely,

~ = ~oA

occur,

about

K is a club,

which

cases

in

does not

a general

kind.

pO has

allow

assertion

theory,

where

not i n v o l v i n g

(on some

it may

that

special

of G, where

is to be a model

is the functor

coherence

of K

In favourable

incompatibility

covarlant

n ° a l; and that we choose

of our theory

T of K);

is to prove

G in G ~ / ~

{pO} of objects

the m o r p h i s m s

by i n d u c t i o n

corollary

is of a certain

that we begin with

family

that

of some

with

that

"cut").

of proofs

proof

proof

theorem,

(B,p,P,~)

therefore

of Gentzen's

of the objects

an inductive

The purpose

asserts

ones by a set of p r o c e s s e s

the p o s s i b i l i t y

of c o m p l e x i t y

particular well

Such a result

up from simpler

restores

measure

[1].

-

of e x p o s i t i o n

case when

The reader

the family

will

and notation, {pa}

see at once

we shall

is r e p l a c e d

that

the proof

prove

the

by a single carries

over to

I

the more

general

difficulties we called

case;

become

horrlfic.

{pa} a f a m i l M

be distinct;

we avoid

The r e a d e r

advisedly;

if G c o r r e s p o n d s

and if we take

it only because

there

should

further

is no r e a s o n

to the theory

p1 = p2 = ® while

the n o t a t i o n a l observe

that

for the pO to

of m o n o i d a l

i I = 1 and i 2 = 2, then

categories, a model

of

- 198

our t h e o r y which

is a b i c l o s e d

cate~or~

both A@- and - @ B have r i g h t

that the pairs difference; different

(pa

names,

- that

to p o s i t

then t h e y ' l l

is,

a monoidal

adJoints.

i a) are d i s t i n c t

if we want

-

We m i g h t

- but

two right

it d o e s n ' t adJoints

be i s o m o r p h i c ,

category

in

as well s u p p o s e really

for A@-,

but that w o n ' t

make

any

with affect

our

result.

We f u r t h e r

simplify

the s l n g l e - c a t e g o r y this m a k e s ing that have

"active"

is a m o d e l

B I,

theory of type unit

..., B k ) :

A ÷

the last.

We spoke

loosely

G in w h i c h

.... B k, -):

first

by s u p p o s -

one; we take

P to

A l+k ~ A, and then a m o d e l

A

(k + 1)[--

and counit

A + A.

in d e s c r i b i n g

IPI admits

to be of the d e s i r e d

a right

equational

... - +}, part

in every

of our t h e o r y

adJoint form

as a

IQI; if we want

(~,p,~,c) we must

hom".)

our

we must make

Q,

also m a k e the

of the a d J u n c t i o n ,

of the data; that

a model

of the data;

..., Bk,

P(A,

e: P ( Q ( B 1 . . . . , Bk, A),

asserting

the n o t a t i o n

(Think of F as "@" and Q as " i n t e r n a l

d: A ÷ Q(BI,

part

f r o m the p r o o f that

A °p x ... x A °p x A ÷ A, c o n t r a v a r i a n t

except

of

IPi:

-

adjoint

IQI is a f u n c t o r

model

is the

IPI

of one e l e m e n t

A of G such that

IQI(B I,

variable

in

be c l e a r

we s i m p l i f y

k ~ O, so that

IPI(-,

has a right

A to c o n s i s t

it will

Finally

variable

1 + k where

of our theory

Here

again

no d i f f e r e n c e .

the

type

case;

by t a k i n g

and we must make

they

constitute

B l,

BI,

..., Bk)),

..., B k) ÷ A,

the t r i a n g u l a r

an a d J u n c t i o n ,

part

axioms

on d and e,

of the axioms.

-

Formal description

2.

of Q~ G together with one extra element Q

type.

The free discrete

an object !; has objects T(Y1, objects Q(YI' T(~,

-

of the theor~

B, then, consists of the appropriate

199

We identify

-'', !).

T E G wlth

The discrete

club of objects

of K is obtained from T by imposing the following relations "substitution" T(S1,

T(S1,

by B has

..., Yn ) for T E G, YI E T; and has

"''' Yk+l ) for YI E T.

..., ~) and Q with Q(~,

club T generated

p: the

..., S n) in Q~ K, when T, S i E G, is equal to the

..., Sn) of G; and the ~ of K Is equal to that of G. Call an object of K prime If it Is either ! or else of the

form Q(YI'

..., Yk+l ).

Then It is clear from the above that every

object K of K has a unique expression

(2.1)

K = T(X1,

where T E G and the factorizatlon

o f K.

In particular

that

Z = Q(YI'

...,

X1 a r e

of ~ is

Yk+l ) i s

...,

prime.

(The p r l m e ~(~);

!(Z).)

as Xn )

We c a l l

(2.1)

factortzation and the We c a l l

prlme

of T e G is

prlme Xl,

the

T(!,

factorizatlon

...,

Xn t h e

''',

!);

of

prlme

factors of K. D consists

of appropriate composite ls

with for

that

G wlth two extra elements

d:

! ÷ Q(!,

e:

P(Q,

graphs.

fg in

1T in G if

of ~

Finally

K when f ,

T E G; t h a t

I n G; a n d t h a t

an adJunctton.

!,

...,

!,

"",

!)

the

g E G is

P), ÷ !,

relations their

substitution d and e satisfy

~ assert

composite

that

i n G; t h a t

the 1T t n

In K of morphlsms In G agrees the

triangular

equations

g

-

200

-

Since K Is a fortlorl a G-category, ¢: G ÷ K of G-categories necessarily

faithful;

sending ~ to ~.

there Is a unique map

I conjecture that ~ Is

in the absence of a proof of this, however, we

shall now for simplicity replace G by Its image H under $.

The

kernel-congruence

~ of $ is clearly a club-congruence,

again a covariant

club wlth the same objects as G; we identify H with

a subcategory

so H = G/~ Is

of K.

The type n[+, +, ..., +] of an object of K that lles in H wlll usually be abbreviated to n.

An object of K of type 0[ ], or an

object of H of type 0, Is sald to be constant.

The prime factorlzatlon

of a constant T of H Is T ( ) . 3.

Central morphisms Define a central morphism of K to be one of the form

X(Xl,

..., Xn): T(X~l , .... X~n ) + S(Xl,

..., X n) where T, S E H,

where the X i are prime, and where x: T ÷ S in H with graph Fx = ~. is clear that the composite of central morphlsms every identity morphlsm Is central;

is central,

so the centrals

and that

form a subcategory

of K. Lemma 3.1 X(Yl,

I f x: T ÷ S i__qnH with graph ~, then

..., Yn): T(Y~I , ..., YEn ) ÷ S(YI,

..., Yn ) Is central whether

the YI are prime or not. Proof

Let the prime factorlzatlons

YI = RI(XI' x(Y1,

"''' Xm 1)'

..., Yn) = x(R1,

of the Yi be

"''' Yn = Rn(''" Xm)" .... Rn)(X1,

Then

..., Xm) , and x(R1,

..., R n) is

in H. Lemma 3.2 R(Zl,

I f zl: YI ÷ Zi is central for each i, and If R E H, then

..., Zn): R(Y1,

..., Yn) ÷ R(Zl,

It

..., Zn) is central.

-

Proof

Let zI = x I (Xl,

201

-

..., Xml): TI(X~II,

...) ÷ SI(XI,

where xi: T i ÷ S i is in H and the X i are prime. R(Zl,

..., Zn) = R(Xl,

..., Xn)(Xl,

...) etc.,

Then

..., Xm) , and R(Xl,

..., Xn) is in

H.

Observe that if z: Y ÷ Z is central then Y and Z have the same prime factors, 4.

Statement

to within order. of the theorem

For Y, Wi, Z 6 K write (4.1)

~: K(P(Y, W1,

w for the adJunction:

..., Wk) , Z) = K(Y, Q(W 1 . . . . , W k, Z)).

Thus ~(f) is in fact the composite (4.2)

Y ÷ Q(WI, d

.... Wk, P(Y, Wl,

where d stands for d(Y, Wl,

.... Wk)) ~ Q(W1, Q(I ..... l,f)

..., Wk, Z)

..., Wk).

For Z 6 g and for fi: Yi ÷ Wi in K (l~isk), write
..., fk >, or just
..., fk > for short,

(4.3) P(Q(Wl,...,Wk,Z),Y1,...,Yk)

÷ P(Q(WI,...,Wk, Z),WI,...,W k) ~ Z

where the first morphism is P(I, fl' e(W I, ..., W k, Z).

for the composite

..., fk ) and where e stands for

As a still shorter notation

for (4.3) we shall

also use (4.4)

: P(Q(W,

Z), Y) ÷ Z.

Define the set of constructible smallest set satisfying

morphisms

of K to be the

CI-C4 below:

C1.

Every central morphism is constructible.

C2.

Let T 6 H with type n ~ 2, and let fi: Yi ÷ Zi be constructible

for l~i~n.

Suppose

Yi and Z i are not both constants

for each i that of H (they are

-

allowed

to be constants

the following

K

composite

~ T(YI,...,

constructlble, composite

K~T(Y) a C4.

m T(ZI, ..., fn )

l, let f: P(Y, W1,

be constructible

..., W k) + Z be

and let u 6 n.

for l~J~k.

composite, (4.4)

4.1

for (4.3),

Every m o r p h i s m

§6 of [ 3].

