Category Seminar: Proceedings Sydney Category Theory Seminar 1972-1973

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

420 Category Seminar Proceedings Sydney Category Theory Seminar 1972/1 973

Edited by Gregory M. Kelly

Springer-Verlag Berlin.Heidelberg • New York 19 74

Prof. Dr. Gregory M. Kelly Department of Pure Mathematics University of Sydney New South Wales 2006 Australia

Library of Congress Cataloging in Publication Data

Category Seminar. Proceedings Sydney Category Seminar 1972/1973. (Lecture notes in mathematics ; 420) Includes bibliography and index. 1. Categories (Mathematics)--Congresses. 2. Functor theory--Congresses. I. Kelly~ G. M., ed. II. Title. IIl. Series: Lecture notes in mathematics (Berlin) ; v. 420. QA3.L28 no. 420 [QA169] 510'.8s [512'.55] 74-19483

AMS Subject Classifications (1970): 18 A15, 18 A35, 18 A40, 18 C15, 18D05, 18D15, 18D99, 18E35

ISBN 3-540-06966-6 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06966-6 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 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE

The 1972,

Sydney

Category

Theory

when the c a t e g o r y - t h e o r i s t s

conceived

the idea of m e e t i n g

collected

in this v o l u m e

has

emerged

represents although typing,

As editor take to turn

enough

were r e c e i v e d The papers

the c o m m o n are

theme

set theory,

in d i m e n s i o n

structure

is categories

monoidal monad, fields,

closed

and

If m o n o i d a l of c a t e g o r y papers

are

these

level.

His

since been which may

first adapted

the time

it would

delays

I owe a p o l o g ' first

with

than our local

structure.

category

two

a lot r i c h e r

structure; enriched

is the

because

of the

of sets with

and m o n o i d a l

categories,

categories,

categories

theorist

proximity;

If c a t e g o r i e s

theory

then the a n a l o g u e

categories

then B r i a n Day with two types

c l o s e d ones, two

about

by more

to the c a t e g o r y

closed

concerned

between

and the

what

bearing

his groups,

a rings,

are to the algebraist.

theory,

ially m o n o i d a l

the w r i t i n g - u p

to B r i a n Day, w h o s e

pure

although

with

categories,

and m o d u l e s

then

by i". But

so on, b e c o m e

done,

than a year ago.

to sets,

"increase

that

"1972/1973"

the w o r k was

for the u n f o r s e e n

is that of cat e$ories

of pure

The papers

of 1974.

sanguine

b e l o w are c o n n e c t e d

seen as a n a l o g o u s

analog u e

the m i d d l e

and e s p e c i a l l y

more

in Sydney

The t i t l e - d a t e

of it, along with

into theorems;

of

a large part of the m a t h e m a t i c s

I was e x c e s s i v e l y

ideas

in the m i d d l e

day each week.

the time at w h i c h

us past

ies to S p r i n g e r - V e r l a g papers

for a w h o l e

e i g h t e e n months.

the final details have brought

was born

at the three u n i v e r s i t i e s

represent

from its first accurately

Seminar

are

in some

is our

field

sense

theorist.

of structure

and a d j u n c t i o n s

- and with

all this m o r e o v e r

paper

a new a d j o i n t - f u n c t o r

by M i k k e l s e n

be seen as r e p l a c i n g

"enriched"

theorem,

topos

process

and espec-

the r e l a t i o n s

at the

to the e l e m e n t a r y by a t w o - st e p

In fact his

- monoidal

structures; gives

the "fields"

which

has

situation,

and

the t r a n s f i n i t e

IV

tower

construction

to c o m p l e t i o n s earlier

and to m o n o i d a l

work on m o n o i d a l

on the r e f l e x i o n closed

of

structures

and gives

a great

the e m b e d d i n g great

of A p p l e g a t e

value

and Tierney; completions.

closed

monoidal

structures

closed

perceive

applications.

for coherence

closed

problems,

of t o p o l o g i c a l

the d e f i n i t i v e

My own papers As e q u a t i o n a l the c a t e g o r y

with

of sets

equational on the

structure

2-category

In this

setting

the

general

category

from a knowledge

a subclass

of doctrines

I call

clubs,

or more

and a smaller

interplay

equational

gives

between

some

coherence

results

available

do well

to read

alone

are used

results

in the club

on any

2-category.

is that

of finding

My first

concrete where

except

looks

representation

the coherence paper

level,

for

by

the

and my third

getting

the fuller

The reader

§i and

at

problem

concerns

case by specialization. last,

the d o c t r i n e

paper

and adjunctions;

at the d o c t r i n e

the first paper

of

generally

My second

structures

things

a doctrine,

subclass

as a w o r d - p r o b l e m .

species

on

called

a very

can be f o r m u l a t e d

to a m o n a d

also

of its algebras.

that admit

al~ebra.

if we want

with a given

to a 2-monad,

problem"

cartesian

quickly

to u n i v e r s a l

- or on a more

"coherence

§i0,

would

which

in the last two papers.

The e l e m e n t a r y

2-categorical

papers

and for Street's

Street

goes

is c o n t a i n e d

on in his papers

one sense his papers

background

needed

in a joint

to look more

are the most

of

theorems.

correspond

explicitly

what

at certain

species

of categories,

one,

to bicategories;

The reader will

in some sense

correspond

and

categories;

into a m o n o i d a l

to look deeply

- so c a t e g o r i e s

categories,

His third discusses

extensions

of his

his

to study m o n o i d a l

of any g i v e n

groups

combines

of functor

category

spaces.

elegance

correspond

algebras

like t o p o l o g i c a l

structures,

many

concrete

second

for functor

subcategories

of a n o n - m o n o i d a l

categories

His

on r e f l e c t i v e

and his fourth uses his techniques closed

along with a p p l i c a t i o n s

general;

deeply

both

for my

expository

paper.

at 2-categories.

in another, the most

In

fundamental.

If the study of various

led to category theory,

that of structures borne by a category leads

i n e x o r a b l y to 2-category theory. inside any 2-category, ant.

Street's

representable

structures borne by a set has

Much of category theory can be done

and the arguments are then often more t r a n s p a r -

first paper looks at some things that can be done in a 2-category - e s s e n t i a l l y the same thing as a finitely

2-complete one, except that it need not contain a terminal object; which is seldom needed. at this level,

In p a r t i c u l a r he studies fibrations and b i f i b r a t i o n s

along with such things as p o i n t w i s e Kan extensions.

second paper, w h i c h uses the first,

investigates

2-categories with a

structure so r i c h that we can imitate those arguments, Yoneda lemma, that depend upon the hom functor. 2-category of categories,

His

i n c l u d i n g the

Even applied to the

it provides new proofs and thus contributes

the "elementary"

theory of categories;

ordered objects,

it throws new light on elementary topoi.

to

applied to the 2-category of

The i n v e s t i g a t i o n s of Street and Kelly, at least ~, are to some degree tentative,

and they m e n t i o n m a n y o u t s t a n d i n g problems which may

be of interest to others. Because of the time the volume has been in preparation, believe

I

it a P p r o p r i a t e to give dates of r e c e p t i o n for the papers

(although I don't quite know what it means to "receive" my ~wn). order is that of the table of contents. 1973; Apr. Street:

1973; Feb.

1974; Feb.

July 1973; Feb.

1974.

Day:

Feb.

1974. Kelly-Street: Kelly:

Nov.

The

1973 revised May Oct.

1973; Jan.

1973.

1974; May 1974.

G.M. Kelly 19 July 1974

Vl

TABLE OF CONTENTS

B r i a n Day

On a d j o i n t - f u n c t o r

factorization

i

On closed categories of functors II

2O

An e m b e d d i n g theorem for closed categories

55

Limit

65

spaces and closed span categories

G.M. Kelly and Ross Street

Review of the elements of 2-categories

75

Ross Street

Fibrations

and Yoneda's lemma in a 2-category

104

E l e m e n t a r y cosmoi

134

On clubs and doctrines

181

Doctrinal a d j u n c t i o n

257

G.M. Kelly

Coherence theorems

for lax algebras

and for d i s t r i b u t i v e

laws

281

ON A D J O I N T - F U N C T O R F A C T O R I S A T I O N by

This note contains an a l t e r n a t i v e a p p r o a c h to a result of Applegate and Tierney S--~T:

C ~ B

(~2J

and t3]) which states that an a d j u n c t i o n

over a suitably complete category B can be f a c t o r e d

through the full s u b c a t e g o r y of B d e t e r m i n e d by the objects are "orthogonal" to all the m o r p h i s m s

in B which

inverted by the functor S: B ~ C.

It is o b s e r v e d that a slight s t r e n g t h e n i n g of the completeness hypothesis on B gives a simple proof of this result. The f a c t o r i s a t i o n of the given adjoint pair takes place in two stages,

the first of which is a w e l l - k n o w n epic-monic

the given a d j u n c t i o n unit.

f a c t o r i s a t i o n of

This produces a full reflective

subcategory

B' of B having the p r o p e r t y that the class of objects which are orthogonal to any given class of morphisms B' of S is r e f l e c t i v e

in B'.

inverted by the r e s t r i c t i o n to

The combined result contains a theorem

of Fakir fll] which a s s o c i a t e s to each m o n a d T on B, the idempotent monad which inverts the same morphisms as T. case, where

For the relative

V-based

V is a complete symmetric m o n o i d a l closed category,

result is closely related to a theorem by W o l f f /19~

the

§5.6 using co-

completeness h y p o t h e s e s on B. Some of the o b s e r v a t i o n s made here are implicit in 121 and 13|. However,

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

of category c o m p l e t i o n to epic-monic

isation, and to r e l a t i v e categories of q u a s i - t o p o l o g i c a l not d i s c u s s e d in the r e l a t i v e this article, ation,

V-based version 191 •

the concepts of category,

etc., are a s s u m e d to be r e l a t i v e

m o n o i d a l closed category

functor,

Thus,

factor-

spaces, was throughout

natural t r a n s f o r m -

to a suitable symmetric

V; this category is assumed to be locally

small with respect to a given c a r t e s i a n closed category S of "small"

sets and set maps. The work r e l a t i n g closure" G.M.

of the c a t e g o r y

Kelly,

isations of the

of S--~T:

notations

and are as given

and [2]

contexts

cartesian

jointly The

with

general-

and the r e l a t i o n s h i p

C ~ B to a m o n o i d a l

in the early parts

§i

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

§2

Factorisation

§3

Categories

§4

Examples;

§5

The f a c t o r i s a t i o n

closed

structure

on

BOP),

[i0].

of M-adjunctions.

completions

cribles. and m o n o i d a l

system

FACTORISATION

m in a V-category

is monic in B if,

closure.

for left adjoints.

THE P R E L I M I N A R Y

B(B,m)

called a strong monic

of E i l e n b e r g - K e l l y

are s t a n d a r d

factorisation.

that a m o r p h i s m

if the m o r p h i s m

theorem

are as follows:

of relative

§i

e in

was done

and the r e p r e s e n t a t i o n

section-headings

Recall

in other

"minimal

in §4.

The basic

The

spaces

in view of [I]

given here are useful

factorisation

2.5 and to the

of t o p o l o g i c a l

but not p u b l i s h e d

B is d i s c u s s e d

monic

to Example

in

V for each B C B.

for each

~(e,,,l)

I

B(l,m)

B(e,l)

A monic

epic e in B (that

the square

B(l,m)

B is called monic

is,

in B

m is for each

3

is a p u l l b a c k easily

diagram

established.

any s t r o n g m o n i c

which

(~,~):

(E,M)

Thus fix

initial

S--~T:

is epic

The category for at least

(a)

E = (all epics

(b)

E = (all strong

(E,M)

throughout.

factorisation Let

be two versions

system b e t w e e n

B' be the full

B E B for w h i c h

a morphism The proof

B is assumed one

to have

of the f o l l o w i n g

canonical two cases:

strong monics

in B)

in BY and M = (all monics

in B~

(E,M)

(a) and

could be taken (b)

subcategory

in the

sense

of B d e t e r m i n e d

but we shall to be any proper of El31

52.3.

by the objects

An object

B E B is in B' if and only

if there

B ~ TC in M. is clear.

Proposition Proof.

C and B and an a d j u n c t i o n

~B E M.

Proposition i~. exists

53) are

and if a composite

which may be compared

In fact,

I151

then so is g.

in B) and M = (all epics

(cf.

in B is a strong monic,

is an isomorphism~

data are categories

C ~ B.

will

any e q u a l i s e r

is a strong monic

- factorisation

there

The usual p r o p e r t i e s

For example,

fg of two m o r p h i s m s The

in V.

1.2

The

inclusion

The r e f l e c t i o n

B' c B has a left adjoint.

sends B E B to the image

of ~B;

let

nB B

......

~ TSB

B denote

the

isomorphism Thus,

for

gram is

factorisation because each

B'

it ~ B',

an isomorphism

of is

an epic

the in

nB with

top

V, a s

n~ E E a n d m ~ M.

in

arrow

C with in

required:

the

left

inverse

following

T h e n Sn~ i s eSB

" Sm.

pullback

dia-

an

B(n',l)

B(~,B,)

.........

1

^ B(B,TSB')

B(B,B')

t

B(n',I)

~ B(B,TSB')

^

C(SB,SB')

Moreover,

.

. . . C(Sn',l)

~ C(SB,SB').

because S: B ~ C inverts

ion, there results

an adjoint

the unit ~' of the reflect-

triangle: ^

B

=

B'

C This choice

of

category

process M. of

terminology, called

In

is

clearly

other

words,

B, a n d

is

a closure B'

category

an adjunction

such

has

operation

the

(E,M)-factorisations

equivalent as

for

S'---tT

to with

B". unit

given as

Following in

a substandard

M shall

be

an M-adjunctiqn.

Ex~ple the category S: [A°P,s]

1.3. [A°P,s]

Let A be the category of finite sets, of all functors

"spanning"

from A °p to S, and let

~ S be "evaluatlon-at-singleton".

category of "simplicial subsets).

complexes"

The category

to S.