Let a be central.

Zj ÷ Wj

Then the

is constructible:

~ T(V I...V u...v n ) ~ L -

the theorem

in §6 below.

The reader making two roles which

(B',p',D',G'),

a comparison

are separated

that we treat differently

T E B'; in [3|

objects

C4.

The objects

of K constructed

of types

fun, adj,

(apply a functor

by

that @ in

the role of P,

of type

to the

l, it would be

the only functor

in B' of

should also notice

type

~2, and those

of

in C3; both can occur in

of H of type 0 are of course

The morphlsms

inspired

if G corresponded

T of H'with

l; the first appear in C2, the second

observe

namely

The reader

type

of formation

should

here;

Clearly

type n~2 is @, which is also our P.

be respectively

It is clearly

and if B' had no objects

in C2 to suppose

g

of K is constructible.

and also that of T in C2 and C4.

sufficient

and let fj:

T(1... ...i)

We prove

theory

Let

where we have used the abbreviated

a

[3] plays

Then the following

..., W k, Z ) ) ~ L . b

K--~T(V I . . . P ( Q ( W , V u ) , Z ) . . . V n)

Theorem

~L. b

..., V n) + L be constructible,

following

Then

is constructible:

Let T E H with type n~l,

notation

..., Z n)

and let a and b be central.

m T ( Q ( W l, T(~(f))

g: T(V I,

Let a and b be central.

is constructible:

T(fl,

Let T E H with type

-

of K!).

Yn )

a C3.

202

covered

by C1.

in C2, C3, C4 will be said to

and av, to suggest

T; take an adjoint;

their methods

apply evaluation).

-

the case when there

In

there are of course

203

-

are many P~ with right

construction

adjoints

Qm,

rule C3 e anc C4 e for each e.

5. K is a club Define

r(1)

inductively

r(T(YI,

..., Yn )) = r(Yl)

r(Q(Y l,

..., Yk+l )) = r(Y I) + ... + r(Yk+ l) + I;

is central,

+ ... + r(Y n) for T E H,

this is a rank on the objects

club generated

with the equivalence

rank.

of K by setting

= i,

in the first instance discrete

a rank for the objects

by

B = ~H

relation

u {Q},

but it is clearly

p and defines

K and L have the same prime

of the free

a rank on ~

factors

K.

and hence

The only objects

of rank 0 are the constants

For a morphism

f: K + L of K set r(f) = r(K)

compatible If K + L

the same

of H. + r(L).

Then it

is clear that: Lemma

5.1

The m o r p h i s m

i; that constructed has rank

> r(fj)

constructed

in C3 has rank

in C2 has rank > r(f);

for each j and also

The category

K is augmented

over ~ ,

are simple

incompatibles

just by discarding

Our present category

purpose

~', whose

is better objects

T ÷ e consist

one new element

~.

in ~ is ~.

constructed

graphs.

served,

whose

We recall

objects

if n and ~ are compatible

however,

by introducing

a new

types, together

but with

of n and ~ in ~' is their composite

simple

That this is an associative

are types

any closed loops that occur.

of all the simple graphs

The composite

in C4

that we compose

are still the mixed-variance

whose morphisms

for each

> r(g).

and whose morphisms in ~

and that

> r(fi)

graphs;

composition,

in all other cases it making

~' a category,

-

from the fact that,

follows

then n is compatible

204

-

if ~ is compatible

with ~ and

~ with n~; and conversely.

It is clear from the c o n s t r u c t i o n that it can be given an a u g m e n t a t i o n composables

are incompatible

with n and ~n with ~,

of K in §4.2 and §4.3 of [2]

F' over ~'.

Then to say that no

in K is to say that F'f is never , for

fEK. The graph of a central m o r p h i s m is a pure p e r m u t a t i o n

(that

is, any mates under it occur one in the domain and one in the codomain); nature

it is therefore

of the constructions

does not have augmentation does.

compatible

with anything at all.

The

C2-C4 shows that the constructed m o r p h i s m * unless one of the h y p o t h e s i s - m o r p h i s m s

Lemma 5.1 n o w gives an immediate

proof,

by i n d u c t i o n on r(f),

that F'f 9 * for f E K; thus

Theorem 5.2

Composable morphisms

0f K have compatible

6raphs;

g is a

club.

6.

Proof of the main theorem

Lemma 6.1

If h is constructlble

and u,v are central then vhu is

constructlble.

Proof

Since constructlbles

from the form of C1-C4 for

vh

that

unless h is produced

clearly follows by induction

Lemma 6.2

are closed under composition, hu

is constructible.

it is clear

The same is clear

by C4; so the constructibillty on r(h),

I f K, L are constants

of vh

in view of Lemma 5.1.

of H, any constructible, h: K ÷ L i s

central.

Proof

Immediate

Lemma 6.3

from Lemma 5.1.

Le__~t T e H with type n z i, and let fi: Yi + Zi b_~e

-

for l~i~n.

constructible

Proof m+l,

Let those m+2,

Then

i for which

..., n say.

205

-

T(fl,

...,

Yi and

fn ) is constructlble.

Z i are both c o n s t a n t s

of H be

Then

T ( f l , . . . , f n) = T ( f l , . . . f m , 1 , .... I) T ( 1 , . . . , l , f m + l , . . . , f n ) . The r l g h t - h a n d discard

factor

it by Lemma

6.1.

T(fl'

"'''

fm'

l,

T(Zl,

...,

Zm,

Zm+l,

S(f I . . . . , fm): S = T(I, by C2;

...,

is central

...,

I): T(Y1,

...,

!, Zm+l,

"''' Ym'

Zn) 6 H.

if m = 0 it is the identity It remains

of C1-C4, 6.4

i E m+l.

this admits

Let S be an object

Zm+l'

Then

S(M 1 ... K

Zn) ÷

form ...,

Zm) where

to prove

h is.

Using

proof

is c o n s t r u c t i b l e

central

and

that,

if S 6 H has type

Lemma

3.2 and the form

by i n d u c t i o n

of H, of t~pe m+l where

L e t h: K ÷ L and t: S(M 1 ..... M i _ I , L ,

constructlble.

"'''

and therefore

an i m m e d i a t e

3.2, and we can

If m ~ 2 this

therefore

wherever

6.2 and

factor

Z n) is of the

...,

l, S(h) Is c o n s t r u c t i b l e

(6.1)

The r e m a i n i n g

S(Y 1 . . . . , Ym ) ÷ S(Zl,

constructlble.

Lemma

by Lemmas

Mi

on r(h). m ~ o, and let

, ..., M m) ÷ N b_~e

the composit e

... Mm)-

~ S(M 1 ...

s(1

...

h

L .~. M m)

~N

i)

...

t

is constructible. We shall to P r o p o s i t i o n

prove

6.4 of [3].

An instance o f therefore

constructlble.

therefore

constructible

the form
Lemma

I,

...,

T = i and g = i).

6.4 in the next Assuming

a morphlsm

it we prove

(with T = I).

If h is c o n s t r u c t l b l e by Lemma

Theorem

by Lemma

6.3.

An instance

constructible its e x p a n s i o n If h and fl'

4.1.

3.1 and

of d is of the form ~(I)

i>, and is t h e r e f o r e

for T 6 H is c o n s t r u c t i b l e

it is analogous

the main

x of H is central

An instance by C3

section;

and is

of e is of by C4

(with

T(l...h...l) "''"

fk are

-

constructible

so is Q(fl'

of a c o m p o s i t e T = ~).

h,

and the l a t t e r

The c o n s t r u c t l b l e s

the c o n s t r u c t l b l e s

m = 0 of L e m m a 6.4,

7.

-

"''' fk' h),

for this is the image u n d e r

is c o n s t r u c t l b l e

therefore

of d, of e, and of the m o r p h l s m s since

206

contain

of H; since

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

the c o n s t r u c t l b l e s

by C4

(with

all e x p a n d e d these

generate

composition

constitute

instances K, and

by the case

the whole

of K.

P r o o f of Lemma 6.4

In the s i t u a t i o n

(7.1)

of L e m m a 6.4,

= r(K)

+ r(L)

write

+ r(N)

+ Z r(M i)

and write

(7.2)

o o = r(K) + r(L).

The p r o o f

is by a d o u b l e

constructlble situations that

induction;

for all such

situations

constructlbillty Is central. their mode

of

S(I

(6.1)

of c o n s t r u c t i o n by C2,

f a c t o r S(1

also i g n o r e

... h

for all

the r e a d e r §5.

when h Is,

We s i m p l i f y

of

cases

according

by o b s e r v i n g

a, w h i c h m e r e l y

(6.1);

l:

h

Is of type

composite

puts

and that we can

by C2 or C3) by a b s o r b i n g

Similarly

we can ignore

factors

b

occurlng

~a.

Wlth h as In C2,

to

that,

at the end of t.

Case

the

by L e m m a 6.1 if e i t h e r h or t

... a ... l) In front

... i) i n t o t.

as we said in

or C4, we can ignore

b (when h Is p r o d u c e d

o, and also

We r e m i n d

h and t, we c o n s i d e r

by C2-C4.

C3,

to be

... I) Is central

is i m m e d i a t e

For n o n - c e n t r a l

If h is p r o d u c e d

... b

wlth a lower

then r(Y) = r(Z),

Since by L e m m a 3 . 2

S(1

(6.1)

w l t h the same o but w l t h a l o w e r o . o

If Y ÷ Z is central,

a central

we s u p p o s e

but i g n o r i n g

a and b,

(6.1)

becomes

the

-

207

-

. 1. . Xl" . "Yn "'Mm)R(l...fl...fn...l) • R(MI...Zl...Zn...Mm)~N, (7 .3). R(M t where R E H is S(l, ..., i, T, ~

..., ~).

Consider the composite

(7.4) R(MI...ZI...Zn_ 1 Yn...Mm) ~R(MI...ZI...Zn_IZn...Mm) R(l...fn...l)

rN. t

This is again of the form (6.1); but its a has only a contribution r(Z l) where the e of (6.1) itself, now written as (7.3), has a contribution r(Yl) + r(Zl)(we are using the fact that n ~ 2).

So the

of (7.4) is lower unless r(Y1) = 0, that is, unless Y1 is a constant of H.

But in that case Z 1 is not a constant of H, and then the ~o of

(7.4), namely r(Yn)+r(Zn) , is less than that of (6.1) = (7.3), namely r(Y l) + ... + r(Y n) + r(Z I) + ... + r(Zn).

So by induction (7.4) is

constructlble. Repetition of this argument if n > 2 yields the constructibillty of

~R(l...fn_ 1 ... i), where E is (7.4); and hence ultimately

of (7.5)R(M1...Z 1 Y2...Yn...Mm) ~ R(M1...Z 1 Z2...Zn...Mm)--~N. R(l...f2...fn...1) t Writing t' for (7.5), we can write (7.3) as

(7.6) R(MI...Y 1 Y2"''Yn "''M)mR(l...fl.. .~)R(MI'''ZI Y2"''Yn "''M-)---~N'm t' The ~ of (7.6) is less than that of (6.1) = (7.3) unless r(Z n) = 0; but in that case the o o of (7.6) is les~ than that of (6.1). induction (7.6) is constructible, Case

2:

h

is of type

So by

as required.

zv

With h as in C4 but ignoring a, we can write (6.1) as t S(l ... g ... l) R(1 ... ... 1), where R = S(1 ... T ... l) 6 H.

-

The

208

composite t' = t S(l...g...l)

-

is constructlble

having a lower c than (6.1) since r(g) < r(h).

by induction,

Then (6.1) is

t' R(1......1) which is constructible by C4. Case 3 :

h is of t~pe adj

We take h to be G(~(s)):

G(A) ÷ G(Q(B,C)),

where B stands for

B l, ..., B k.

We may as well take G = ~, for we can replace S by

s(~

~).

...

s

...

(7.7)

h

Thus

~(s): A ÷ Q(B,C),

=

where (7.8)

s: P(A,B) ÷ C

is constructible. prime;

Next, we may as well suppose the M i in (6.1) to be

for if their prime factorlzations

Just replace S by S(R 1 ... ! ... Rm). (7.9)

S(l...h...l):

S(MI...A...Mm)

are M 1 = Rl(Xl,

...) etc, we

So the codomain of

~ S(MI...Q(B,C)...Mm )

is already expressed in its prime factorlzatlon. Whether t is of type

~un, adj,

or gv, its first factor is a

central a, which we shall write for the moment as (7.10)

a: S(MI...Q(B,C)...Mm)

Let the prime factorlzations

~T(Yl,

..., yn) .

of the Yi be Y1 = RI(XI''')

T(YI--.Y n) = T(RI...Rn)(XI...Xm+I).

etc., so that

Since a is central the primes X i

are Q(B,C) and the M i in some order, and a is X(Xl...Xm+l) x: S ÷ T(RI...Rn)

in H with some graph 6.

for some

Suppose for simplicity that

equates Q(B,C) with the prime factor X 1 of Yl" so that Y1 = RI(Q(B'C)' by

X2 . . . . )"

Set U = Rl(A, X 2 . . . . ) and define p: U ÷ Y1

-

(7.11)

209

-

p = Rl(h , X2, ...): U ÷ YI"

We have the commutative diagram

S(MI...A...M m) S(1...h...1) I

x(A,X2-..Xm+ I) T(RI...Rn)(A,X2...Xm+I) ~I T(Rl'''Rn)(h'l''''l)

S(MI...Q(B,C)...M m)

T(RI...Rn)(Q(B,C),X2...Xm+I);

x(Q(B,C),X2...Xm+ I) that is, a I

T(U, Y2...Yn)

S(MI...A...M m)

(7.12) [ S(1...h...1)

I

T(p, I, ...I)

S(MI...Q(B,C)...M m)

T(YI' Y2 " ' ' Y n )'

a

where we have written a' for the central x(A, X2...Xm+l). More generally now, allowing for the fact that ~ may equate Q(B,C) with any prime factor of any Yi' we have expressed aS(1 ... h ...I) by (7.12) as T(I ... p ... l)a', with a' central and with (in place of (7.11)) P: U ÷ Y

i

of the form

(7.13)

R(X 1 ... A ... Xq) R(I ... h ... -i) R(XI...Q(B,C)...Xq)

for some R 6 H.

Let us express the ~ of (6.1) in these terms; since

the Xj in (7.13) are some of the Mj; since the prime factors of the Yu for u ~ I are the rest of the Mj; and since K = A and L = Q(B,C) -the o of (6.1) is (7.14)

o = r(A) + r(Q(B,C)) + Z r(Xj) +

Z r(Yu) + r(N). u~i

-

Since we can ignore the c o n s t r u c t i b i l i t y is c o n s t r u c t e d

Case

-

a' by Lemma 6.1, we are reduced

of t T(I

the a, which we have

for, and wltho t the b, w h i c h we can ignore.

3(a):

t is of type ~a~

T(f I .,. fi p ... fn ). is constructible.

. I. . (7 .15). R(X

This is c o n s t r u c t i b l e

(6.1), and its o is strictly

r(Yj) + r(Zj) with j # i.

fip

q) ..... fi ~ less than

Zi;

(7.14);

for

~2, we have lost at least one c o n t r i b u t i o n Hence

(7.15) is c o n s t r u c t i b l e

by induction.

t is of type gd#

Then n = I, so T(I so that t T(I composite

by Lemma 6.3 p r o v i d e d

A . . X q ) R ( l . . . h . . . l )~ R ( X I . . . Q ( B , C ) . . . X

r(N) = ~ r(Zu) and, n being

Case 3(b):

... p ... i) is

But fip is

this is of the form

... p ... i) is Just T(p);

... p ... I) is T(~(f)p).

and t = T(~(f)),

By the n a t u r a l i t y

of ~, the

~(f)p is ~(g) where g is the composite

(7.16)

P(U,W)

~

P(Y,W)

P(P,I) Setting H = P(R,I

~

Z.

f

... I) E H (recall that W stands for Wl,

this equal by (7.13)

(7.17) is of the form (6.15, and its ~ is less than

r(N) = r(Q(W,Z)) by induction,

= r(W) + r(Z) + i.

whence

..., Wk) ,

tb

(7.175 H(X I. . ..A. . . . X. q. ,. W ) H ( I . . . h . . . I~ 5H ( X I . . . Q ( B , C ) . . . X q , W )

C3.

We

to the cases.

Then t = T(f I ... fn) and t T(I

But

to p r o v i n g

... p ... i) where t: T(Y 1 ... Yn ) ÷ N

by C2, C3, or C4 but without

already a l l o w e d proceed

210

t T(I

f ~ Z.

(7.14) because

Hence g = (7.175 is c o n s t r u c t i b l e

... p ... I) = T(~(g))

is constructible

by

-

211

-

Case 3(c) : t is of t~pe zv We take t = g T(I ... ... I) as in C4, with a omitted. Here YI' "''" Yn = VI' "''' P(Q(W, Vu) , Z), ... V n.

We must distin-

guish three subcases: Subcase (i) Suppose the codomaln Yi of p is one of the Vj for J # u; for simplicity suppose it is YI = Vl" g T(I ...... l) T(p,l,

Then t T(p,l,

... l) = g T(p,1,

..., I) =

..., I) T(1......1).

This will be constructlble by C4 if g T(p,l,..., i) is. (7.18)

T(U'V2'

The latter is

"''' Vn)T(p,l, .......... ..,I)~T(VI' V2' "''' Vn) .... g ~-N;

that is, setting H = T(R, ~, ..., i),

(7.19)

H(XI'''A'''Xq'V)H(I...h[. .~) H(XI'''Q(B'C)°''Xq'V) -'----mPN'g

where V = V2,

..., V n.