B' is the

(sets equipped with certain finite

If M = {all monies

in B'} then B" remains

valent to B', but if M is changed to {all strong monies is equivalent

let B be

equi-

in B'} then B"

§2 Under

additional

factorisation following is first

F A C T O R I S A T I O N OF M - A D J U N C T I O N S completeness

of §i reduces

the given a d J u n c t i o n

form of a d j o i n t - f u n c t o r established

a category

theorem

for o r d i n a r y

C is M - c o m p l e t e

that C has the f o l l o w i n g

hypotheses,

set-based

limits

to one

in w h i c h

is applicable. categories.

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

inverse

the a d j o i n t - f u n c t o r

The t h e o r e m We

of monics

and M contains

the

say that in C such

each monic

so

formed: (a)

equalisers

(b)

pullbacks

(c)

all i n t e r s e c t i o n s

A functor limits

of morphisms.

of M-monics

(i.e.

inverse

of M-monics

T: C ~ B is M - c o n t i n u o u s

M-images).

with a common

if it preserves

these

codomain. inverse

in C. Theorem

T: C - ~ B exists that,

of pairs

2.1.

If C is an M-complete

has a left adjoint

a "bounding"

family

for each C E C and

if and only (BB:

then a functor

if T is M - c o n t i n u o u s

and there

B E B) of m o r p h i s m s

in B such

B ~ TCB;

f E B(B,TC),

category

there

exists

a commuting

square:

BB B

~ TC B

i TC

~ TD Tm

with m C M. Proof. and the perties.

family

If T has a left

adjoint

then T is clearly

(nB; B E B) of a d j u n c t i o n

Conversely,

a left adjoint

for some m o r p h i s m factors

uniquely

with m E M and

B B factors

~B: B ~ TSB. through

let

(p,q)

~B"

has the r e q u i r e d

S: B ~ C is c o n s t r u c t e d

taking h: SB ~ C B to be the i n t e r s e c t i o n n: M ~ C B such that

units

M-continuous

through

by

in C of all the M - s u b o b j e c t s

Tn.

Moreover,

Then

B B factors

each m o r p h i s m

To see this,

be the p u l l b a c k

pro-

f E B(B,TC)

let Tg.~ B = Tm.f:

in C of

(gh,m).

as Th.~ B

B ~ TD,

BB

TC B

Tg

TC

TD

' Tm

Then q E M and n B factors definition

of SB.

TSB as T ( p q - 1 ) ~ B . be the

equaliser

through

Tq so q is an i s o m o r p h i s m

Thus

f, w h i c h

factors

This

factorisation

in C of (p,rq).

so e is an i s o m o r p h i s m

through

is unique.

Tp,

by the

factors

If f = Tr.~B,

Then e C M and ~B factors

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

of SB.

through

This

let e

through

completes

Te

the

proof. Remark. V, §6 and §8) from T h e o r e m exist

Several of Freyd's

2.1 under

of the adjoint

in C and are p r e s e r v e d

that

C is c o t e n s o r e d

the c o t e n s o r i n g ;

this

which

2.2

the i n c l u s i o n

if and only

if there

by T: C ~

version

hypothesis

(cf. [18]

Chapter

may be r e c o v e r e d that enough products

B.

provides

2.1 we shall

simply

and that T: C ~ B p r e s e r v e s

a V-adjunction

by [16|§4.1.

are b a s e d on the following:

Let C be an M-complete

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

C C B is M-continuous. exists

each B E B and

B C E M for each C E C. follows

Then

an e n d o f u n c t o r

~: i ~ S such that

This

theorems

of T h e o r e m

transformation

Proof.

statements

as a V-category

assumption

Our a p p l i c a t i o n s Theorem

functor

the a d d i t i o n a l

To o b t a i n a V-based assume

standard

C is r e f l e c t i v e

in B

S: B ~ B and a natural

BB factors

from T h e o r e m

of B for

through

an object

2.1 and the fact

that,

of C for

for

each m o r p h i s m

f: B ~ C in

B, with c o d o m a i n

Tf.B B ~ ~c.f by the n a t u r a l i t y

of

8, and

C E C, we have

~C E M for all C C C by

hypothesis. For a g i v e n class denote

the

full

subcategory

the t e r m i n o l o g y Z-orth0$0nal

of f13])

if B(s,B)

if B is Z - l e f t - c l o s e d Now suppose proper

V-category)

Let TS:

for all

(~,n):

2.~.

S--~T:

(E,M)

An object

forming

E is

s E Z (that is,

C ~ B is an M - a d j u n c t i o n

on B and let E denote

B Z c B has

for a

the class

of

B Z is closed

(and is c o t e n s o r e d

as a

a left M-adjoint.

under

limits

endofunctor

(and cotensoring)

on B.

Because

of T h e o r e m

(orthogonally)

closed

in V for all B E B E then of a subset

of Z onto a complete

of B w h i c h

B E B is called

in

TC E B Z

2.2 are

satisfied

is an M-adjunction.

the closure

Corollary

(following

of 114|).

C E C, and ~B E M, the conditions

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

are

by S.

B ~ B be the desired

The class

B, let B Z

in V for each

If B is M-complete

Clearly

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

in B w h i c h

to Z.

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

and if Z c Z then

Proof. B.

of all objects

orthogonal

system

in B i n v e r t e d

~orollary

in a V-category

in the t e r m i n o l o g y

that

factorisation

morphisms

Z of m o r p h i s m s

2.4.

contain

B

f ~ ~.

of E reflects

sublattice. The

in the sense

class

Thus

of

of subclasses

2.3 implies:

of all full r e f l e c t i v e

as a full r e f l e c t i v e

if B(f,B)

The o p e r a t i o n

the class

Corollary

that

subcategory

subcategories forms

a complete

lattice. Example spaces

2.5.

and continuous

determined

by all the

functors

are u s u a l l y

conta±ns

Top

Let

Top

maps.

denote Let

subfunctors called

the category

B be the

subcategory

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

"eribies").

as a full r e f l e c t i v e

full

Then

subcategory.

of all t o p o l o g i c a l of

functors

B is locally

ITop°P,s~

(these S-small

Furthermore,

~nd

B ~ B' if

we take S:

tTop°P,s]

M ~ (all monics

~ S to be e v a l u a t i o n at the one-point space and

in B).

Let T: B ~ B be the monad d e t e r m i n e d by the

functor w h i c h evaluates each crible at the one-point space. is the class of all bijections classes of bijections, -topological

spaces,

§3

in B and, by i n v e r t i n g a p p r o p r i a t e

one obtains r e f l e c t i v e

subcategories of quasi-

limit spaces, and related structures,

the "minimal extension of

Top"

Then

including

discussed in Eli.

CATEGORIES OF RELATIVE CRIBLES.

Suppose h e n c e f o r t h that the given symmetric m o n o i d a l closed category V ~ (V, ®, I-,-I ,...) is S-complete and admits all intersections of M-subobjects,

where M is fixed as either the class of mon-

ics in V or the class of strong monics in V. theses imply that

V has canonical

ponding class E of epics in

These completeness hypo-

(E,M)-factorisations

V(cf. /15|

P r o p o s i t i o n 4.5).

Categories of relative cribles are a practical M-adjunctions.

for the corres-

source of

Given a category C, each functor M: A ~ C generates

the ordinary c a t e g o r y ~o of "M-cribles" An M - c r i b l e is a functor t r a n s f o r m a t i o n F ~ C(M-,C)

or "M-preatlases"

(|2J

§2).

F: A °p ~ V for which there exists a natural each of whose components

is in M.

A

m o r p h i s m from F to G of M-cribles is a natural t r a n s f o r m a t i o n from F to G. We shall call the functor M: A ~ C extendable M) if the limit C(MF,C)

fA|FA, C(MA,C)I

exists in V and has a r e p r e s e n t a t i o n

for each M-crible F. P r o p o s i t i o n 3.1.

If M: A ~ C is extendable then A

enrichment to a V-category A and M: A ~ C is a left Proof. A(F,G)

(with respect to

~ ~ A

o

admits

V-adjoint functor.

For each pair of M-cribles F and G, define

IFA,GAI

(cf. 151

§4).

This limit exists in V by virtue of

the M - e m b e d d i n g

/A [ FA, GA]

fA [ i,m]~

[FA, C(MA,C)] ~ C(~F,C) A

with the f o l l o w i n g lemma. Lemma.

If mAB:

S(AB) ~ T(AB)

is a natural family of M-monlcs

b e t w e e n two functors from A °p ® A to V then the end of S exists

in V

if the end of T exists. Proof.

As in [9] P r o p o s i t i o n 111.2.2,

the end fA S(AA)

c o n s t r u c t e d directly as the i n t e r s e c t i o n in fA T(AA)

is

of all the pull-

back diagrams:

[ A

PA --

,- S(AA)

T(AA)

=- T(AA).

The components mAB are all r e q u i r e d to be monic the induced family of morphisms

fA S(AA)

~ S(AA)

in order that

should be natural in

A. The functor category A inherits equalisers, pullbacks M-monics,

and i n t e r s e c t i o n s of M-monics

from V.

However,

for A to be

c o t e n s o r e d we shall in general suppose that C is cotensored; because the pointwise

cotensor /X,F]

of

this is

of X E V with F 6 A is then an

M-crible by virtue of the M - e m b e d d i n g

[~FI

[ l ~ m ] [X, C(M-,C)] ~ C(M-, [X,C]).

The category A also inherits

(E,M)-factorisation

from V and, by Prop-

osition i.i, the relative Yoneda a d j u n c t i o n M---~T: C ~ A is an M-adjunction.

In other words, F: A °p ~ V is an M-crible if and only

if the a s s o c i a t e d natural t r a n s f o r m a t i o n F ~ C(M-, MF) has components in M. If the functor M: A ~ C is M-faithful

in the sense that each

10 component of the canonical t r a n s f o r m a t i o n A(-A) ~ C(M-,MA) then every r e p r e s e n t a b l e

is in M,

functor from A °p to V is an M-erible.

Thus

there is a dense Y o n e d a e m b e d d i n g YA: A ~ A with respect to w h i c h plays the role of the funetor category [A,V]; however A is a well-defined

V-category,

even when A is large.

P r o p o s i t i o n 3.2. then so is M: A ~

If M: A ~ C is extendable and M-faithful

C.

Proof.

To prove that M is extendable,

M-cr~ble with t r a n s f o r m a t i o n ~F: KF ~ C(MF,C) M-faithful,

let K: ~op ~ V be an

in M.

Because M is

the category A contains all the r e p r e s e n t a b l e functors

from A °p to V.

Thus we can substitute F = A(-,A)

in ~ and obtain an

M-monic

K(A(-,A)) -- C(M(A(-,A)),C) ~ C(MA,C). This makes K(A(-,A)), M-crible

r e g a r d e d as a functor in A E A °p, into an

so there exists a r e p r e s e n t a t i o n C(M(K(A(-,A))),-)

Thus, on d e f i n i n g ~

~ f [K(A(-,A)), A

= M(K(A(-,-))),

c(~,-)

C(MA,-)]

we obtain

~ fA[X(A(-,A)), c(~,-)l

fA[ fF~F ~ FA, C(MA,-)[ by the r e p r e s e n t a t i o n theorem,

~[KF, fA[FA, C(MA,-)]] = ] [KF, C(MF,-)],

F

as r e q u i r e d for M to be extendable. M-faithful,

To verify that M: A ~ C is

consider the following commutative

square:

11

MFG ]A [FA,GAI

..

:,....C(MF,MG)

[ l,m] A

fA [ FA,C(MA,C)] ~f [ Z,C(1,r~)]

fA[FA,C(~,~Q)]

A where

m: MG ~ C c o r r e s p o n d s to m: G ~ C(M-,C)

MFG E M because

f [l,ml A

E

M.

under adjunction.

Then

This completes the proof.

While the process of forming M from M is clearly not a closure operation, forms a monad.

the p r e c e d i n g result leads us to ask w h e t h e r it

For a fixed category C, the M-faithful e x t e n d a b l e

functors M: A ~ C may be r e g a r d e d as a category M(C)

in which a mor-

phism from M: A ~ C to N: B ~ C consists of a functor ¢: A ~ B and a natural i s o m o r p h i s m N~ ~ M.

Each such m o r p h i s m induces a r e s t r i c t i o n

functor ¢*: B ~ A which maps G E B to G~ ~ A; this functor preserves limits but in general does not commute with the a u g m e n t a t i o n s M and into C. Example iant "endofunctor" isomorphism)

3.3.

The c o n s t r u c t i o n of M from M becomes a covar-

(composition being p r e s e r v e d only to within an

on M(C) if we replace ¢* by its left adjoint ~.

V = S, the existence of ¢ follows from T h e o r e m 2.1. exists as a left

For a general V,

V-adjoint if C is cotensored relative to V.

r e s u l t i n g " p r e c o m p l e t i o n monad"

on

YA: A ~ A as its unit and Y~: ~ ~ A

The

M(C) has the Yoneda e m b e d d i n g as its multiplication.

In the case where M: A ~ C has a right adjoint, alent to the c a t e g o r y of all cribles of A. suitably complete

For

A is equiv-

If, in addition,

A is

(see P r o p o s i t i o n 4.5) then the Yoneda e m b e d d i n g Y

has a left adjoint which serves as an algebra structure for A with respect to this m o n a d on M(C). Conversely,

if S: A ~ A is the structure functor for any

12

a l g e b r a of the p r e c o m p l e t i o n monad then S is left adJoint to Y: A ~ with a d j u n c t i o n unit ~ , transformation;

where A: ~A ~ Y~ is the canonical natural

this follows from T h e o r e m 2.2.

i n d i c a t e d to the author by Anders Kock

§4

EXAMPLES;

E x a m p l e 4.1.

(cf. [17]).

C O M P L E T I O N S AND M O N O I D A L CLOSURE

As in §3, the base category V is S - c o m p l e t e and

admits all intersections

of M-subobjects

Let M: A ~ C be an M - f a i t h f u l functor whose direct

The role of ~ was

for the given choice of M.

extendable functor and let K: K ~ A be a

limit colim K exists in A.

Then M preserves

colim K if and only if M inverts the canonical t r a n s f o r m a t i o n s: colim A(-, Kk) ~ A(-, k Because each r e p r e s e n t a b l e there exists a largest

colim Kk). k

functor from A °p to V is orthogonal

to s

(relative to M) full r e f l e c t i v e subcategory S

of ~ for which the Yoneda e m b e d d i n g A C A factors through As and preserves colim K. This is the basis of many c o m p l e t i o n processes. ular,

In partic-

if M: A ~ C is a strongly c o g e n e r a t i n g and c o l i m i t - p r e s e r v i n g

extendable full e m b e d d i n g into a cotensored category C then the facto r i s a t i o n of §2 yields:

z

A

C, M

where E is the class of all m o r p h i s m s a dense,

strongly cogenerating,

inverted by M.

Thus one obtains

continuous and cocontinuous e m b e d d i n g

A -~ AZ and the functor A~ ~ C reflects isomorphisms.

This gives an

alternative proof of [9] T h e o r e m 111.3.2. Remark. functor

The process of e x t e n d i n g an M-faithful extendable

M: A ~ C to M: A -* C and then forming the category of

13

fractions of A with respect to the class of all m o r p h i s m s inverted by defines a m o n a d on the category M(C) functors over C. monad"

of M - f a i t h f u l extendable

This monad is a quotient of the " p r e c o m p l e t i o n

on M(C) d e s c r i b e d in Example 3.3. The following two examples concern the r e l a t i o n s h i p of the

f a c t o r i s a t i o n s of 51 and §2 to a given m o n o i d a l

structure on the cat-

egory B, and to the q u e s t i o n of m o n o i d a l closure c o n s i d e r e d in monoidal

l o c a l i s a t i o n [ 8].

be symmetric

The m o n o i d a l

for n o t a t i o n a l

simplicity,

structure on B is a s s u m e d to however the results can be

e s t a b l i s h e d in the more general setting of b i c a t e g o r i e s of B $ n a b o u [4]), b i c l o s e d bicategories, We recall from [7] called a normal reflective

and their localisation.

that a full reflective subcate~ory

enrichment to a m o n o i d a l adjunction.

(in the sense

subcategory of B is

if the a d j u n c t i o n admits The existence of such an enrich-

ment implies that the r e f l e c t i n g functor preserves tensor products when B is m o n o i d a l

closed,

and,

is equivalent to the subcategory being

closed under e x p o n e n t i a t i o n in B (by [ 7] T h e o r e m 1.2). Example has

4.2.

Let S--iT: C ~ B be an a d j u n c t i o n in which

([,M)-factorisations,

r e f l e c t i o n B~-~B.

as in §i, and let P: B ~ B' denote the

If B has a m o n o i d a l structure then P is a m o n o i d a l

l o c a l i s a t i o n in the sense of [81 A,B E B.

if P(A @ n{) is an i s o m o r p h i s m for all

By P r o p o s i t i o n 1.2, P(A @ q~) is the unique m o r p h i s m m a k i n g

both the f o l l o w i n g diagrams commute:

A®B

, ~,,,

I®T] '

m

P(A@B)

,,- T S ( A ® B )

P(I@q')

A@PB

= n'

where

B

q' E E and m E M.

TS(I@q') : TS(A@PB)

P(A~PB) m

Thus an obvious

sufficient c o n d i t i o n for

P(l@q') to be an i s o m o r p h i s m is that A @ ~ for all A,B E B.

E E and TS(A @ ~ )

C M

14 In the case where the m o n o i d a l structure on B is closed, the condition A ® n~ E E for all A,B E B is a u t o m a t i c a l l y Moreover,

if S--~T:

satisfied.

C ~ B is a m o n o i d a l adjunction then S n e c e s s a r i l y

preserves tensor products,

so TS(A @ ~ )

is always an isomorphism.

Thus the category B' becomes a normal reflective

subcategory of B.

This example is related to the situation discussed in [8] where B = [A°P,v]

for a small m o n o i d a l category A over V, and B is

a s s i g n e d the convolution structure: F @ G = fAA'FA ~ GA'

@ A(-, A @ A')

IF,G] = fA IFA, a(A~-)1 If M: A ~ C is a functor into a cocomplete category C then the category A of M-cribles is a normal reflective subcategory of |A°P,v] if C(M(A@-),C) ition I.i).

is an M-crible

for all A E A and C E C (by [8]

Propos-

This coincides with the p r e c e d i n g situation if C is

m o n o i d a l closed and M preserves tensor products. Example 4.3

Suppose

(s,n): S--~T:

C ~ B is an M - a d j u n c t i o n

in w h i c h B has equalisers,

pullbacks of M-subobjects,

sections of M-subobjects.

Then, by Corollary 2.3, B Z g B has a left

adjoint if Z is any class of morphisms (orthogonally)

and all inter-

in B inverted by S.

If Z is

closed then the r e f l e c t i o n functor coincides with the

p r o j e c t i o n of B onto the category of fractions of B with respect to Z. Suppose that B has a m o n o i d a l closed structure and let Z ° = {s E Z; B@s E Z for all B E B} denote the m o n o i d a l interior of Z with respect to this structure. P r o p o s i t i o n 4.4. by S--~T:

If Z is a class of morphisms

in B inverted

C ~ B then BZO is a normal r e f l e c t i v e subcategory of B.

Proof.

The left adjoint of BZO c B exists by Corollary 2.3.

To verify that BzO = {C E B; B(s,C)

an i s o m o r p h i s m for all s E Z °} is

closed under e x p o n e n t i a t i o n in B, choose objects B E B and C C B Because

s@B E Z ° for each s E Z ° we have that B(s@B,C)

Z O"

~ B(s,[ B,C] ) is

15

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

for all

s E Z o.

Thus

[B,C]

is o r t h o g o n a l

to Z °, as

required. It follows Z is closed. subclasses closed

Thus,

embedding

of A itself

such an adjoint

M-subobjects

is the

A(-,

R~F)

adjoint

structure to M).

This

R and

A criterion

if A is closed

of M-subobjects,

is a corollary

satisfied

admits

a unit

it admits category

of limit

The

embedded

w h i c h all example

functor.

reflective spaces

into a r e f l e c t i o n

is i n v e r t e d

a monoidal

for the e x i s t e n c e

of

lie in

Here

under

extendable the f o r m a t i o n of

2.2.

The c o m p l e t e n e s s

T: ~ ~ ~ m a p p i n g

F to

components M.

is that where

A = Top and M: Top -~ S

Top is c a r t e s i a n closed and conThe

cartesian

closure

extension"

discussed

in [ i] and

embedding

into the

cartesian

closed

of Fischer [12].

THE F A C T O R I S A T I O N

general

unit

and if

in A.

by A and the functor

to M is the "minimal

a normal

§5

of B which

and i n t e r s e c t i o n

of T h e o r e m

n: i ~ T with

~ A(-,RMF)

respect

whose

on ~ induces

Top as a full r e f l e c t i v e subcategory. with

of

of

and extendable

If M: A -~ C is an M - f a i t h f u l

adjoint

is the u n d e r l y i n g - s e t

Top

subcategory

if

following:

4.5.

The m o t i v a t i n g

tains

lattice

can be r e f l e c t i v e l y

in A, then A is r e f l e c t i v e

are

F ~ C(M-,~IF)

a left

closed

pullbacks

Proof. hypotheses

on the class

to the

if M: A ~ C is M-faithful

A-*A has

a right

of equalisers,

subcategory

(relative

Proposition with

restricts

closed

closure".

by M then each m o n o i d a l

functor

operation

of Z, and each r e f l e c t i v e

In particular, the Y o n e d a

Z ° is o r t h o g o n a l l y

the m o n o i d a l - i n t e r i o r

B E as a r e f l e c t i v e

in a "monoidal

closure

that

of Z = {f; Sf isomorphism}

subclasses

contains

from this result

process

followed

S Y S T E M FOR LEFT ADJOINTS.

of f a c t o r i n g

an adjoint

pair of functors

by an i s o m o r p h i s m - r e f l e c t i n g

embedding

of

16

has a global interpretation.

We

consider the "category"

Adj

for w h i c h

an object is a category which is M-complete with respect to a suitable

(E,M)

(see §§i and 2) f a c t o r i s a t i o n syst~m left-adjoint

functor;

the left adjointso

on it and a m o r p h i s m is a

c o m p o s i t i o n of morphisms is Just c o m p o s i t i o n of

The class of r e f l e c t i o n functors is d e n o t e d by R

and the class of i s o m o r p h i s m - r e f l e c t i n g left adjoints is denoted by N. P r o p o s i t i o n ~.i.

(R,N)

To w i t h i n natural i s o m o r p h i s m of functors,

forms a f a c t o r i s a t i o n system on Proof.

We use several facts from [13]

r e f l e c t i o n is an "epimorphism" adjoint, b e i n g faithful, functors;

Adj. §2.3.

and every i s o m o r p h i s m - r e f l e c t i n g left

is a "monomorphism"

to within i s o m o r p h i s m of

thus the f a c t o r i s a t i o n will be "proper".

left-adjoint

Moreover,

functor on an M - c o m p l e t e category, where

proper f a c t o r i s a t i o n (by §i and §2).

Clearly every

(E,M)

every

is a

system, has a f a c t o r i s a t i o n of the r e q u i r e d form

Finally, we verify that if a diagram of left adjoints: S A

~

F

B

~P//

H

J

C

~ D M

with S ~ R and M E N,

commutes to within an i s o m o r p h i s m then there

exists a left adjoint P such that PS ~ F and MP ~ H. (~,B): F--~G, M--~N,

and H - ~ K

and M reflects isomorphisms, Q = SG.

be the adjunctions.

Let

(~,~): S--4E,

Because HS ~ MF

F factors through S as P = FE.

Let

To verify that P--~Q it suffices to verify that G factors

through E; that is, that the m o r p h i s m ~GC: GC --~ESGC is an i s o m o r p h i s m for all C E C.

But,

for all C C C,

FqGC: FGC ~ F E S G C is an i s o m o r p h i s m because M reflects isomorphisms and MF~Gc ~ HSqGc which is an i s o m o r p h i s m because Sq is an isomorphism.

Define

PC to be

17

the composite: PC

ESGC

GF(ESGC)

~,GC

~-~GFGC

.

G(FnGc )-i Then pC.~GC adjunction implies

= I by naturality (a,6):

nGc'Pc

F--~G.

= i.

of 6 and the triangle

Because

axioms

E is a full embedding,

This completes

the proof.

for the

this

18

REFERENCES

[1]

Antoine, P., Extension minimale de la catSgorie des espaces topologiques, C.R. Acad. Sc. Paris, t, 262 (1966), 1389-1392.

[2]

Applegate, H. and Tierney, M., Categories with models, Seminar on Triples and Categorical Homology Theory, Lecture Notes 80 (Springer 1969), 156-244.

[3]

Applegate, H. and Tierney, M., Iterated cotriples, Reports of the Midwest Category Seminar IV, Lecture Notes 137 (Springer 1970), 56-99.

[4]

B@nabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar I, Lecture Notes 47 (Springer 1967), 1-77.

[5]

Day, B.J. and Kelly, G.M., Enriched functor categories, Reports of the Midwest Category Seminar III, Lecture Notes 106 (Springer 1969), 178-191.

(6]

Day, B.J., On closed categories of functors, Reports of the Midwest category Seminar IV, Lecture Notes 137 (Springer 197o), 1-38.

[71

Day, B.J., A reflection theorem for closed categories, J. Pure and Applied Algebra, Vol. 2, No. i (1972), i-ii.

[8]

Day, B.J., Note on monoidal localisation, Bull. Austral. Math. Soc., Vol. 8 (1973), 1-16.

[9]

Dubuc, E.J., Kan extensions in enriched category theory, Lecture Notes 145 (Springer

[io]

1970).

Eilenberg, S. and Kelly G.M., Closed categories, in Proc. Conf. on Categorical Algebra, La Jolla, 1965 (Springer 1966), 421-562.

[11]

Fakir, S., Monade idempotente associ@e & une monade, C.R. Acad. Sc. Paris, t. 270 (1970), 99-101.

19 [12]

Fischer, H.R., Limesr~ume,

Math. Annalen, Bd. 137 (1959)

269-303. [13]

Freyd, P. and Kelly, G.M., Categories of continuous

functors

I, J. Pure and Applied Algebra, Vol. 2, No. 3 (1972), 169-191. [14]

Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, Springer-Verlag,

[15]

Kelly, G.M., Monomorphisms, Austral.

[16]

Berlin, 1967.

epimorphisms,

and pullbacks, J.

Math. Soc., Vol. 9 (1969), 124-142.

Kelly, G.M., Adjunction for enriched categories,

Reports of the

Midwest Category Seminar III, Lecture Notes 106 (Springer 1969), 166-177. [17]

Kock, A., Monads for which structures are adjoint to units (to appear).

[18]

Maciane, S., Categories for the working mathematician, ~erlag, New York, Heidelberg,

[19]

Wolff, H., F-localisations 1973, 405-438.

Springer-

Berlin, 1971.

and F-monads, J. Algebra, Vol. 24,

ON C L O S E D C A T E G O R I E S OF F U N C T O R S II

*

by Brian

In many product basic

examples

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

generating

free

tensor

formal

product

or, more

can be e x t e n d e d biclosed

of this

a l o n g a dense

types

of dense

Yoneda

functors there type

functor

embedding on

A.

corresponds

C C B.

adjunction tiation

of

to this

N: A °p ÷ C is a dense embedding

of

are c o m b i n e d

A °p in [A,S]

*The r e s e a r c h here r e p o r t e d R e s e a r c h Council.