This is of the form (6.1), and its a is less

than (7.14) because we have replaced the contribution r(Y u) = r(Q(W,Vu)) + r(Z) by r(Vu) , which is strictly less; hence it is constructible by induction. Subcase (ii) Suppose the codomain Yi of p is P(Q(W,Vu),Z ) and that the prime factor Q(B,C) of Yi is Q(W,Vu). P(h,l): P(A,Z) ÷ P(Q(B,C),Z). of

Then U is P(A,Z), and p is

It is easy to verify the commutativity

-

T(VI...P(A,Z)...V n)

212

-

T(1...P(l,f)...l) ...................... ~ T ( V I . . . P ( A , B ) . . . V n )

~

T(1...s...1)

T(I-..P(h,I)...I) I T(VI...P(Q(B,C),Z)...V n)

T(VI...C...Vn), T(l......l)

where h = ~(s) as in (7.7).

Then g T(l......l) T(I...p...I)

g T(l...s...l) T(l...P(l,f)...1).

is

First, g T(1...s...l) is of the

form (6.1), and it is constructible by induction because its =

Z

r(Vj) + r(A) + r(B) + r(C) + r(N) is less than (7.14) since

j~u

r(Q(B,C)) = r(B) + r(C) + I.

Next, writing g' for g T(I...s...I),

and

H for T(~...P...I), we have to prove the constructibility of

We prove this step-by-step as in Case l; only we need not, as there, suppose k a 2, or use the side induction on Oo; for E

r(V~) + r(A) + r(Z) + r(B) + r(N)

J~u

is already less than (7.14), r(Z) being the ~ r(Xj) of (7.14). Subcase

(iii) Suppose the codomain Yi of p is P(Q(W,Vu),

Z) and that the

prime factor Q(B,C) of Yi is a prime factor of some Zj, say Z I. Z~ for Z 1 with the prime factor Q(B,C) replaced by A.

Write

Then p is

P(1,r,l...l), where r: Z~ ÷ Z 1 is of the form R(1...h...1):

R(X~...A...) ÷ R(X~...Q(B,C)...).

In this case (7.14) becomes (7.2o) = r(A) + r(Q(B,C)) + r(Q(W,Vu) ) + ~ r(Xj) +

~ r(Zj) + ~ r(Vj)+r(N). j#l J#u

-

213

-

It is easy to verify the commutativity of T(VI...P(Q(W,V u), Z{, Z2... )...V n) ...

> ...

i)

T(Vl""" P(Q(W'Vu)'Zl' Z2"'" )'" "Vn)T(1... ... ~ T(VI' ""' 'vn) where ~ = fi r , f2' "''' fn"

Thus t T(l...p...l) = g T(I...<~>...I) is

constructible by C4, provided that fl r is constructible.

But fl r is

the composite (7.21)

R(X{...A...)

....... r-- R(X{...Q(B,C)...) R(l...h...l)

which is of the form (6.1). r(Q(W,Vu)) > r(Wl).

r

WI,

fl

The s of (7.21) is less than (7.20), since

Hence (7.21) is constructible by induction.

This completes the proof. REFERENCES [ 1]

G. Gentzen, Untersuchungen ~ber das Logische Schliessen I,II, Math. Z 39(1934-1935), 176-210 and 405-431.

[2]

G.M. Kelly, An abstract approach to coherence (in this volume).

[ 3]

G.M. Kelly and S. Mac Lane, Coherence in closed categories, Jour. Pure and Applied Alg. I(1971), 97-140.

[4]

G.M. Kelly and S. Mac Lane, Closed coherence for a natural transformation (in this volume).

[5]

J. Lambek, Deductive systems and categories I. Syntactic calculus and reslduated categories, Math. Systems theory 2(1968), 287-318.

[6]

G. Lewis, Coherence for a closed functor (in this volume).

A NEW RESULT OF COHERENCE F O R D I S T R I B U T I V I ~ Miguel L. Laplaza University of Puerto Rico at Mayaguez Received May 3, 1972 INTRODUCTION Let C be a category with the additional structure given by the following data: i) ii) iii)

iv)

TWO functors, 0, ®:C X ~

> ~.

Two objects, U, N, called the unit and null objects. For any objects, A, B, C of ~, natural isomorphisms, ~A,B,c:A®(B@C)

> (AeB)®C,

YA,B:A®B----~B@A,

~,B,C:A®(BeC)

> (A®B)eC,

y~,B:AeB

kA:U®A----~A ,

PA:A®U---~A,

k~:NeA---~A,

p~:AeN

k~:N®A -

p~:A®N----~N.

> N,

f~ BeA,

>A,

For any objects, A, B, C of ~, natural monomorphisms, 6A, B, C :A@ (B®C) - - ~ (A®B) • (A®C), 6# • (AeB) ®C ----~(A®C) e (BeC) A,B,C" In [5] we have given a coherence theorem for this situation

that can be summarized as in [4] or as follows. Let X be the s e t ~ x l , x 2 ,°'',Xp,n,u } , A_ the free { +, "~ -algebra over X and G the graph consisting of all the following formal symbols for x,y,z 6 A,

-

~x,y,z:X(yZ)

• (xy)z,

215

-

~' :X + x,yrz

(y + z)

Ax:UX _

• x

,

A':n + x x

% x,

Px:XU

- x

'

Px' : x

~x

% yx

,

7~,y:X

7x,y:Xy

k*:nx x

+

n

• (x + y) + Z,

'

+ y

~y

+ x,

~n,

p~:xn----~n,

their

formal

inverses,

6

indicated

by the u p p e r

:x(y + Z)

x,y,z

We c o n s t r u c t H the u n i q u e jections

•xz

+ yz,

"x.

the free {+, "~-algebra,

extension

of the g r a p h

are { + , ° l - m o r p h i s m s .

w i t h at m o s t one e x c e p t i o n ,

-1, and,

~ xy + xz,

6# :(x + y) z x,ytz 1 :x x

index

H, o v e r G and we take on

structure

An e l e m e n t only elements

of G in w h i c h

of H is an i n s t a n t i a t i o n of G of type 1 --

in its e x p r e s s i o n . instantiations

(*)

We d e n o t e

of G.

Yl-

"Y2

the p a t h s

....

if,

are i n v o l v e d x

by T the g r a p h c o n s i s t i n g

We d e f i n e d

the pro-

of all the

as the s e q u e n c e s ,

~Ym+l

'

w h e r e ~. E T, i = 1 , 2 , ' o o , m . 1 Each

map,

f:X

tended

in a n a t u r a l

arrows

and o b j e c t s

product

)Ob

of ~.

of the v a l u e s

of r e p l a c i n g ,

x of X by f(x),

f(u)

= U,

f(n)

= N, can be ex-

w a y to a m a p of the g r a p h T o n t o the g r a p h of the

to a m a p of the p a t h s result

C such that

every

The v a l u e

of a p a t h can be d e f i n e d

of the steps and so we have e x t e n d e d into the a r r o w s in e a c h p a t h

as the

the m a p f

of C that can be d e f i n e d

(*), any o c c u r r e n c e

• by ® and e v e r y + by e.

as the

of an e l e m e n t

-

Our paper path which

216

[5] has studied

insure

the conditions

that its value only depends

end of the path.

For this we introduced

let A* be the free ~+, ~ -algebra mutativity element

for

element

na = an = n for a 6 A*.

regular

the support,

if Supp(x)

~* each of which herence

b, C is coherent

upon the origin

the concept

of

is a product [5] states

and the

and com-

to +, null

condition,

The identity map of X defines Supp:A

of the

of regularity:

° relatively

u, and the additional

a ~+,~-

> ~*, and an element x of A is

can be expressed

result of

on the origin

over X with associativity

° and +, d i s t r i b u t i v i t y

n, identity

morphism,

-

as a sum of different

of different that

if P,Q:a

and a is regular,

elements O>b

elements

of X.

of

The co-

are paths

from a to

then P and Q have the same value

in C. Suppose that P,Q:a

O>

that C satisfies

the coherence

b are two paths.

conditions

of

[5] and

We want to study the conditions

under

which P and Q have the same value

in C.

We will define

for each path

a finite

called

the distortion

and prove

sequence

of permutations

that if P and Q have the same distortion, value.

The method used to define

we construct

a category

the distortion

D with a structure

ditions

i) to iv) of this introduction,

quences

of permutations

and define

any path in Arr D defined The possibility by Saunders paper were though

their

author

is deeply

of type given by the con-

whose arrows

a map,

g:X

are finite

se-

> Ob D; the value of of the path.

of a result of the above type was suggested communication.

largely by the methods

subsequent

is the following:

from g is the distortion

Mac Lane in a private inspired

then they have the same

development

indebted

used in

has dimmed

to Professor

The ideas of this [i] and

[3], al-

that connection.

The

Mac Lane for his assistance.

-

i.

S o m e Preliminary We are going

217

Concepts

to explain

on Permutations some concepts

in the definition

of the category

also some results

intended

for this we will define

-

of distortions

tained

a category

some of the required

in the lemmas below;

straightforward

(and long)

of the categorical

way,

[, coherent

results.

taking

definitions

to the product

of Chapter V of

[6].

i)

the definitions

independently

categories

it is

("PROPS")

in the definitions

by a simple routine.

g:A n

If(x), (f + g)

iii)

of Sets whose objects number.

(for any natural

are the

We define

two

numbers m, m',n,n'

> An,):

7 (x)=qm'

if 1 ~ x ~ m,

+ g(x - m),

if m < x ~ m + n,

If x = i +

(j-l)m, with 1 < i < m, 1 < j < n, then,

(f x g)(x)

= f(i)

+

[g(j)

- 1]m'.