E7].

[A,S]

structure

adjoint

structure under

followed

is the

set-valued

on EA,SI.

is, the left biclosed

the two basic

on a small

category The

A

second

to a full then the

all e x p o n e n -

of B. in the

f o l l o w i n g manner.

then it can be d e c o m p o s e d

was

A

a monoidal

The first of all

if C is closed

functors

functor

on a category

by c o m b i n i n g

structure

that

if and only

two results

a

when a m o n o i d a l

structure

in [4] and

If B has a m o n o i d a l

by the i n t e r n a l - h o m

from the

consider

N: A °p + C to produce

question

biclosed

functor;

is

C.

To each p r o m o n o i d a l a monoidal

on a

example

groups

We shall

by a s k i n g

a promonoidal functor

A simple

groups.

construction

the tensor

structure

of two a b e l i a n

A °p into the category

is m o n o i d a l

The

product

considered

is the r e f l e c t i o n

embedding

in the category.

on the category

We give an answer

categories

of a m o n o i d a l

of two free a b e l i a n

generally,

structure

biclosed

as an e x t e n s i o n

of the tensor

generalisation

structure

of m o n o i d a l

set of objects

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

Day

by a r e f l e c t i o n

supported

by a grant

from

If

into the Yoneda [A,S]

to

from the D a n i s h

21

C p r o v i d e d C is sufficiently complete w i t h respect to A.

However,

this over-all c o m p l e t e n e s s h y p o t h e s i s on C is g e n e r a l l y u n n e c e s s a r y in order to produce a m o n o i d a l biclosed

structure on

case w h e n N itself is a r e f l e c t i o n and

C, as it is in the

A °p is m o n o i d a l biclosed.

Thus we answer the original q u e s t i o n by first c o n s i d e r i n g a c o m p l e t i o n C* of C and finding conditions under which this c o m p l e t i o n is m o n o i d a l biclosed. completed

The structure we obtain on C is simply the trace of the structure C*.

The p r o c e d u r e produces many known c o n s t r u c t i o n s of tensorproduct functors and i n t e r n a l - h o m functors. the tensor product of algebras closed category, by L i n t o n lution"

it produces

for a m o n o i d a l m o n a d on a m o n o i d a l bi-

and gives a conceptual

[18] and Kock [16].

In particular,

e x p l a n a t i o n of c o n s t r u c t i o n s

It also produces the canonical

structure on a functor category

[A,B] w h e n A

"convo-

is a p r o m o n o i d a l

category and B is a suitably complete m o n o i d a l b i c l o s e d category. One advantage of this a p p r o a c h is that the coherence of the structure p r o d u c e d on C follows from the coherence of C* which, turn,

follows from the coherence results already e s t a b l i s h e d in [4]

and E7].

Completions can also be used to provide a concept of "change

of V-universe"

in the case where all categorical algebra is based on a

fixed symmetric m o n o i d a l closed category V. " V-structure" large)

in

on any category [A,B]

This enables us to put a

of V-functors

V-category A to a suitably e n r i c h e d category

from a (possibly B.

This,

in

turn, makes the relative c o m p l e t i o n process a v a i l a b l e for large categories

V-

(and u l t i m a t e l y leads to a r e d u c t i o n in algebraic computa-

ion) . The c o m p l e t i o n process is used in sections

5,b, and 7 to exam-

ine m o n o i d a l b i c l o s e d structures on categories of functors from a promonoidal

category to a m o n o i d a l b i c l o s e d category.

discuss b i c l o s e d categories of c o n t i n u o u s the work of B a s t i a n i - E h r e s m a n n

[2].

In section 6 we

functors and relate this to

Finally,

section 8 contains a

22

proof of the r e p r e s e n t a t i o n theorem for monads. The u n e x p l a i n e d notations and t e r m i n o l o g y used in this article are standard, and familiarity with the r e p r e s e n t a t i o n t h e o r e m is assumed [7],

(cf.

[ii] §i).

This m a t e r i a l is a development of results in [4],

[8], and [9] and is based on a doctoral thesis by the author

and [6]).

([5]

The thesis was supervised by Professor G.M. Kelly at the

U n i v e r s i t y of New South Wales.

The author has also b e n e f i t e d from

several d i s c u s s i o n s with A. Kock and R. Street.

The s e c t i o n - h e a d i n g s are as follows:

§i

R e f l e c t i o n in closed functor categories.

§2

The c o m p l e t i o n process.

§3

M o n o i d a l closed completion.

§4

MonQidal monads.

§5

B i c l o s e d functor categories.

§6

B i c l o s e d categories of continuous functors.

§7

C o m p l e t i o n of functor categories.

§8

Denseness

presentations.

§i R E F L E C T I O N IN CLOSED FUNCTOR C A T E G O R I E S The formulas needed for m o n o i d a l b i c l o s e d functor categories and r e f l e c t i o n of m o n o i d a l biclosed and [7].

S denotes

V = (V,®,I,[-,-],...)

structures are r e c a l l e d from [4]

"the" cartesian closed category of small sets and is a fixed symmetric m o n o i d a l

with small limits and colimits.

closed base-category

All concepts of categorical algebra

are h e n c e f o r t h assumed to be relative to this V unless o t h e r w i s e stipulated. An a d j u n c t i o n

(~,~): S--iT:

C ÷ B is called a m o n o i d a l reflect-

ion or normal r e f l e c t i o n if T is a full embedding and the a d j u n c t i o n

23 data admits monoidal biclosed

enrichment.

If B = (B,@,I,/,\,...)

category then S--~T is monoidal

following pairs of conditions notation)

is a monoidal

if and only if one of the

is satisfied

(where T is omitted

from the

for all B,B' E B and C E C: n: C/B ~ S(C/B)

]

n: B\C ~ S(B\C) q\l: SBkC ~ B\C

1

l/n: C/SB ~ C/B

]

S(q@l):

S(B@B')

~ S(SB@B')

]

S(!@q):

S(B'@B)

~ S(B'®SB)

I

~ S(SB®SB').

}

S(n®D): S(B@B')

A subcategory A C B is called class of functors ~ ( A , - ) ;

A E A} jointly reflects

r e f l e c t i o n theorem for monoidal 1.2)

strongly generating

biclosed

states that the above conditions

in B if the

isomorphisms.

categories

The

(cf. [7] Theorem

are equivalent

to either of the

pairs n: D/A ~ S(D/A)

]

n: AkD ~ S(i\D)

J

S(n~l):

S(B~A)

~ S(SB@A)

1

S(l®n):

S(A@B)

m S(A®SB)

J

(I.i)

for all B E B, A C A, and D E D, where A strongly generates B and D strongly

cogenerates

C.

If B is a monoidal

biclosed

category and C c B

category we say that C is closed under e x p o n e n t i a t i o n C/B have isomorphs generally functors

in C for all B E B

a stronger

and C e C.

condition than requiring

-\- and -/- have restrictions

to

is a full subin B if B\C and

Note that his is

that the internal-hom

C.

Given a small category A, each functor F e [A,V] expansion: F ~

[A FA @ A(A,-)

: A-~V

has an

24

Thus

each m o n o i d a l

is isomorphic

biclosed

structure

to a structure

of the

on the functor

following

F@G =

lAB FA @ GB @ P(AB

G/F =

lAB [P(-AB),[FA,GB]]

category

form:

-)

(1.2)

(1.3) F\G =

f

AB

[ P(A-B) ,[ FA,GB] ]

where P(AB-) = A ( A , - )

® A(B,-)

P: A °p @ A °p @ A ~ V

is the

A monoidal

biclosed

I and a s s o c i a t i v i t y

~ A(i,-)

structure

The e x t e n s i o n called

of p r o m o n o i d a l

on

categories,

x(F)

= [ =

AB

[A

B.

such a c o l l e c t i o n

structure

of A with

IA(A,B), [IA,

components

A@B whose

for A and

§3), where

"o" denotes

(A,P,I,~,k,0)

on A to

[A,V]

is

V and it is an internal

hom

[ P ( A B C ) , [FA % GB,HC]] ABC

small p r o m o n o i d a l

product

by P and

with:

[FA,GB]]

[I,FA]]

and I are r e g a r d e d

and O - d i m e n s i o n a l

is d e t e r m i n e d

A.

the c o n v o l u t i o n

[A,V](F,G)

([4]

of a p r o m o n o i d a l

P(FGH) = [

P, Hom,

axioms

We call

of @ on [A,V].

m P(ABX) op(XC-)

: IXoP(AX-) coherence

functor

isomorphisms:

p = 0A

a promonoidal

of those

[A~V]

on

~ A(A,-)

composition.

tensor

functor"

: IXoP(XA-)

profunctor

Two

The r e s u l t i n g

k = kA

suitable

where

structure

P(AX-)oP(BCX)

satisfying

sometimes

"structure

and identity

= ~ABC:

[A,V].

in

,

as the

2-dimensional,

of p r o m o n o i d a l categories

(P,Hom,I)

The usual

1-dimensional,

structure.

A and

B have a p r o m o n o i d a l

are the r e s p e c t i v e

isomorphism

tensor

of categories:

products

25

[A~B,V] then

becomes

an

~ [A,IB,VJ! isomorphism

of

To combine convolution monoidal

biclosed

structure

tive embedding where egory.

monoidal

categories.

with reflection,

and let S--~T:

C contains

The Yoneda embedding

biclosed

let [A,V]

C ~ [A~V]

have a

be a full reflec-

P as a strongly c o g e n e r a t i n g

A°p~[A,V]

is strongly

subcat-

generating,

thus

we have: Proposition

i.i.

The adjunction

S--~T:

C ~ [A,V]

is monoidal

if and only if the functors G/A(A,-)

= /B [P(-AB),GB]

A(A,-)\G

: ~[P(A-B),GB|

have isomorphs

in C for all A E A

Proof, B = [A,V].

and G E P.

This is a r e f o r m u l a t i o n

of Condition

(i.I) with

For example, G/A(A,-)

= /B'B [P(-B'B),[A(A,B'),GB]]

/B

[P(-AB),GB]

by (1.3)

by the r e p r e s e n t a t i o n

theorem

applied to B' E A. Remark. bicate@ories

The concept

of convolution

in the sense of B@nabou

[3].

can be extended to include If

A = {Axy ; x,y E 0bj A} is a V-bicategory, [A,V]~

is a biclosed

with hom-categories

{[Axy,V] ; x,y E Obj A}

bicategory

This consideration,

by formulas

in turn,

The extension of the results straightforward The reflection

Axy , then

analogous

leads to the concept

(1.3).

of a V-pro-bicategory.

in this article to V-bicategories

exercise once this conceptual theorem for biclosed

of the "one-object"

to (1.2) and

theorem above.

is a

framework is established.

bicategories

is an exact analogue

A form of this theorem has been

2B

introduced by J. Meisen [19] in the study of relations Remark on symmetry.

The concept of a symmetric p r o m o n o i d a l

category is defined in [4] §3. result-with-symmetry

in categories.

This produces an (obvious) analogous

for each result verified in the sequel.

§2

THE C O M P L E T I O N PROCESS

For this section the base category V will be assumed to have all i n t e r s e c t i o n s of m o n o m o r p h i s m s as well as all small limits and colimits. To each small category C we will assign a c o m p l e t i o n C* which is taken to be the largest full reflective contains the r e p r e s e n t a b l e subcategory.

subcategory of [c°P,v] which

functors as a full strongly c o g e n e r a t i n g

This coincides with the left-adjoint

f a c t o r i s a t i o n of the

c o n j u g a t i o n functor

[c°P,v] through

a reflection

~ [C,V]°P; followed

F ~ [ [FC,C(C,-)], C by an i s o m o r p h i s m - r e f l e c t i n g

embedding,

denoted: [ c°P,vJ

;-

C*

[ C,V] op The r e s u l t i n g embedding of C into its c o m p l e t i o n will be denoted by: E ~ E(C):

C ~ C*.

The basic p r o p e r t i e s of C* can be verified by e x a m i n i n g the c o t r i p l e - t o w e r c o n s t r u c t i o n of C* (Appelgate and Tierney

[i] and Dubuc

[i0]) or by using a direct method of a d j o i n t - f u n c t o r f a c t o r i s a t i o n ([9], Example 4.1).

There is a closely related t h e o r e m in W o l f f [22],

§5.6. The c o m p l e t i o n C* can be d e s c r i b e d e x p l i c i t l y as the full subcategory of [c°P,v]

of functors F such that ~

[Sc,FC]

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

27

whenever

s is a m o r p h i s m in [c°P,v]

inverted by conjugation,

C* is equivalent to the c a t e g o r y of V-fractions of [c°P,v]

As such,

with respect

to the class of m o r p h i s m s inverted by conjugation. Remark.

The assignment C ~ C* is c a n o n i c a l l y functorial

w i t h i n isomorphism)

(to

on functors S: C ~ B which have a right adjoint,

and the image S*: C* ~ B*, S*E = ES, has a right adjoint. P r o p o s i t i o n 2.1.

If N: A ~ C is a dense functor then

EN: A ~ C* is dense. This fact is established in [8] Corollary

3.2.

C* is equivalent to the full subcategory of [A°P,v] that ~ [ S A , F A ]

More precisely,

of functors F such

is an i s o m o r p h i s m w h e n e v e r s is a m o r p h i s m in [A°P,v]

inverted by the r e s t r i c t e d - c o n j u g a t i o n

[A°P'v]

~ [C'v]°P;

functor:

F ~ [A [FA,C(NA,-)] .

Thus the r e f l e c t i o n from [A°P,v]

to C* at F E [A°P,v]

has the value:

A

EN(F) = I FA.ENA, and this is isomorphic to E(f A FA.NA) w h e n e v e r the r e f l e c t i o n ~ P A . N A of F exists in

C.

The second m a i n o b s e r v a t i o n of this section is that the completion process provides a "structural change of V-universe".