It is easy to prove that + and × are associative, A 0 = @ and unit object

relatively

natural

A I.

Moreover

in

ap,

x

(Am + An ) = A

P

x A

m

+ A

p

x A

n'

Am a n d An .

The permutations

t

P,q

and ~

P,q

w i t h null

x is left distributive

to +, and the relation,

Ap is

in a

Am + A n = Am+ n, A m × A n = Amn,

ii)

object

are con-

sums and products;

where n is any natural

f:Am----~Am,,

[5],

P is a full subcategory

We will omit all the details

+ and × by the conditions

and any maps,

These results

of direct

Let P be the full subcategory

functors

of D:

in the sense of

and permutation

that can be completed

sets An= ~l,2,°°°,n~,

state

[5] can be used in an effective

The category

closely related

and proofs

We will

we note that they can be proved

framework.

of Sets with suitable

D.

later

to ease the proof of the coherence

such that the coherence' theorem of way to prove

that will appear

defined

by

-

i) tp,q (x) =

i~

-

÷ x, if 1 < x < p, -- -

ii)

218

p,

if

p

<

x

<_ p

+

q,

if x = i + (j-l)p, with 1 <_ i <_ p, 1 <_ j <_ q, then, ~p,q(X)

= j + (i-1)q,

are natural isomorphisms, tp,q:Ap + Aq-----~Aq + Ap,

_p,q:Ap

x Aq

~ aq

x AP .

We can define now a natural isomorphism, a m,n,p # :(a m +

by the commutativity

An)

x nP

>

(Am

x

Ap)

+

(n n x

of the diagram,

(Am + An) x A p I~

>

6# m,n,p

(Am x Ap) + (An X Ap) I~ +~

m+n,p

~m,p

Ap x (Am + A n ) that is,

ap) ,

-

n,p-

(Ap × Am) + (Ap x An)

6~

m,n,p = ( % , m + ~p,n) % + n , p " If we take for ~, ~' , k , p, k' , p ' , ~*, p*, 6 the identities,

and 7Ap,Aq = tp, q, 7Ap,Aq = ~p~q~

6 p~Aq'As

situation given by the conditions

i) to iv) of the Introduction.

proof of the coherence of P is easy

p,q,r~ we have the The

(see the final remarks of ~l of

[5], and observe that the functors + and × are the direct sum and product). For all the following lemmas we make the following conventions: Pn is the permutation group of An, i n the identity map of A n , p,q,r and s are any natural numbers,

the symbols Ap will be represented

simply by p when they occur in subindices and the parenthesis and the symbol x will be omitted when no mistake can arise.

-

LEMMA i.i. i)

219

-

If a, a' E Pp and ~, ~' q Pq~ then,

(a + ~) (a' + ~') = U u' + • $',

ii)

(a X ~)(~'

iii) iv)

X ~') = a u' X ~ ~',

tp,q(a + ~) = ( ~ + U)tp,q ~p,q(a X ~ )

= ( ~ X a)

~p,q"

The different parts of this lemma are concrete cases of the naturality of +, x, tp,q and % , q

respectively.

The proof is routine.

For the proof of the next lemmas the method is similar; each states the commutativity of a diagram that can be proved by means of the coherence theorem of [5] applied to P.

It will be sufficient to

show those diagrams in the first three lemmas. LEMMA 1.2. i)

tp+q, r = (tp, r + lq)(ip + tq,r),

ii)

pq,r = ( p,r x lq)(lp x

Proof:

q,r )

The relation i) is the commutativity of the diagram,

P + (Aq + a )r I

a'

> (ap + Aq) + Ar

p,q,r

,

>Ar

Yp+q,r

+ (ap + Aq) I ~*

ip+Yq,r

r,p,q

AP + (Ar + Aq)

~' p , r , g > (Ap + Ar ) + Aq

Y~,r+lq

~(A r

+

+ Ap)

q.

The diagram for ii) is similar. LEMMA

1.3.

tpq, pr Proof:

=

ip

x

tq, r"

The lemma states the commutativity A

P

A P

x (Aq + Ar)' 1 pXyq ,r X(Ar + Aq)

6p,q, r

of the diagram,

> A X A + A X A p q ~y p r

~ pq,pr 6 ~A X A + A X A p r p q p,r,q

-

LEMMA

1.4.

( % , q + ~r,p)~p+q,r(tp,q Proof:

220 -

X ir ) = tpr,qr ( % , p + ~r,q)~p+q,r.

The lentma states the commutativity of the diagram,

p,q

r

q

Ap) dr

6#

q r

p+q,r

p r

pr,qr 6

Ar(Ap + ~q) ~

dr pA + dr d q

Yr

'P

+7r,

q>dpAr + Aq ~ r

LEMMA 1 •5.

[lps + ( % , q + ~s,r)~q+r,s ] (~s,p + % , q + r ) = [(~s,p + ~ s , q ) % + q , s LEMMA

"Cs,r)"

1.6.

(%,pr + 17s,qr)'Cqr+pr,s[(~r,p+~,q)%+q,rXls LEMMA

+ lrs] (%-s,p+q +

] = (~rs,p+~rs,q)~+q,rs

1.7.

1p x [ ('~s, q + ~s,r)~q+r,s] = ( % , p q + ~s,pr)~pq+pr,s . LEMMA 1 •8 + tpr,q r + lqs) (%+s, p + ~r+s,q)%+q,r+s * C r,p ÷

2.

r,q> p÷q,r ÷

÷

The Category Of Distortions our next aim is to describe the category of distortions,

D,

which is coherent in the sense of [5]: the distortion of a path will be an arrow of D determined by a method requiring largely the use of the structure

of D. We will give the definition of D and its structure

but many details are to be omitted: a routine allows one either to

-

2 2 1

-

check them or reduce the situation to one of the lemmas of § 1. Let D be the category whose objects are the finite sequences of natural numbers

(the empty sequence @ included),

we will abbre-

viate the finite sequence (al,a2,°'*,ar) by (a°) and identify each natural number with the sequence consisting of that number alone. Consequently in the expressions (a.) and (b.) it is not supposed that ° ranges over the same set of indices.

The length of an object

of D is the usual length of a finite sequence.

The set of arrows

between sequences of different lengths is empty, and otherwise the definition is given by,

D[(a°), (b°)] = {(u; ~l,°°°,~r)la E Pr' ~i E Pai , a i = ba(i) ~ where r is the common length of (ao) and (b°).

I

Hence the existence

of an arrow from (a°) to (b°) implies that the first sequence can be obtained by a permutation of the second. The composite of arrows is defined by,

(a'; (Z~, °°°, ~r)(U; ~i' "°°' (Zr) =

(~'O; ~'a(1) ~i' ~'a(2)~2 ' ''0 ' ~'U (r)~r)

"

Thus we have defined a category D where the identity of the object (a1, "'', a r) is (lr; lal, ".', far) or simply (lr,1) if we make the convention of representing any element of type (a;

°'° , lnr) by ( u; 1) when no misunderstanding can arise.

lnl,

Note that the identity of the empty sequence is 1 I, the identity map of A 1 . The functor e:D × D

>D

is defined on the objects by juxta-

position, that is,

(aI, "'', a r) • (bI, °'°, b s) = (al, °'', ar, bl, °'', bs). and on the arrows by using the functor + defined in § i in the following way:

- 222 -

(~;~i''"

quence

,~r ) •

( o ' ; ~ , • " ,~s' ) =

The

functor

@.

A natural

• is a s s o c i a t i v e

of

(a.)

by,

and

, p ,

(bo)

k' , and

> (b.) •

where

.}

9' the

is a c o h e r e n t

is, D is a s y m m e t r i c

The functor

(al,

object

the empty

se-

of c o m m u t a t i v i t y ,

(tr,s;l),

for ~',

, y , [2]; that

•

l, .... ,~) .

Ca•)

,

r and s are the lengths

respectively.

If we take

and

:(a.)

7(a•),(bo)

(b.)

w i t h null

transformation

Y(a.),(b.)

is d e f i n e d

(a + o ' ; ~ l , ' " . ~ r , ~

®:D × D---~D

•••, a r) •

(b I,

identities situation

monoidal

for •

that of

(see

on the o b j e c t s

[7]

[8]).

by,

(a I + bl,a 2 + b I,

°••,

ar + b I ,

a I + b2,a 2 + b 2,

"'',

ar + b 2 ,

•

and on the arrows

in the sense

category

is d e f i n e d

"••, b s) =

we c o n c l u d e

•

°

•

•

a I + bs,a 2 + b s,

*'•,

•

a r + b s),

by,

(a; ~i' ~2' "••' ~r ) • (a'; ~{, a~, •••, ~ ) : (o x o,; : l + ~{' ~2 + ~{' ~l + ~ '

~2 + ~ '

"'"

=r + ={'

•

"" ' ~r

+ ~F

=l + ~s' r ~2 + ~'s o •'• t ~r + ~s) w i t h the d e f i n i t i o n Intuitively be d e s c r i b e d transpose between

of × given the a c t i o n

as the r e s u l t

of the m a t r i x

the e l e m e n t s

this d e s c r i p t i o n

of

(a.)

in ~ i. of ® on the o b j e c t s "reading

by

(b.)

by columns"

the a c t i o n

A slight

of • on a pair

(bo) can

the p r o d u c t

(using the a d d i t i o n

of the two m a t r i c e s ) .

explains

(a.) and

of the

as o p e r a t i o n

modification of arrows

and

of

- 223 this proves almost immediately that ® is left distributive relatively tO @. The functor ® is associative with unit object 0 (the sequence with the element 0 only) and both the products are the empty sequence @.

(a.) ® @ and @ ® (a.)

A natural transformation of commutativity, > (b.) ® (a.)

7(a°), (b°) : (a.) ® (b°)

,

is defined by 7(a.),(b.) = (

,q; tal,bl ' ta 2 ,b I' "°" ' tap,b I tal,b 2' ta2,b 2, "'', tap,b 2 •

•

•

°

tal,b s' ta 2 'bs'

•

•

''° , t ap,b s) ,

where r and s are the lengths of (a°) and (b°) respectively. If we take for ~, k, p the identities we conclude that {D, 0, ~, y, k, p, 0} is a coherent situation in the sense of [7] and [2]; that is, D is a symmetric monoidal category for ® (see [8]). As we have pointed already, ® is left distributive relative to ® but not right distributive.

A natural transformation,

6#(a.), (b°), (c.) : [(a.) • (b°)] • (c°)

) [(a.) ® (c°)] • [(b.) ® (c.)],

can be defined by the commutativity of the following diagram, where we have omitted the symbols 0:

> (a.)(c.)e(b.)(c.)

[ (a.)e(b.) ] (c.) 6# (a.) I

Y(ao)S(b ) , (c)

(c.) [ (a°)e(b°) ]

,

(b.) , (c°)

JY (a.),(c.)e¥(b.),(c.) (c.) (a.)e(c.) (b.)

so that we need not check the naturality of the definition.

A simple

-

computation

224 -

proves that if the lengths of

q and r, respectively,

6#(a.),

(a.),(b.)

and

(c.) are p,

then

(b°) (c.) = [ ( ~ '

r

,p

+%,q)

~ p + q , r ;I]

If we take for 6, k* and p* the identities we have the structure given by the conditions

i) to iv) of the Introduction

easy to prove that D is coherent As we have pointed

in the sense of

in the Introduction

element x i of X = {Xl,X2,

[5].

if we fix for each

°'°, Xp, n, u~ an object f(x i) of D, each

path whose steps are instantiations arrow of D.

and it is

of T takes a value that is an

When we take for each i,f(x i) = 1 (the sequence with

unique element i), the value of the path is b ~ d e f i n i t i o n tion of the path.

We shall denote by d i s t ( ~ )

the distor-

the distortion of the

path ~ .

~3.

Some Complements

on the Construction

For a given set, X = {Xl,X2, the free {+,

of this paper or ~ 2 of [5]).

change the set and take X' = {xl,

x~,

>X

is a map such that k(u')

category with the structure referred

f'(n')

>0b

~, f ' : X ' - - - ~ O b

by A' and T'

~, two maps such that, f'(u')

M r

~ X o

~

O

b

C

Suppose

= u, k(n') = n, C a

in the Introduction

= f(n) = N, and that the diagram

k

If we

"°', X'p,, n', u'} we adopt the

of denoting the above constructions

now that k:X'

f:X

°'°, Xp, n, u S_ we can construct

"}- algebra over X, A, and from it the set of instantia-

tions, T (see the Introduction

convention

of the Formal Paths

and = f(u) = U,

225 -

-

is commutative.

The map k can be extended

in a natural way to maps

from A' to A, from T' to T and from the paths w i t h steps in T' to the paths w i t h steps in T: the action of these maps can be described tuitively

as the result of replacing

f(x i) , u' or n', respectively. represent

matter

The maps

f and f' define values

in T and T' respectively

to prove that k preserves

given a path,

of xi,u or n by

The same symbol k will be used to

any of these extensions.

for the paths with steps

any occurrence

in-

the values

and is a simple

of the paths;

that is,

.Tq P :a I

~ a2

such t h a t ~ i q T, 1 = 1,2,

as+ 1

"'', s, then the value of the path,

k (~'i).. k(P) :k(a I)

k(~"2 ) k(a2)

,

k(."s ) % . . . .

k(as+ I)

,

is the same as the value of P. If we apply the preceding tion we have proved

remarks

the following

lemma,

to the case of the distorwhere X and X' are of the

type just described: LEMMA 3.1.

If k:X'

• X is a map such that k(u')=u,k(n')=n,

then for each path P' with steps LEMMA 3.2. paths whose

X' = { x ~ , P', Q':a' i) ii)

inverses,

x~,

k(u')

iii)

There

= P, k(Q')

,

a

map k:X'

= Q, k(a')

two of

to find a set,

> X and two paths,

= a, k(b')

= b.

of the unit element u' in the

of the paths P' and Q'. of P and Q there

then in the vertices I ,

k(P')

= n.

If in the vertices

n

Then it is possible

is no occurrence

vertices iv)

=dist

w i t h steps in T', such that:

= U, k(n')

k(P')

dist(P')

of T that are instantiations

y and y'

"'', x'p, n' , u'

O>b',

8

Let X, A and T be as above and P , Q : a - ~ - ~ b

steps are elements

~, ~', their

in T'

is no occurrence

of P' and Q' there

of n,

is no occurrence

of

226 -

-

Proof: obtained

Take X ~ =

adding

a new element,

k(x i) = x i for i = 1,2, Then

for e a c h e l e m e n t

and no o c c u r r e n c e the e l e m e n t s of ~, ~', ~',

Xp+l,

of a A there

to X and d e f i n e

is o n l y one a'

an e l e m e n t , ~ ,

inverses,

subscripts

¥ or y', =~,

by s u b s t i t u t i n g

Note

result

= a

is true for

is one and o n l y one element, of u in the e x p r e s -

that all these

Xp+ 1 for u.

= n.

is an i n s t a n t i a t i o n

w i t h no o c c u r r e n c e

of ~ ' .

taking,

such that k(a')

of T that

there

k:X'----~X

= u and k(n)

of the u is in a', and a s i m i l a r

in _T' such that k(~')

obtained

"'" , X p , X p _ 1 , n, u} to be the set

"'', p, k ( X p + I) = u, k(u)

of T: g i v e n

their

sion of the

Xl,X2,

elements

The s t a t e m e n t

can be

of the lemma

is

n o w clear. LEMMA

3.3.

Let X and A be as a b o v e and a an e l e m e n t

of A w

w i t h no o c c u r r e n c e set X', a map, k(a')

k:X'

= a, k(u') Proof:

of X' o c c u r 3 of

of u or n in its e x p r e s s i o n . >X

and a r e g u l a r

= u, k(n')

a' of A'

a

such that

= n.

We will c o n s t r u c t

at m o s t o n c e

element

Then t h e r e e x i s t s

a' in such a w a y that the e l e m e n t s

in the e x p r e s s i o n

of a' and so by P r o p o s i t i o n

[5] a' is regular. The p r o o f of the lemma can be d o n e by i n d u c t i o n

of a, that

is the n u m b e r

formal d e f i n i t i o n s t r u c t b y the kl:X ~

of the n o r m

induction

~ X, k 2 : X ~

u n i t and null e l e m e n t s

see 9 3

hypothesis

>X

a~ and a~ are r e g u l a r

extension

of o c c u r r e n c e s

such that,

elements;

of

of e l e m e n t s [5]).

lal, n o r m

of X in a

(for a

If a = a I + a 2, con-

the sets X i and X~, k l ( a ~) = al, k(a~)

we can s u p p o s e

on

the m a p s = a2, w h e r e

that X { N X ~

= [u,n~

,

!

of X, and we take X' = X ~ U X 2 and for k the

of k I and k 2 to X'.

If a = ala2,

a similar

construction

can

be used. LEMMA beginning

3.4.

of ~3, ~ : a

of ~, e', t h e i r k(a')

= a.

Let X , X ' , k : X '

~ b an e l e m e n t

inverses,

Then there

> X, A, A', of T that

~ and [' be as in the is an i n s t a n t i a t i o n

y or y', and a' an e l e m e n t

exists

an e l e m e n t ~ '

of ~'

of A'

such that

such that k(~')

=I~.

- 227 Proof: element

The proof can be done by induction

of type ~, ~', their

we will give details this implies that,

inverses,

only in the case

a' = x'(y'z'),

and it is sufficient

to t a k e ~ '

on

lal.

If~is

an

y or 7', then it is easy to do; ~=

k(x')

~

: then a = x(yz),

x,y,z

= x, k(y')

= y, k(z')

and

= z,

= ~x',y',z'"

Suppose now that ~ = ~i + ~ 2 ' ~ l : a l "----~ bl' ~ 2 : a 2 ~ b 2 " Then a = a I + a 2, and ~his implies k(a~) ~

= a2; using

such that,

The c a s e ~ = ~ l ~

the induction

k(~)

k(~)

a' = a~ + a~, k(a~)

hypothesis

we can d e t e r m i n e ~

= aI , and

= ~ F 2 , and we can take ~ ' = ~ 1 +~'2"

2 is similar.