Consider

the case V = S and C = V and let S* be a larger c a r t e s i a n closed category of sets c o n t a i n i n g S and V, and any other categories that we want to regard as "small", as internal category objects. be the c o m p l e t i o n of V with respect to S *.

Let W = V*

The basic p r o p e r t i e s of V*

(verified in [83) are as follows: Property i. Because

V °p is a symmetric m o n o i d a l category,

the

functor category [ v°P,s *] is symmetric m o n o i d a l closed and this structure extends the original structure of

V.

closed, the reflective embedding W C [ v°P,s*]

Because

V is monoidal

is monoidal,

hence the

embedding E: V ~ W preserves tensor product and internal hom.

This

28

implies

that

the

symmetric

embedded

in

W-Cat

internal

hom. Property

and dense,

V-Prof

(cf.

into

between

them,

closed

tensor

E:

is strongly

V ~ W Thus

consisting

is embedded

bicategory

V-Cat

category

preserves

coends.

Introduction),

the b i c l o s e d

profunctor

The e m b e d d i n g

it preserves

[3],

V-profunctors

and the embedding

2.

hence

monoidal

W-Prof

is fully

product

the biclosed

cogener-ating bicategory

of V-categories

(but not

fully

and

and

on morphisms)

in such a way as to preserve

composition.

Property

3.

completion

IA,VJ*

equivalent

to

V-promonoidal essentially

Because

E:

with respect

[A,WI

to W of a functor

(as v e r i f i e d

structure

unique

V -* W is strongly

in [8]

on a small

W-promonoidal

§4).

V-category

structure

cogenerating, category

This

IA,VI

ensures

is

that

A corresponds

on A when

the

each

to an

A is r e g a r d e d

as

a W-category. Because sections

W has all S * - s m a l l

of m o n o m o r p h i s m s ,

be an a r b i t r a r y hypotheses

products filtered

and colimits

the given base monoidal

closed

and all

inter-

category

V can be assumed

to

category

(with no completeness

on it).

Remark cartesian

symmetric

limits

2.2.

closed

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

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

([7] C o r o l l a r y colimits

2.1).

closed

category

[v°P,s *]

from

then W is

to W preserves

Thus

finite

products

in any c a r t e s i a n

closed

category,

commute

finite

with

because

this

is

so in [ v°P,s *] .

§3

MONOIDAL

CLOSED

Let N: A °p ~ C be a dense let W be a c o m p l e t i o n S* of sets w h i c h W-promonoidal

functor

of V with respect

contains S, V, A,

category.

COMPLETION

and

between

two

to a c a r t e s i a n C.

V-categories closed

Let A = (A,P,I,...)

and

category be a

29

P r o p o s i t i o n 3.i.

The reflective e m b e d d i n g C* C [A,W]

is monoi-

dal if and only if the functors

[P(-AB), C(NB,C)] B

(3.1)

[P(A-B), C(NB,C)] B have isomorphs in C* for all A E A and C c C. Proof.

This is a restatement of P r o p o s i t i o n 1.1, with the

strongly c o g e n e r a t i n g class 9 C C* being Remark.

{C(N-,C);

C E C}.

The objects C E C in (3.1) could be r e s t r i c t e d

further to lie in any subcategory 9 C C for which strongly cogenerates

C*.

For example,

is the limit of some functor

{C(N-,D);

D E 9}

if D c C and each object C E C

(depending on C) with object values in 9,

then the embedding 9 C C followed by E: C ~ C* strongly cogenerates We shall consider the ~pecial case where the functors have isomorphs in C, and let C(N-,H(AC)) representations.

and C(N-,K(AC))

The i n t e r n a l - h o m and t e n s o r - p r o d u c t

C*.

(3.1)

be their

operations on C*

then provide functors -/-: C ® C °p ~ C*, -\- C °p @ C ~ C*, and -@-: C @ C ~ C* w i t h values:

D/C = fA [C(NA,C), C(N-,H(AD))] C\D = f

[C(NA,C),

(3.2)

C(N-,K(AD))]

A and AB

C @D = [

(C(NA,C)

C(NB,D) ) -Q(AB)

(3.3)

where A Q(XY)

= ~

P(XYA).ENA.

The identity object of C* is: I = S A IA.ENA. Remark.

Note that, by the c o n s t r u c t i o n of the m o n o i d a l bi-

closed c o m p l e t i o n C*, the r e p r e s e n t i n g objects H(AD) are isomorphic

(3.4)

and K(AD)

in C

in C* to the exponentials ED/ENA and ENA\ED respectively.

30 §4

Let that

is,

T = (T,~,q)

a monad

where

( T , T , T °) w i t h r e s p e c t natural

MONADS

be a m o n o i d a l T:

monad

on a m o n o i d a l

B ~ B has a m o n o i d a l

to w h i c h

p: T 2 ~ T a n d

functor

q:

category

B;

structure

i ~ T are m o n o i d a l

transformations. Let

egory

MONOIDAL

B(T)

exists

denote

over

the

the b a s e

category category

B(T)(CD)

~

of T - a l g e b r a s V when

B(CD)

over

V has t h e

.... B ( [ , 1 )

~

B.

This

cat-

equalisers

B(TC,D)

B(TC,TD) for

all

K(T)

T-algebras

has

(C,~)

and

The monoidal

axioms

a monoidal

structure; 0:

is d e f i n e d

on objects

(D,~) on

K(T)

(as

(T,~,q) the

constructed imply

that

tensor-product

® K(T)

in the

[10]

or

Kleisli

[16]). category

functor

~ K(T)

as it is in B a n d o n m o r p h i s m

objects

by t h e

components :

A$-

K(T) (B,C)

B(B,TC)

.......

- - - - - - - - ~ B ( A®B, AOTC ) ~A 0 B(I,I)

K(T)(A®B,AOC)

B(A@B,T(AOC) )

and

K(T)(A,C)

-

B(A,TC)

~

$ B

B(A®B,TCOB)

-®B where

I and

co~nutativity

p are

the

o f the

, ,

K(T)(A®B,COB)

~ B(ASB,T(COB)),

B(l,p) left

and right

triangles:

actions

o f B on T d e f i n e d

by

31

A @ TC n @ i

TA @ TC

$ ® 7

TA @ C

T(AeC). It

is easily

verified =

that z.TX.p

t h e i , n t e r c h a n g e law =

u-To-1

holds and this is equivalent to @ being a V-bifunctor. If T is a symmetric m o n o i d a l m o n a d on a symmetric m o n o i d a l category B then the interchange law corresponds to the " c o m m u t a t i v i t y law" of Kock [15].

The r e s u l t i n g m o n o i d a l

structure on K(I)

is then

symmetric. Lemma 4.1.

If a T - a l g e b r a is e x p o n e n t i a b l e

exponents are e v a l u a t i o n w i s e Proof.

in B then the

T-algebras.

If (C,~) is a T-algebra and B is an object of B then

the exponents C/B and B\C have algebra structures defined as the exponential transforms of the morphisms: B @ T(B\C)

1

>,, T(B @ ( B \ C ) ) \ T

e TC

~ ~

C

/Te T(C/B)

@ B

- T((C/B)

@ B)

P where e denotes the r e s p e c t i v e e v a l u a t i o n transformations. axioms for these structures follow d i r e c t l y Thus,

from those for ~.

if all T-algebras are e x p o n e n t i a b l e in S then we obtain

adjoint actions of B °p on P r o p o s i t i o n 4.2.

B(T).

The c o m p l e t i o n B(T)* is m o n o i d a l biclosed if

each T-algebra is e x p o n e n t i a b l e in Proof.

T h e algebra

Let

~.

(C,~) be a T-algebra.

Then the adjunctions:

32

B(-~B,C)

~ S[-,C/B)

B(B~-,C)

~ B(-,B\C)

provide adjunctions: B(T)(F(-@B),C) ~ B(T)(F-,C/B) B(T)(F(B@-),C) ~ B(T)(F-,B\C) where F: B ~ B(T) is the free-algebra

functor.

,

If these isomorphisms

remain natural when F is extended to the dense comparison N: K(Y) ---~B(Y) ion 3.1.

(see §8) then B(Y)* is monoidal

By Proposition

the adjunctions)

biclosed by Proposit-

8,2, it suffices to show (working out one of

that the composites:

N(T(C/B) @ B)

are equal.

functor

N(~I):

N((C/B)

~ B)

On composing both composites

Te ~

TC

, ~ ~ C

with ~ and filling in the def-

initions of @ and ~C/B' we obtain a commuting diagram pair: T(C/B)®B n@l

~, T(T(C/B)®A)

I

T 2 (C/B) ®B

N(~@I)

T(C/B)@B

,~,

¢((C/B)~B)

iTe

I

{C/B

C

TC

where ~C/B is the adjoint-transform the adjoint-transform T(C/B)

N(T~@I)

n~

of ~C/B"

,

Then the lefthand

side is

of T2(C/B)

~ E

T(C/B)

~

~

C/B

T~

and these composites This completes

are equal because ~C/B is an algebra

structure.

the proof.

Under the hypothesis

of Proposition

the trace of B(T)* exists on B(T). there are natural transformations

First,

4.2 we can determine when for each T-algebra

(C,~),

33

Te.~:

C/B

Te.p:

B\C

-~ TC/TB m TB\TC,

corresponding to the transformations: (C/B)

~ TB

TB ® (B\C) Proposition 4.3.

k ~ T((C/B)

@ B)

Te

TC

P ~ T(B @ (B\C)) ~ T C .

The category B ( T )

is biclosed in B(T)* if and

only if the equalisers: - ~ B\C

B~C

[\C

~- TB\C

TB\ TC and

CtB

-, C / B

C/[

~ C/TB

TC/TB exist in B for all T-algebras Proof.

B(T).

Because K(T) is monoidal,

The equalisers

B(T)* C [K(T)°P,w]

the unit object NI lies in

are precisely the ends (3.2) by Proposition 8.2.

Proposition 4.4.

algebras

(B,~) and (C,~).

If the reflective

embedding

is monoidal then the tensor product in B(I)* of two

(C,~) and (D,~) lies in B(T) if and only if the coequaliser of

the reflective pair: T(TC~TD)

T2(C~D)

T(C~D)

exists in B(T). Proof.

The pair has a common right inverse T ( n ~ ) ;

that is, it

is reflective.

Moreover,

the coequaliser in B(T)

is then the joint

c o e q u a l i s e r of the pairs: T(C@TD)

Tk ~ T2(C®D) ~ TO

T(C®D)

T(TC@D)

.

By P r o p o s i t i o n 8.2, this coequaliser is the iterated coend

(3.3);

C ® D = [AB(B(T)(NA,C) @ B(T)(NB,D))-N(A@B) in B(T). this

B e c a u s e E:

completes

the

B(T) ~ B ( T ) * p r e s e r v e s

and r e f l e c t s

colimits,

proof.

The p r e c e d i n g

propositions

provide

an a l t e r n a t i v e

earlier work by Kock [16] and Linton [18].

approach

to

The use of c o m p l e t i o n leads

to a significant r e d u c t i o n in coherence computations. Another result of Kock [17], T h e o r e m 2.6, may be e s t a b l i s h e d using the c o m p l e t i o n method.

Namely,

if the base c a t e g o r y V is cartes-

ian closed and B is cartesian closed over V then the category B(Y) is c a r t e s i a n closed if the functor T: B ~ B

preserves finite products.

This follows i m m e d i a t e l y from the fact that K(T) has the c a r t e s i a n monoidal structure,

hence the c o m p l e t i o n B(Y)* c [K(T)°P,w]

is c a r t e s i a n

closed. As m e n t i o n e d earlier,

the Kleisli c a t e g o r y K(T) of a symmetric

m o n o i d a l monad T has a symmetric monoidal ted structure on B(Y)* is symmetric.

structure, hence the comple-

For a nonsymmetric m o n o i d a l monad

on a symmetric m o n o i d a l closed category B the tensor product might exist on B(T) but have no symmetry. Exampl e .

Let V = S and let ~: ~ ~ ~

of the discrete category ~ egory

of finite integers into the simplical cat-

4, both with the o r d i n a l - s u m m o n o i d a l

is given the c o n v o l u t i o n m o n o i d a l ed on [~°P,s]

be the m o n o i d a l inclusion

by the (monadic)

structure.

If

[N,S]

structure then the structure g e n e r a t -

adjoint pair

35

¢

i [¢°P,l] : [~ °P,s]

is isomorphic to the c o n v o l u t i o n m o n o i d a l structure extends the n o n - s y m m e t r i c

§5

,-[ ~,S] structure on [b°P,s] .

o r d i n a l - s u m structure on

This

~.

BICLOSED FUNCTOR CATEGORIES

When A and B are two V-categories the category [A,B]

of all

V-category if V is suitably compl-

V-functors from A to B exists as a ete; that is, if the end: [A,BI (F,G) = f A

B(FA,GA)

exists in V for all F,G C [A,B]. Lemma 5.i.

If the category B admits the V - t e n s o r - c o p o w e r s

A(A,-).B for all A E A and B E B then the Yoneda functor

[A,B];

N: A°P@B ~

N(AB)

= A(A,-)-B,

is dense. Proof.

[

[[A,B](N(AB),F), [A,B](N(AB),G)] A,B [A,B [B(B,FA),B(B,GA)I by d e f i n i t i o n of N,

f

B(FA,GA)

A by the r e p r e s e n t a t i o n t h e o r e m applied to B ~ = [A,BI(F,G), Remark. T h e o r e m 1.33)

as required.

This result was e s t a b l i s h e d by F. Ulmer

for the case where

V is the symmetric m o n o i d a l closed

category Ab of a b e l i a n groups and group homomorphisms, tensor product of abelian groups. eral Yoneda functor N: A°P®B

~

(|20],

with the usual

In fact, Ulmer considers a more gen[A,BI, N(AB)

= A(A,-).MB,

36

where M: B ~

B is an Ab-dense functor.