LEMMA 3.5. steps

='~i'

that,

Let X, A and T be as before and P a path with

in ~, P:a

~ aI

that are instantiations occurrence

~

.'"

~ an ,

of ~, ~', their

inverses,

of u and n is in the expression

there exists a set X',a regular

7, and 7'

If no

of an element a of A, then

element a' of X', a map k:X'

>X

and a path, p,:a,

such that,

k(u')

for i = 1,2,

~i

, a, 1

= u, k(n')

"'', n.

Proof:

3.5 we will

in general,

we can deduce

a, n

'

= a, k(a I) = a i and k(~i)=~i of X', k and a' only de-

of the other components consequence

of the path P.

of Lemmas

~ ? b are two paths allowing

find two paths,

b' M b".

~n,

the choice

This is an immediate

Note that if P,Q:a Lemma

= n, k(a')

Moreover

pends upon a, independently

fF2 . . . .

P':a'

O>

us to apply

b', Q':a'--~-~b",

Our next aim is to find conditions

that b' = b"; as we will

3.3 and 3.4.

see,

where,

under which

this is the case when P

and Q have the same distortion. We now need an auxiliary algebra map,

over ~*~

sh:A

concept.

Let S be the f r e e ~ + , ~ -

, the set with only one element.

> S defined

by the conditions:

The shape

is the

228 -

-

i)

For x £ X,

ii)

sh(x)

x = a + b ~

iii)

sh(x)

x = ab - ~ i sh(x)

It is i m m e d i a t e have

= *. + sh(b).

= sh(a)sh(b).

that,

for e v e r y a' of A'

= xh(a)

t a k i n g X,X'

sh k(a')

and k:X'

> X as before,

we

= sh(a')

m

LEMMA P:a

~>

b, P':a

i)

Let X, A and T be as b e f o r e O>b'

are paths w i t h

a is a sum of e l e m e n t s

ii)

sh(b)

iii)

steps

and s u p p o s e

that

in T such that

that are p r o d u c t s

of e l e m e n t s

of X,

= sh(b'),

dist(P)

iv)

then,

3.6.

= dist(P'),

P and P' are

sequences

of i d e n t i t i e s

~, ~',

inverses,

y and y',

their

and i n s t a n t i a t i o n s

of

b = b'. Proof:

lation

sh(b)

additive

N o t e that u s i n g = sh(b')

implies

or m u l t i p l i c a t i v e

A d e c b' =

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

of § 3 of

[5], the re-

t h a t b and b' have the same n u m b e r

components

and

if, Adec b =

(bl,b2,''°,br),

' 2, ' °°" 'b r'' '~ then b i and b i' have the same n u m b e r (bl,b

multiplicative

components.

A l l this c a n be p r o v e d Suppose M d e c b! = 1

Moreover,

immediately

bij = bij, '_

using

i = 1,2,''',r,

Proposition

by i n d u c t i o n

and a p p l y i n g

If we p r o v e 1 of

again

of

A p t b = A p t b' and M p t b = M p t b'.

t h a t for e a c h i, M d e c b i =

' ' "'" b' ). (bil'bi2' ' in i

of

on

Ibm.

(bil,bi2,''',bini), that for e v e r y pair ,

[5] we have that b i = b i' for that p r o p o s i t i o n

we have

that

b = b'. To p r o v e can be c o m p u t e d to prove

that

that bi~J = b!. it will be s u f f i c i e n t to s h o w that bi~J ~3 by m e a n s of a and d i s t ( P ) , and in fact we are g o i n g

if A d e c

a =

(al,a2,'°',ar),Mdec

ai=(ail,ai2,''',aim

) l

and dist(P)

=

(~; ~i' ~ 2 ' ' ° ' ' ~ r )' then mj=no(j)

ai, j = b u ( i ) , ~ i ( j ) . additive reduced

components to p r o v i n g

and

N o t e that a and b h a v e the same n u m b e r of as a c o n s e q u e n c e the a b o v e

of c o n d i t i o n

statement

iv).

and we are g o i n g

So we are to do this by

- 229 induction

on the number of steps in the path P. Q

P--a where,

Adec c =

that,

R

O > c --~->b,

(Cl,C2,''°,Cr),

dist(Q)

Suppose

=(~;~l'°'"~r

Mdec c i = )' dist(R)

(Cil,Ci2,°'°,Cin.), 1 = (w;71,''°,y r),

then,

(u; ~i ) = dist(P)

= dist(R)dist(Q)

=

(w; 7 i ) ( ~ ; ~ i

)

= (w~; ~(i)~i )' that is, u = w ~ ,

e i = y~(i)~i

for i = 1,2,''',r,

and by the induc-

tion hypothesis,

aij = c T ( i ) ' ~ i (j) = bw[~(i)]' Hence we are reduced when P = ~ : a a u c t i o n on

~b lal.

Y~(i)~8i (j)] = bu(i)'

ei(J)"

to the case when P has only one step,

that is,

is an e l e m e n t of T and this will be done by inIf~=

b = b' + b", ~ ' : a '

~'

+ ~",

then we can suppose,

~ b',~[":a"

a = a' + a",

% b",

(u ,el, Adec a'

=

Adec b' =

(al,a2,''',ap)

, Adec

a"

=

(ap+l,ap+2,''',ap+q)

(bl,b2,''',b p) , Adec b'. = .(bp+l,bp+2, . .

and by the induction

hypothesis,

,bp+q)

if i < p,

aij = a~. = b' = b ' ( i ) , e ~ (j) 13 a' (i) '~i' (j) i

'

and for i > p,

aij = a"l-p,j = b"u" (i-p) e" ,, . (j) = bp+u" (i-p) ,ei_p(3) ' i-p Hence,

dist(~)

= dist(q

. . . .,ep,e . . 1 ,~,''',e'q) and for i < p,

=

') • dist

(~")

(~; ~l,~2,''°,~p+q)

=

(a' + u . .,el,e ... 2 , ,

, ,

- 230 -

aij = bo' (i),~.~(j)

= b(o'+a") (i),~!1(j)

= ba(i),~i(j)

'

and for i > p,

= bu(i),~i(j)

a13 .. • . =. bp+~..(i_p),~i_p( . . j) = b (a'+u ..) (i),~i_p(3) If ~ = then,

taking

O' ~",

a = a'a",

into a c c o u n t

condition

dist(~')

Adec

a =

(I;~'

(I;~")

+ ~")

,

,

a' =

(a~)

(a"), Mdec

aI =

(all, a l 2 , ° ° ' a l n )

a" =

(a~) =

Mdec

a' =

(all,'°°,alp), (b l) =

Mdec

(b), Mdec

hypothesis,

i > p, ali = b l , p + ~ . ( l _ p ).

=

=

(a), Adec

Adec

induction

dist(~")

= dist(~)

(a I) =

Adec b = and by the

iv),

-- (i;~'),

(I;~)

b",

• b' , ~ " :a"

b = b'b", ~ ' :a'

a" =

bI =

=

(a')

,

(al,p+l,°'°,aln)

(bll,°'°,bln)

,

if i ~ p, ali = b l , ~ . ( i ), and

F r o m the above

remarks

it follows

for that, .

= ~' + ~" and for i < p,

ali = bl,~' (i) = bl,~(i) and for i > p,

ali = b1,p+~.(i_p) So we are r e d u c e d identity ~is

to the case w h e r e ~ i s

or an e l e m e n t

of type

of type u, ~' or their

the r e s u l t

is immediate.

then by c o n d i t i o n

= bl,~(i)

iv),

~, ~',

inverses,

Suppose

an e l e m e n t their

inverses,

dist(~)

now that

of G, that

~=

y or Y'

is an i d e n t i t y 7a.,a.:a'a"

is an If and

~a"a';

-

Adec

a =

Adec

a' =

Adec

Then,

dist(~)

=

(a) =

(b), Mdec

=

The o n l y r e m a i n i n g

(all,al2,''°,aln)

(all,al2,°°',alp)

a" =

,

(al,p+ l,°°°aln)

(ll;tp,n_p),

b =

can be c h e c k e d

aI =

a' =

(a"), M d e c

(~l,l;tp,n_p)

and the r e l a t i o n

-

(al) , M d e c

(a'), M d e c

a" =

Adec b =

231

(a I ,p+l' . . 'ar'al' . . . .

,ap)

immediately.

case

is w h e n

I 7a.,a,.:a

~=

!

• a" + a' .

+ a"

If, Adec

a =

(al,a2,°°°,ar),

Mdec a i =

(ail,ai2,°'°,ain.)

,

1

A d e c a' =

(al,a2,°°°,ap),

Adec

a" =

(ap+l,ap+2,°°°,ar)

,

we have, dist(~)

=

(tp,r_p;1),

and the r e l a t i o n

4.

can be c h e c k e d

where will whose

(ap+l,''',ar,al,a2,°°°,ap)

immediately.

The R e s u l t of C o h e r e n c e We are n o w g o i n g

will

Adec b =

suppose

that C is the c a t e g o r y

the c o n c e p t s suppose

represented

that X and f:X

construction

referred

r e s u l t and

to in the

for this we

Introduction,

by X, A and T are also d e f i n e d . > Ob C are

is not d e t a i l e d

v a l u e on Ob C by m e a n s COHERENCE

to p r o v e our c o h e r e n c e

fixed,

is d e f i n e d

We

so that any p a t h

over T and takes

its

of f.