In order to investigate the existence of a m o n o i d a l b i c l o s e d structure on

[A,B],

first take W to be the c o m p l e t i o n of V with respect

to a c a r t e s i a n closed category S* of sets which contains A, B, V, and S, as "small" category objects.

Note that the a s s u m p t i o n that

A,B

exists as a V-category can be avoided by the use of such a c o m p l e t i o n W.

Let A = (A,P,J,...)

be a W - p r o m o n o i d a l

structure on A and let

B be a m o n o i d a l b i c l o s e d category relative to structure on B provides a promonoidal

V.

T h e n the m o n o i d a l

structure on B °p and c o n s e q u e n t l y

A ® B °p has the t e n s o r - p r o d u c t of p r o m o n o i d a l structures given by the expressions:

P((X,B),(Y,C),(-,-)) I(X,B) P r o p o s i t i o n 5.2.

~ PIXY-)

~ B(-,B~C)

~ JX ~ B(B,I).

The c o m p l e t i o n

[A,BI*

of [A,B]

with respect

to W is monoidal biclosed if the functor S [P(XYA),B(-,GA)] : B °p ~ W A is r e p r e s e n t a b l e Proof.

for all X,Y E A and G ~ [A,B] . The first functor of (3.1), with the object

(a,b) E A °p @ B m a r k i n g the variable position,

becomes:

f [ P( (a,b), (A,B), (X,C)) ,[ A,BI (N(XC),a)l XC

= #

[P(aAX) ~ B(C,b~B),/ B(A(X,Y).C,GY)] XC Y

by the d e f i n i t i o n s of N and A ~ B °p,

[ [P(aAX), X by t h e

representation

theorem

[ [P(aAX), X because

B is

biclosed,

B(b®B,GX)] applied

B(b,GX/B)]

Y @ A a n d C E B,

37

B(b,H(A,B,G)(a)) by the representability hypothesis,

[ B(A(a,X).b, X

H(A,B,G)(X))

by the representation theorem applied to X C A,

[A,B] (N(ab), H(A,B,G)) by definition of N,

=

as required.

A similar computation reduces the second end in (3.1) to

the required form. Henceforth we assume that the hypothesis of Proposition 5.2 is satisfied, with natural isomorphisms:

~ [P(XYA),B(-,GA/B)]

~ B(-,H(Y,B,G)(X))

[ [P(XYA),B(-,B\GA)] A

a B(-,K(X,B,G)(Y))

1

(5.1)

for chosen representations H and K. Proposition 5.3.

The category

[A,B] is biclosed in [A,B]* if

and only if the ends: [AH(A'FA'G)

}

f ~(A,~A,a) A

exist in

[A,B] for all F,G E [A,B]. Proof.

The internal-hom values (3.2) can be reduced as foll-

OWS:

G/F = [

[[A,B] (N(AC),F), [A,B] (N-,H(A,C,G))] AC [ [ A,B] ( A ( A , - ) . C , F ) ,

[ A, B] (N-,H(A,C,G))]

AC by the definition of N,

f

[B(C,FA), [A,B] (N-,H(A,C,G))] AC by the r e p r e s e n t a t i o n theorem a p p l i e d to A E A, [

[A,B] (N-,H(A,FA,G)) A

38 by the r e p r e s e n t a t i o n

theorem applied to C c B,

[A,B] (N-, ~AH(A,FA,G)) ~A H(A,FA,G) Similarly,

if and only if the end

exists in [A,BI .

we obtain:

F~G ~ [A,B] (N-, and this completes

~ K(A,FA,G)), A

the proof.

Thus the resulting

internal-hom

Q/~ = ~H(A,FA,a)

functors on [A,B]

}

have values:

(5.2)

F\O = ~ K(A,FA,G) A and, by the definitions natural

of H and K in Proposition

5.2, there exist

isomorphisms:

G?N(AB) ~ H(A,B,G) N(AB)\G ~ K ( A , B , G ) . The internal-hom

[A,B]*

functors have an identity object

if the identity of

lies in [A,B] ; this identity has the value: AB

I = f

by

I(AB)-EN(AB)

(3.4),

= ~AB(jA @ B(B,I)).(A(A,-).B) --- J.l in [A,B]*,

by the r e p r e s e n t a t i o n

theorem applied

to A E A and B E B. The tensor product

in [A,BI*

of two functors F,G e [A,B]

the value: F @ G = fXBYC(B(B,FX)

@ B(C,GY)).Q(X,B,Y,C)

by

fxY Q(X,FX,Y,aY) by the r e p r e s e n t a t i o n

theorem applied to B,C E B, where AB

Q(X,FX,Y,GY)

= f

(P(XYA) ~ B(B,FX@GY))-EN(AB)

by d e f i n i t i o n of the promonoidal

category A @ B °p,

AB = ~

(P(XYA)

@ B(B,FX@GY)).E(A(A,-).B)

(3.3),

has

39 by d e f i n i t i o n of N,

[AB(p(XYA) @ B ( B , F X ® G Y ) ) - ( A ( A , - ) . E B ) because E preserves tensoring,

P(XY-).(FX@GY) by t h e r e p r e s e n t a t i o n

theorem applied

t o A E A and B E B.

This establishes the following: P r o p o s i t i o n 5.4. r e s t r i c t i o n to [A,B]

The m o n o i d a l

structure on [A,B]* admits a

if the coend

F ~ G = [XY P(XY-).(FX@GY) exists in [A,B]

(5.3)

for all F,G E [A,B], and the identity object J.l of

[A,B]* lies in [A,B] . The coherence of the m o n o i d a l and biclosed structures on [A,B]

induced

is a consequence of the coherence of the m o n o i d a l biclosed

structure on [A,B]*.

In the case where B = V, the formulas

(5.3) reproduce the original c o n v o l u t i o n structure

(1.3) and

(5.2) and (1.2) on

[A,V]. Remark. A

It is s t r a i g h t f o r w a r d to verify,

using

(5.3), that if

and A' are two small promonoidal categories and if B is a sufficien-

tly complete and cocomplete m o n o i d a l b i c l o s e d category then the canonical i s o m o r p h i s m of categories:

[A @ A',B]

~ [A,[A',B]]

admits enrichment to an isomorphism of m o n o i d a l b i c l o s e d categories, where A @ A' has the t e n s o r - p r o d u c t p r o m o n o i d a l

structure and each

funetor category has the "convolution" m o n o i d a l b i c l o s e d structure defined by (5.2) a n d (5.3).

4O

§6

BICLOSED

CATEGORIES

If A is a p r o m o n o i d a l egory

for w h i c h

functor

category

of [A,BI ctors.

the

subcategories

is assumed

to be

C.

For each

that

full

in [A,V] .

on the

subcategory

internal-hom categories

of

for those

full

functors

Again,

fun-

from

the base

cat-

category

C let C m denote

ortho~onal

the class

of all m o r p h i s m s

the full

subcategory

of C oonsis-

to Z (following

terminology

of

[12]);

is,

Let

~ denote

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

the ortho~onal

= {s E cm;c(s,C)

in V for all s E Z}.

c l o s u r e of Z in cm;

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

that

is,

in V for all C E CZ}.

C Z = C U = C~ for all Z C U c ~. For a given m o n o i d a l

Z c C m is called m o n o i d a l s @ C E Z; that both

by these

exist

cat-

V.

C Z = (C E C;C(s,C)

Then

a given

this q u e s t i o n

Z of m o r p h i s m s

biclosed

(5.2)

of the Z - c o n t i n u o u s

Z C C m let C Z denote

ting of the objects

by

[2] on closed

us to examine

which consist

class

For a given

defined

all e x p o n e n t i a t i o n

leads

of [A,B]

A to B for a given

in

functors

The work of B a s t i a n i - E h r e s m a n n structures

FUNCTORS

and B is a m o n o i d a l

then we can ask w h e t h e r

is closed under

sketched

egory

category

internal-hom

|A,B]

OF C O N T I N U O U S

is,

structure

if C E C and

on the c a t e g o r y s E Z imply

if Z is stable under

C, a class

that

the m o n o i d a l

C @ s E Z and

action

of C on

sides of Cm. Remark.

with respect respect

It was

established

in [B]

§i that

to C then the category

C(Z -I)

of V-fractions

to Z (as c o n s t r u c t e d

such that

each m o n o i d a l

in W o l f f

functor

has a unique m o n o i d a l - f u n c t o r

[21])

on C w h i c h

if Z is m o n o i d a l of C with

has a m o n o i d a l inverts

factorisation

through

structure

the elements

of Z

the p r o j e c t i o n

C ~ C(Z-I). Proposition a monoidal

biclosed

6.1.

If A is a strongly

category

generating

subcategory

C and Z C C m then the f o l l o w i n g

are

of

41

equivalent: (a)

A E A and

(b)

C Z is c l o s e d

(e)

Thus

are

C/s and

isomorphisms lie

in

C.

If B C C

C(s,B\C)

are

and

C(A@s,C)

~ C(A,C/s)

for all A ~ A and

for all

B ~ C and

s E Z and

~ C(B,C/s)

s E Z, w h i c h

and

s E Z by

(a).

C E C Z because

and

C(s,C/B)

implies

that

A

m C(~s\C)

B\C

and

C E C Z.

s E ~ then

isomorphisms

(a) b e c a u s e

T h e n we can

[A,B] Z C [A,B]

[A,V~z

"Z-continuous" all

Hence

C(B@s,C)

for all

~ C(s,B\C)

C E C Z by

(b).

and

Thus

B @ s

full

sub-

such

that

A C C.

C be a f u n c t o r

Z C IA,V] m.

E

s @ A E Z.

in Z~ as r e q u i r e d .

(c) ~

B(B,F-)

isomorphisms

for all

(c).

~ @ B are

category

If C E C Z t h e n

isomorphisms

C.

~ C(s,C/B)

Let

exponentiation

in C Z for all B ~ C and

(b) ~ C(s@B,C)

(b).

are

s\C are

generates

C/B b o t h

and

(a) ~

~ C(A,S\C)

strongly

under

A ® s E Z and

is m o n o i d a l .

Proof. C(s@A,C)

s E Z imply

category

form

(as is done

consisting

for all B E B.

model

of the

of A in B.

of the

[A,V]

form

in [13],

§8.1)

functors

F E

Such a functor Clearly

and

let the

[A,B]

F is i n t e r p r e t e d

as a

[A,B] Z = [ A , B | U = L A , B ] z

for

Z c U c ~. We now

egory

over

which

the

category

suppose

V and

that

internal-hem

convolution

functor

B is a m o n o i d a l functors

is a small

biclosed

defined

by

promonoidal

category

(5.2)

exist

over

cat-

V for

on the

functor

6.2.

If Z is a m o n o i d a l

category

|A,V]

then

class

of m o r p h i s m s

IA,B] Z is c l o s e d

under

in the exponen-

in [A,B] . Proof.

have:

A = (A,P,J,...)

[A,B] . Proposition

tiation

that

For

all

s C Z, B E B, F E [A,B]

and

G E [ A , B ] z , we

42 J" [ s x , B ( B , ( G / F ) ( X ) ) ] X / [Sx,B(B , f H ( Y , F Y , G ) ( X ) ) ] X Y

by (5.2),

-~ ~ / [ S x , B ( B , H ( Y , F Y , G ) ( X ) ) ] Y X by interchanging

limits,

/ / [Sx, YX /

(/ Y

by interchanging

/ [P(XYA),B(B,GA/FY)]] A

by ( 5 . 1 ) ,

[ / X s X ® P(XYA),B(B®FY,GA)]) A

limits and using the tensor-hom

adjunctions

of V and

B, f (f [(s ® A(Y,-))(A),B(B®FY,GA)]) Y A

(*)

because

(s @ A(Y,-))(A)

= /

XX'

s X @ A(Y,X') @ P(XX~) by ( 1 . 2 ) ,

X

/ by t h e

representation

theorem

end o v e r Y E A o f i s o m o r p h i s m s dal

and G E [A,B]z

i n [A,B] Z f o r

all

Corollary

applied because

by hypothesis. F ~ [A,B] 6.3.

s X ® P(XYA)

The m o r p h i s m ( * )

s @ A(Y,-)

E Z since dually,

is monoidal

If Z is a class of morphisms

6.1.

in the convolution

Thus [A,B]z,

is closed under exponentiation Thus,

if the Z-continuous

A form an exponentially

biclosed

then so do the Z-continuous oidal biclosed

in

category

an

Z is monoi-

F\G b o t h l i e

in [A,B]

models

for which

functor cate-

in [A,V]

then

which coincides with by Proposition

in V of a promonoidal

subcategory

models

[A,V]

in [A,B] .

If [A,V] Z is closed under exponentiation by Proposition

is

and G E [A,B]z.

then [A,B] Z is closed under exponentiation

Proof.

[A,B]~,

~ E A.

Thus G/F a n d ,

[A,V] Z is closed under exponentiation gory [A,V]

to

of the convolution

6.2. category [A,V]

of A in any suitably complete mon-

B which is based on V.

43

This result in [2], Chapter III.

contains an a l t e r n a t i w For example

be the simplicial category. gories and functors

app:~oacn to c o n s t r u c t i o n s

(c~~. [2] ~12), let V = S and let

Then the category Cat of small ca~e.-

is fully embedded in [~°6~S]

r e f l e c t i v e subcategory.

as a c a r t e s i a n cl6sed

This implies that C~t is defined by a (carte-

sian) m o n o i d a l class of m o r p h i s m s

in [A°P,s]]

m o r p h i s m s inverted by the reflection.

namely,

~he class of all

It ne~ ~ follows from C o r o l l a r y

6.3 that the category [~°P,B] Z of all category objects in B is closed under e x p o n e n t i a t i o n in [~°P,B]

w h e n e v e r [~°P,B]

-hom functors

if the m o n o i d a l b i c l o s e d

(5.2).

Moreover,

c o n s i d e r e d in §5 exists on [~°P,B]

admits the i n t e r n a l structure

and the e m b e d d i n g

[~°P,BIz C [~°P,B 1 has a left adjoint then this a d j u n c t i o n admits m o n o i d a l e n r i c ~ e n t the r e f l e c t i o n theorem of §i.

by

Note that, by P r o p o s i t i o n 6.1, Z may be

r e p l a c e d by an~ class of m o r p h i s m s in [A°P,s]

w h i c h defines Cat as its

class of orthogonal objects. Remark.

The general q u e s t i o n of the e x i s t e n c e of a left-adj-

oint functor to an i n c l u s i o n of the form |A,B] Z C [A,B] studied in some de

L2.

If

L3.

If @~ C(x)

Example

~

This class is stable

TONz

The category

spaces and continuous maps.

[3]) that a limit space is a set

xcX~

C(x)

bEB

a set

C(x) of "convergent"

satisfy the axioms

X

filters

:

C(x).

@>.~ and ¢c C(x)

3.2.

and ~

Let Z

then @E C(x)

C(x)

be the

then 0aCe C(x)

(larger)

generated by the i d e n t i f i c a t i o n maps

class of m o r p h i s m s

f:A

÷

B

with the following

property: for each point

bEB

and open covering

exists a finite set is a n e i g h b o u r h o o d category

TONz

{Xl,...,~n}CA of

beB

<x>

c

of

f-lb

there

fGxlU .-ufGxn The r e s u l t i n g

and ¢ c C(x)

then 0 E C(x)

if there exists a set of filters

{ @X ; ~ A {V X ; XcA {V~.I ;

:

C(x)

@ E C(x)

The class

XeA}

is equivalent to the category of those limit spaces

L2. If @~¢ L3'

such that

(see [5] T h e o r e m i).

which satisfy the following axioms LI.

{G X ;

and @~c and

} such that each set

V X E @~} contains a finite subset

i = l,'',,n} Z of m o r p h i s m s

interior of the r e f l e c t i o n from smaller than the m o n o i d a l

C(x)

such that

UV~i ~ @.

obtained here is the stable To N

interior

to

Top

(see [5]

§3

and it is strictly for a counterexample).

72

Finally, we note that in order to completely describe the canonical Grothendleck topology on classes

{fa : Am ÷

given space property: ~

B

Top

it is necessary to consider all covering

B ; ~ ~}

in

Top

Such a class of morphisms into a

.

covers in the canonical topology if and only if it has the

for each point

b~B

and open covering

there exists a finite subset

is a neighbourhood of

{Gl ~ ~ A

{ll''"'In } c A

and

such that

AeS}

of

edife(Gll)

b~B . However, it is easy to verify that the introduction

of these covers imposes no new restriction on the limit spaces under consideration. In other words it suffices to take ~

small so that

~

is generated by the

identiflcationmaps already described above.

4.

Stability conditions and relations. In this section we note several "known" facts about general closed span

categories.

Proposition 4.1. and only if

Proof.

SpanC

C

is a closed span category if

is closed as a bicategory.

For each morphlsm

pullback-along-f SpanC

A finitely complete categor~j

and let

fcC(BC)

f, : C/B

is closed if and only if

f*

let

-~ C/C

f* : C/C

C/B denote

denote the left adjoint of

has a right adjoint

result then follows frown the fact that if

+

[f,-]~-

-a f

~f

f~ ~C/B

'

C / c

f . The

exists then it has a

unique llmlt-preserving extension along the comonadlc functor

C/C ~

for all

f* .

f, :

73

Suppose

B

is a

cs-category containing a strongly generating class

such that each

B~B

has a presentation as a coequaliser:

2A

B .

A class

x

all

~

EA

s~Z

~

y and

f : A

Proposition 4.2.

÷

B

Z

of morphisms in

with

=

A ~ A . Let

z(A) if

z

is A-stable if f*s~Z

~

(b)

If Z = Z

(c)

If

Z

consists of epimorphisms then

if

Z

is stable.

Z

for

AeA}

Z(A) = {s: B ÷ A: s~Z and

(a)

then

is

B

A

A-stable.

is stable if

Z

is

A-stable. Z

is stable

The verifications are straightforward. If then

Z

Z

is a stable class of morphlsms in a finitely complete category

is closed with respect to span composition.

{B/AxB ; A,BEB}

categories of fractions of

Thus the class of

with respect to

forms a new

bicategory

(cf. [7] )

suppose

is a closed span category with a proper factorisation system

where

B E

is stable.

M-relations in

B .

with the evident universal property.

Z

Then, on takirg Z = E, Because

SpanB

is closed under exponentiation in

In particular,

E-M

one obtains the bicategory of

is closed the subbicategory of relations SpanB

(by the "several objects" form or

the reflection theorem for closed categories [6] )

and the exponentiation

provides a form of universal quantification (cf. [ll] ).

Bicategories (not necessarily closed) of relations and spans have also been considered in

[12]

by J. Meisen,

74 References

[ I]

Antoine, P., Extension minimale de la cat@gorie des espaces topologiques, C.R. Acad. Sc. Paris, t.262 (1966), 1389-1392.

[ 2]

B@nabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar I, Lecture Notes 47 (Springer 1967), 1-77.

[ 3]

Binz, E. and Keller, H.H., Funktionenr~u~neinder Kategorie der Limesrgume, Annales Acad. Sc. Fen., A.I.383 (1966), 4-21.

[ 4]

Day, B.J., Relationship of Spanier's quasl-topological spaces to k-spaces, M.Sc. Thesis, Univ. Sydney, 1968.

[ 5]

Day, B.J. and Kelly, G.M., On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Phil. Soc. 67 (1970), 553-558.

[ 6]

Day, B.J., A reflection theorem for closed categories, J. Pure and Appl. Alg., Vol.2, No.l (1972), i-Ii.

[ 7]

Day, B.J., Note on monoidal localisation, Bull. Austral. Math. Soc., Vol. 8 (1973), 1-16.

[ 8]

Day, B.J., On adjoint-functor factorisation, Proc. Sydney Cat. Conf., to appear.

[ 9]

Fakir, S., Monade idempotente associ@e ~ une monade, C.R. Acad. Sc. Paris, t.270 (1970), 99-101.

[I0] Freyd, P. and Kelly, G.M., Categories of continuous functors I, J. Pure and Appl. Alg., Vol.2, No.3 (1972), 169-191. [ii] Lawvere, F.W., ed., Toposes, algebraic geometry and logic, Lecture Notes 274 (Springer 1972), Introduction and references. [12] Meisen, J., On bicategories of relations and pullback spans, preprint, University of B.C., (1973).

REVIEW

OF THE E L E M E N T S .............

OF 2 - C A T E G O R I E S

H.'.......

by G.M. The purpose duction

ion,

notions

to avoid

original,

ary facts below

papers

needed

by each of us,

chiefly

to i n t r o d u c e

pasting

that we use

our n o t a t i o n

to give a treatment,

which

seen,

adjunctions

f--J u and f'--Ju'

ality.

and then m e n t i o n the

2-category

i.I

DOUBLE

Both notions first

Cat, Set,

respectively

egory

the notion

categories,

is a c a t e g o r y

elementary

of these

CATEGORIES

of d0uble

the most

of m a k i n g

in CAT;

element-

our papers

the o p e r a t i o n

of

operation

complete

than

arising

from

= (f'b,af)

and of its natur-

of monads

that

but

become

it is really

in a 2-category, available

in

a 3-category).

AND 2 - C A T E G O R I E S due to Ehresmann; category.

the categories

object

is

in the literature

and more

2-category,

(because

and of small

see [6j and [ 71.

We denote

of all categories,

sets. but

Conceptually, it admits

the

by CAT, of all

a double

SET,

sets, cat-

following

description.

It has objects arrows

(bu,u'a)

propemties

are o r i g i n a l l y

We recall

of small

in any

some e n r i c h m e n t s

§i

here

care to read them,

to us simpler

the basic

of 2 - c a t e g o r i e s

as may

and e s p e c i a l l y

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

In §3 we recall

account

together

some notat-

In §2 we use the p a s t i n g

seems

any we have

of substance

in the hope

for such beginners

intro-

by c o l l e c t i n g

In §I we rehearse

partly

constantly.

as a common

and e s t a b l i s h i n g

Nothing

our needs.

2-categories,

self-contained

is to serve

find no c o n n e c t e d

satisfied

about

Str~t

in this volume,

later duplication.

but we could

that e x a c t l y

{~d Ross

of this r e v i e w

to the authors'

some basic

Kellv

A etc.; h o r i z o n t a l

x etc.; and squares

ain functions

sufficiently

~ etc.;

there

indicated

arrows

a etc.; v e r t i c a l

are various

by the diagrams

domain

and codom-

76

a A

~B

A

A

~B

a x

Y

C

C

b

mD

The objects and the horizontal arrows form a category, with identities hA: A ~ A ;

the objects and the vertical arrows form a category, with A $ VA

identities

The squares have horizontal and vertical laws of

A composition, represented by A

x:

a ~B

c --7--

c dE

D

~--F

lz

x

A

a

C

,

~ B

~D

u G

e

~H

;

under each of these laws they form a category, with respective identities A x[

hA

~A

ix

C

A Ix C

,

a

VA I

la

A

a

:-B IVB

In the situation

the result of composing first horizontally and then vertically is to be the same as the result of composing first vertically and then horizontally. The composite

77

A

a ~B

A

is to be ica, identities.

and

Finally

~B

a

similarly

c ~E

r E

c

for v e r t i c a l

the h o r i z o n t a l

composition

and v e r t i c a l

hA ~A

1

identities

hA

A

A

of h o r i z o n t a l

A

--~

~ A

i

,

A ------~ A

hA

,

hA

are to coincide. Examples and in §2.2

1.2.

of double

categories

may be found

in P a l m q u i s t

|21],

below.

A 2-category

K may be thought

w h i c h all the v e r t i c a l

arrows

are

of as a double

identities.

category

A more

direct

in

descrip-

tion is as follows. K has qbjects 1-cells

f: A ~ B etc.,

or 0-cells and

A etc.,

arrows

or m o r p h i s m s

or

2-cells f

A

~ v

_ e ~

B

or

A

$ ~

g (Kelly

tends

to

it is purely

use

g

double

a matter

The objects underlying

category

shows what

is meant,

For them

arrows

for

2-cells,

and the arrows of K, with

Street

form a category

identities

we sometimes

write

fixed A and B, the arrows

composition:

under the

/ A

composite

and

single

ones

-

of taste.)

form a category K(A,B)

The v e r ~ c a l

B

f ~

~

~

~ ~

h above

_

K0, called

IA: A ~ A; when the

the context

K for K 0. A ~ B and the

operation

known

2-cells

between

as v e r t i c a l

~ B.

is d e n o t e d

by ~.a or B.~,

or rarely

78 by 8~ when no c o n f u s i o n introduced

below;

is likely

its identities

with the h o r i z o n t a l are denoted

composite

to be

by

f

f There whereby

is

also

a law of

horizontal

composition

U

B

B

g

V

a 2-cell f

u

uf =

g this

2-cells,

from 2-cells f

we g e t

of

v

composite

2-cells

C;

y*~

vg

is also denoted

by ¥~:

are to form a category, with iA A ~

We r e q u i r e

finally

A

uf ~ vg.

Under

this

law the

identities

.

1A that,

in

the

~ •

situation

...... ~ C ,

h the

composites

(6,8).(~*a)

and

(6.y)*(~.~)

coincide;

and that

in the

situation

we have

lu*l f

=

We also object arrow

f

U

f

u

luf. freely

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

whereby

A or of an a r r o w f is also used as the name i A or its identity

composite

2-cell

if.

In p a r t i c u l a r

the name

of an

of its identity the h o r i z o n t a l

79

A

B

C

f is a l s o

written

D

v

g

as U

A

~

B

~

C

=D

~

f

g V

and denoted

by gyf.

The more

general

basic

above

operations

operation

situations

on 2 - c e l l s

of p a s t i n g ,

can be c o m b i n e d

introduced

by B e n a b o u

to give [i] .

the

The

two

are

V

The

first

and

the

give

of t h e s e

second

meaning

identity

has

2-cell One

to g i v e

to i n d i c a t e

to s u c h

If in a d i a g r a m

one r e g i o n

is m e a n t

the

2-cell

composites

the

vy.6f:

2-cell

Bg.u~:

uf ~ uhg ~vg,

uf ~ v k f ~ vg.

Thus

we

as

s u c h as

no

2-cell

is m e a n t ,

to

f

g

h ~

k

marked which

can g e n e r a l i z e

meaning

to i n d i c a t e

in it, implies

the p a s t i n g

such multiple

it is to be u n d e r s t o o d that

the

vf = hu.

operation

composites

that

as

further

still,

so as

B0

This

is meant

of the

to indicate

a vertical

composite

of h o r i z o n t a l

composites

form

r

there

is usually

a choice

taken,

but the result

simple

cases,

1.3.

is independent

and can be proved

after an a p p r o p r i a t e itions

of the order

of this

in terms

the

choice;

i n d u c t i ve l y

formalization

composites this

is clear

in the general of p o l y g o n a l

are in

case

decompos-

of the disk.

As for examples

of 2-categories,

just as the p a r a d i g m a t i c

category

the

and the

arrows

are

The context merely

in which

functors,

will

the p a r a d i g m a t i c

is SET. 2-cells

The objects are natural

show when CAT is c o n s i d e r e d

the u n d e r l y i n g

category

is meant.

one

are categories,

transformations.

as a 2-category

There

is CAT,

is the

and when

sub-2-category

Cat of small categories. For a m o n o i d a l V-categories, of E81;

V-functors,

a g a i n with

categories

the

themselves,

transformations The a 2-category

category and

V-natural

sub-2-category with m o n o i d a l

in the sense

category

V, we have

of L8J,

K of ordered

when we observe

that

the

2-category

transformations,

in the

V-Cat of small ones. functors

and m o n o i d a l

form a 2 - c a t e g o r y

objects

K(A,B)

sense

Monoidal natural

Mon CAT.

in any c a t e g o r y

the hom-set

V-CAT of

A becomes

has a natural

81

order,

and can therefore For a c a t e g o r y

be r e g a r d e d A, the comma

which

is a category

B together

which

from

(C,C)

(B,B)

a 2-category natural

transformations will

1.4.