THEOREM

If C s a t i s f i e s are p a t h s w i t h the in the c a t e g o r y Proof: the g e n e r a l

the c o h e r e n c e

same d i s t o r t i o n ,

conditions

of

[5] and P , Q : a

~ >b

then P and Q take the same v a l u e

C.

m

The proof will c o n s i s t

situation

of the p a t h s

of d i f f e r e n t

parts

P and Q to o t h e r s

that r e d u c e

easier

to handle.

- 232 -

Part

I:

We r e d u c e

the t h e o r e m

is in the v e r t i c e s

of P and Q.

Take an e l e m e n t

the d e f i n i t i o n

of r e d u c t i o n

d of A-- is a path,

times as p o s s i b l e

d-~d'

instantiations

a reduction

is a p a t h o b t a i n e d

occurrences

of n.

determined of

by d

The v a l u e

a

P O

>b

a

©

>b

of ~',

occurrence

tions, tion

inverses

i0 of

is an i s o m o r p h i s m that,

dist(P')

of D

C>d',

are

a--

©

)b'

5

,

and no i n s t a n -

is in P' or Q'.

If some

and the t h e o r e m suppose

of P' and Q'

is a m o n i c

(the c a t e g o r y

that P' and Q' take the same v a l u e

d

>b

by their v a l u e s

= dist(Q')

of the

Q O

imply the c o m m u t a t i v i t y

the p a t h s

speaking

a

[5]: h e n c e we can

of n is in any of the v e r t i c e s

by r e p l a c i n g

as m a n y

diagrams,

then a' = n, a' is r e g u l a r

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

of

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

of a and b, r e s p e c t i v e l y ,

and as the v a l u e of any p a t h

conclude

elimination

,

t i v i t y of the above d i a g r a m s obtained

applying

[5])and a c c o r d i n g

k*, p* or their

of n

[5]: a r e d u c t i o n

and p*; r o u g h l y

by s u c c e s s i v e

two c o m m u t a t i v e

of n is in a',

is c o n s e q u e n c e occurrence

p',

of

, obtained

of k ' , p ' , ~ *

4 of

R and S are r e d u c t i o n s

tiation

of ~ 4

no o c c u r r e n c e

and the end of a r e d u c t i o n ,

(Proposition

[5] we can c o n s t r u c t

where

to the case w h e r e

that no

The c o m m u t a -

of the d i a g r a m s

or by their d i s t o r of C and the d i s t o r -

of d i s t o r t i o n s ) ,

and that we are r e d u c e d

we can to p r o v i n g

in C. m

The same type of a r g u m e n t and some d e t a i l s Part

II:

can

We r e d u c e

(and will) the t h e o r e m

tions that all the v e r t i c e s ments

will be

applying

in the next two P a r t s

omitted.

to the case w i t h

the a d d i t i o n a l

of P and Q are sums of p r o d u c t s

of X and that no i n s t a n t i a t i o n A rappel

be used

of an e l e m e n t

condi-

of ele-

of 6 or 6# is in P or Q.

d of A is a p a t h d--~->d' o b t a i n e d

as m a n y t i m e s as p o s s i b l e

instantiations

of 6 and

6#

by

(see

- 233 5 of

[5]):

the end of any r a p p e l

elements

of X.

Proposition

mutative

diagrams

7 of

has to be the sum of p r o d u c t s

[5] p r o v e s

the e x i s t e n c e

Q

a

O

>b

a'

p, O

......> b '

,

R and S are r a p p e l s

tiation

~

O

~'

Q, O

of a and b, r e s p e c t i v e l y ,

of k' • p ' , k* , p*, t h e i r

As in P a r t

of two com-

of type,

P

where

of

I, dist(P') = dist(Q')

P' and Q' take the same v a l u e

inverses,

>b

>b'

and no i n s t a n -

6 or 6 # is in P' or Q'.

and we are r e d u c e d

to p r o v i n g

that

in C. w

Part III~ po%hesis

We r e d u c e

the t h e o r e m

that no i n s t a n t i a t i o n ,,,

A normalization by a p p l y i n g 6 of

as m a n y

[5]).

proves

,,

times as p o s s i b l e

P 0

of A is a p a t h d

O>d'

instantiations

of ~ and

and the end of a n o r m a l i z a t i o n ,

by d ( P r o p o s i t i o n

the e x i s t e n c e

the a d d i t i o n a l

hy-

of k or p is in P or Q.

of an e l e m e n t

The v a l u e

are d e t e r m i n e d

,,

to the c a s e w i t h

8 of

[5]).

of two c o m m u t a t i v e

,~ b

p

(see

d-~d',

Proposition

diagrams

obtained

9 of

[5]

of type,

a -

Q ~

a'

~.~

>b

o, a' -where

,/% .,

b'

>

R and S are n o r m a l i z a t i o n s

Q' are i n s t a n t i a t i o n s q u e n c e of the a b o v e reduced Part

,

to p r o v i n g

IV:

We r e d u c e

of ~, ~',

facts

Y and Y' .

= dist(Q')

3.2 to c o n s t r u c t >b'

A conse-

and t h a t we are in ~.

such that k(P')

in the v e r t i c e s

condi-

of P and Q.

a set X',

a map k:X'----~X

= P, k(Q')

of P' and Q'.

,

and P' and

to the c a s e w i t h the a d d i t i o n a l

of u is in the v e r t i c e s

P' , Q' :a ' O of u'

inverses•

is that dist(P')

the t h e o r e m

We can use L e m m a

no o c c u r r e n c e

their

t h a t P' and Q; take the same v a l u e

tion that no o c c u r r e n c e

and two paths•

of a and b, r e s p e c t i v e l y •

>b'

By Lemma

= Q, w i t h 3.1,

- 234 -

dist(P')

Define now f':X'---~Ob

= dist(P)

= dist(Q)

= dist(Q') .

~ by the c o m p o s i t i o n ,

X'

k>x

f >Ob

~;

t h e n the v a l u e s of P and Q are the s a m e as the v a l u e s of P' and Q', respectively. the

same v a l u e

P a r t V: Parts

H e n c e we are r e d u c e d

t h a t P' and Q' t a k e

in C.

We P r o v e the t h e o r e ~ for t h e r e d u c e d c a s e o b t @ i n e d t h r o u g h

I to IV. We are r e d u c e d

to p r o v i n g

are s e q u e n c e s of i d e n t i t i e s verses,

set X' , a m a p k:X' k(P')

Define

the t h e o r e m

= P, k(Q')

f' as in P a r t

t a k e the same v a l u e

of u and n.

>X,

for two p a t h s

and i n s t a n t i a t i o n s

7 and ¥', a n d w h o s e v e r t i c e s

of X w i t h no o c c u r r e n c e

that,

to p r o v i n g

two paths,

= Q, w h e r e

a'

their

in-

a r e sums of p r o d u c t s of e l e m e n t s By L e m m a P':a'

3.5 we can c o n s t r u c t

~/b',

is a r e g u l a r

IV a n d we are r e d u c e d in ~; k(P')

of ~, ~'

P,Q that

Q':a'

= Q,

vf~ b" s u c h

e l e m e n t of A'

to p r o v i n g

= P and k(Q')

a

t h a t P' and Q'

i m p l y by L e m m a

3.1

that, dist(P')

and f u r t h e r m o r e , sh(b')

= sh(b)

shape). i0 of

k(b')

= sh(b")

= dist(P)

= k(b")

(see r e m a r k s

H e n c e w e can u s e L e m m a

[5] p r o v e s

the t h e o r e m

t h a t P', Q':a'

is p r o v e d .

= b.

= dist(Q)

= dist(Q'),

This last fact implies that following

the d e f i n i t i o n

of

3.6 to o b t a i n b' = b" and P r o p o s i t i o n ~ f _~ h , /%

t a k e the s a m e v a l u e

in C and

- 235 ~FERENCES [i]

S. Eilenberg and G.M. Kelly: calculus", J. Algebra,

[2]

G.M. Kelly:

"A generalization of the functorial

3(1966),

366-375.

"On Mac Lane's cond:[ti0ns for coherence of natural

associativities,

commutativities,

etc.", J.Algebra,

4(1964),

397-402. [3]

G.M. Kelly and S. Mac Lane: J. Pure Appl. Algebra,

[4]

M.L. Laplaza:

"Coherence in closed categorieS",

i(1971),

97-140.

"Coherence for categories with associativity~

commutativity and distributivity",

Bull. Amer. Math. Soc., 78

(1972), 220-222. [5]

M.L. Laplaza:

[6]

S. Mac Lane:

"Coherence for distributivity", "Categorical algebra",

(this volume).

Bull. Amer. Math. Soc., 71

(1965), 40-106. [7]

S. Mac Lane:

"Natural associativity and commutativity",

Rice

Univ. Studies, 49(1963), No. 4, 28-46. [8]

S. Mac Lane:

"Categories for the working mathematician",

York-Heidelberg-Berlin,

Springer,

1971.

New