Besides

by 2 - f u n c t o r

oint;

see [141

do not given

use

and by

2-cells

B ~ A, and an arrow of

B ~ C with

CT = B, becomes

C~ : id.

Many

of 2 - c a t e g o r y one:

K is Just

other

treatment

of

§1.2 above, CAT is

CAT-category, determines

transformation:

Similarly

in

the category

a

This d e f i n i t i o n

or "2-adjoint"

assigns

terms,

of L, arrows

what

namely

V-adjunction.

its we shall

CAY-functor

for 2-adjoint,

in the more

domains

and composition. to each object

is not only natural nB.Df ~:

a 2-functor

of K to arrows

of L, p r e s e r v i n g

identity

(1.1)

of

: CAY-adjNote

general

that we

senses

to t h e m by Gray in [lO].

to objects

2-cell

definition

transformation.

"2-natural"

B:

an object

below.

2-natural

for a general

CAT/A,

a: T ~ S: B ~ C to be the

but n o n - e l e m e n t a r y

K(A,B).

In e l e m e n t a r y

have

T:

~: T ~ S for w h i c h

and a 2 - c a t e g o r y

hom b e i n g

and C A T - n a t u r a l

2-cells

the e l e m e n t a r y

closed,

CAT-valued

is a functor

in the papers

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

cartesian

mean

arise

category

with a functor

w h e n we take the

examples

there

to

as a category.

A 2-natural

A of K an a r r o w

but also

2-natural

EB

category;

2-cells

of K to

and all types

transformation ~A or HA: that,

in the

of

n: D ~ E:

K ~ L

DA ~ EA in L, which for

f: A ~ B, we

sense

that,

for each

= DA

Eg

As in general D to the

V, so here where

of L, and

of K

Ef

Dg

V-functor

objects

sense

f ~ g in K, we have Df

DA

L sends

and codomains

in the o r d i n a r y

= El.hA,

D: K ~

the

V-natural

V-functor

transformations

E form not

V = CAT the 2-natural

in other words

from the

only a set but an object transformations

2-CAT is really a 3-category,

of

D ~ E form a i.e.

a 2-CAT-

82

category.

We

follow

transformations

Benabou

[i]

modifications.

in calling

morphisms

of 2-natural

A modification

p: q -* ~ :

K "* L ,

D ~ E:

also written

D

E assigns

to

each object

A of K a 2-cell

pA: nA ~ ~A s u c h t h a t ,

for

f: A ~ B, we h a v e nA

nB

(l.2)

DA ~

DB

~pB

EB = DA

~pA

~B Zn p a r t i c u l a r , gory End

K of

its

2-cells

this

2-category

The what

reverse dual,

Observe

Kelly feels

1-cells

2-category

of K; its

objects

like

by the

that

K c°°p

i s m of c a t e g o r i e s Specializing

transformations

P: n ~ ~.

When working

the r i g h t

definition we h a v e

not

the

to w r i t e

q:

q:

D ~ E; a n d

totally

within

D ~ E and

of a 2-category

K°P(A,B)

2-cells.

duality

thus we reverse

are

2-functors

D:

CAT-adjoint;

K(EB,A)

the

K also deter-

= K(B,A),

We w r i t e

on CAT,

to the

is j u s t

K ~ that

~ L(B,DA)

of [14] same

on

thing

i ~ DE a n d ¢: E D ~ i s a t i s f y i n g

2-adjunction

endo-2-functors

it.

ordinary

the r e s u l t

it c o m e s

are

a 2-cate-

so t h a t

we

K c° f o r the o t h e r

so that

2-cells

but

not

the

1-cells

= K °pc°.

say t h a t they

K, we g e t

2-natural

reserves

but

= K ( A , B ) °p"

to say t h a t

q:

a fixed

are

K °p s h a l l m e a n :

the

To

see t h a t

arrows

non-elementary

induced

KC°(A,B)

taking

are m o d i f i c a t i o n s

p: q ~ ~ if he

mines

~A

endomorphisms

D: K ~ K of K; its

EA - - - - - ~ - ~ S S .

/

adjunction

L a n d E: is,

which

that

there

is 2 - n a t u r a l

V-adjunction to h a v e the u s u a l

i__n_nthe

L ~ K are

to the

2-natural

is

is a n i s o m o r p h in A a n d B. c a s e V = CAT,

we

transformations

conditions;

2-category

2-adjoint

2-CAT,

so t h a t in t h e

sense

83

of §2.1 below. i. 5 .

A great many

notions

2-CAT g e n e r a l i z a t i o n s

within

in the basic a given

isomorphism phisms

notions"

do indeed

indicate

before

this

a systematic

theorem"

shows

replaced

one)

in the

nomenclature

the

[22],

and are

[i].

Indeed

egory,

these

We should

notion

morphism:

choices

has

criterion

2-cells

case

to

in others by the

illustrates notion,

a general

the d e f i n i t i o n

as the norm, replaced

by an

principle

we call the

N".

and

D: K ~ L b e t w e e n IDA ~ DI A instead [24]

morphisms

of b i c a t e ~ o r i e s

point:

to

but

by Street

sense

we

n o t i o n with

of this morphism,

of lax functor

of B e n a b o u ' s

like to

use.

relaxed

Dg.Df ~ D(gf)

here;

(doubtless

however

N is taken

an "op-lax

in B e n a b o u ' s

book

of usefulness,

have b e e n c o n s i d e r e d

a bicategory

"morphism-like"

seem in fact

we have no general

of the sense

sense r e v e r s e d

a special

and this

"relaxed

consider;

notion with e q u a l i t y

is the concept

which

of equalities;

they

isomor-

some

systematically

that we shall

"strict"

by a mere

An example 2-categories,

these

forthcoming

of Gray [12].

c h o s e n a sense by some

same n o t i o n with

this

them to be r e p l a c e d

to go into this

in detail

for the r e l a x e d

offer for our various having

only to

the given

that

willy-nilly

a l l o w in g

was to

to satisfy

in some cases

isomorphism, and "lax N" for the still more equal i t y

a diagram

sense);

be r e q u i r e d

that we must

the original

say "pseudo-N"

(in a given

experience

our i n t e n t i o n

are treated

Where

For where

notion.

It is not such things

V;

and replace

in nature;

a "coherence

for an a r b i t r a r y

or even go further

Moreover

thing,

in

it to commute

then usually

occur

normal

"strict"

to appear

will

axioms".

we can prove

no sense

for any V admit

we can now allow

by a given m o r p h i s m

be the more

simpler

notion,

isomorphism,

or m o r p h i s m s

"coherence

that w o u l d make

2-CAT is a 3-category.

and this b e c a u s e commute

in V-CAT

definable

is itself when

continues

and Roberts

a pseudo-2-cat-

one relaxes to make

sense

some when

84

the domain

and the c o d o m a i n

such do not occur the p a p e r ~25]

paper

of Street

[i0]

of 2-natural

of Kelly and

below,

[15],

pseudo-monads

there

are the lax algebras

their

lax morphisms...

with

strict

and

and i d e n t i t y

where

take the lax n o t i o n "strong

strict

functor"

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

functor"

(with e q u a l i t y

2.1.

Other accounts

pasting (2.2)

arrows

by Gray

[ 2].

[3] ; the 127].

the

Then and

"strict

Even

lax m o r p h i s m s

special

the tensor

N; we then

in

Another

over these,

for them,

case

of

product

In these

cases we

call the pseudo-

N";

thus

"monoidal

"strong m o n o i d a l

CA @ CB ~ @(A@B)),

here;

such lax or pseudo papers

ADJUNCTION

and

"strict

monoidal

in [19],

as we need are

IN A 2-CATEGORY

below, [14],

of v a r y i n g

and [211.

of the a d j u n c t i o n

from R.F.C. ~,s:

notions

below.

of the m a t t e r

for the neat e x p r e s s i o n

An & d j u n c t i o n

in the

CA @ @B = ¢(A®B)).

can be found

b e l o w we learnt

below

Bunge

in nature.

notion

El31.

of our nomenclature.

( p re s e r v i n g

notion

in

transformation

of Bunge

CA @ @B ~ ¢(A®B)),

in the i n d i v i d u a l

completeness,

ones

N" and the strict

§2.

in

as

occur

"2-natural"

as is shown by the

are quite rare

(with a c o m p a r i s o n

functors

by Z o b e r l e i n

aspect

algebras

thing,

functor"

introduced

by Gray)

2-monad)

second

as our basic

We say no more

called

are c o n s i d e r e d

strict

on the nose)

do occur

or the p s e u d o - a l g e b r a s

are the normal

monoidal functors,

-notion

(relaxed

us to the

2-monads,

of al g e b r a s

of lax natural these

(now adopted

corresponding

Lax

but p s e u d o - f u n c t o r s

and are the things

of lax m o n a d

"lax".

and go back to G r o t h e n d i e c k

transformation;

is the notion

brings

book,

is the n o t i o n

"quasl-natural"

This

themselves

in the present

A second example in place

are

degrees

of

The utility

equations

of

(2.1)

and

Walters.

f---4u: A ~ B in a 2-category

u: A ~ B and f: B ~ A t o g e t h e r

with

2-cells

K consists

~: i ~ uf and

of

85

c: fu ~ i s a t i s f y i n g

i

A

(2.1)

the axioms

~A

equals

identity,

B

~, 1

I

equals identity

B We

i

say that

" ~"B

f is left adjoint

we call ~ the u n i t , a n d When

V-natural usual

s the counit,

K is V-CAT

shown by Kelly

in |14]

of a d j u n c t i o n

If ~1,si: a composite

for a symmetric

A(fb,a)

monoidal

fi-~ui:

adjunction

in p a r t i c u l a r

V, it has been with

we get the

K = CAT.

B ~ C is a second

2,c2:

closed

f--4u are in b i j e c t i o n

~ B(b,ua);

when

to f;

of the adjunction.

that a d j u n c t i o n s

isomorphisms

notion

to u, and that u is r isht adjoint

ffl---~ ulu:

adjunction,

we clearly

A ~ C if we define

g~

q2,e2

as the composites i

(2.3)

A

/B~r~l

1'

A

,L B

.....

~A

S

Ul\~ c / fl ~

C

~

.

i

Thus

adjunctions

in K form a category,

with

i,i:

i--~i:

A ~ A as

identities. 2.2. and E

If we look upon are m u t u a l l y

the f o l l o w i n g

(2.1)

inverse

proposition

and

(2.2)

under

becomes

the

as a s s e r t i n g indicated

evident:

that

pasting

the

2-cells

operations,

86

Proposition n',s':

f'--~u':

a bisection

(2.4)

2.1.

Let

A' ~ B'.

between

is t h e

D,s:

Let

2-cells

a: A ~ A'

and

b:

B ~ B'.

2-cells

a

compo,s,i,te

B

X is the

A ~ B and

I: bu ~ u ' a a n d

A

(2.5)

f--~u:

~

A T

~-B

...........

b

~B

i

composite

~:

f'b ~ af,

expressed objects

naturality

~A

as

are

follows. those

the v e r t i c a l tal

arrows

arrows

,

~A'

double ~,s:

is j u s t

category,

f--~u:

the

both

double

are

same

given

2-cells

in

sides

Proposition isomorphism That

between

is to say~

vertical

(2.4).

In the

horizontally

2.2.

The

above

and horizontal.

of horizon-

above.

(In o u r In the

first

b: B ~ B'

and vertical

is a 2 - c e l l category,

X in K

a square

in

(2.5).

composition

or v e r t i c a l l y , is n o w

the

of squares

expressed

between

I and

categories

we h a v e

just

composition

and

In

corresponding

bijection

respects

o f K, a n d

of v e r t i c a l

"oblique".)

double

in q u e s t i o n

the two d o u b l e

the bijection

as

be

In e a c h the

the a r r o w s

~ in K s u c h as a p p e a r s

horizontal

"naturality"

§2.1

A' ~ B '

second

~ may

composition

of

B'

Composition

a: A ~ A'

f'--~u':

X and

are

in K.

shown

sides

is a 2 - c e l l

by p a s t i n g , The

with

and ~',~':

categories,

of K.

arrows

of a d j u n c t i o n s

=

categories.

in K, w h i l e

is c o n v e n i e n t l y

a square

A ~ B,

s u c h as a p p e a r s with

composition

T

i

between

two d o u b l e

horizontal

B T

~

b

are the adjunctions

composition

"vertical"

-'

bijection

Consider

o f K, the

arrows

is the

diagrams

of this

where

AT

~

T

The

is

a

A

B

there

i

..........

i

Then

by: ~ is an

described.

identities,

87

Proof.

As regards

are r e s p e c t i v e l y by

(2.4)

only

that

i,I:

= I by

I--~i:

identities:

A ~ A and i,i:

~ = I and in p a r t i c u l a r

if k = i.

identities,

vertical

For h o r i z o n t a l

identities,

and f--~u c o i n c i d i n g

(2.4)

and

(2.2),

For vertical

that,

with

I--~i:

then

toget h e r

two diagrams

and observe

that

It will mates

under

mention

and

we have

only

to write

f (2.4)

the central usually,

we paste

to get

triangles

s' and ~' cancel

be u n a m b i g u o u s

f--~u and

f'--~u',

out by

if we call

without

(2.1).D k and

explicit

2.3.

l_f f---~u an d f,--~u

then

f and f' are

isomorphic.

Proof.

Let

the mates

of I

A

B

f'--~u

composition

and b.

Proposition

under

D

For h o r i z o n t a l

in context,

the adj.unqtions

of a

canonically

(2.4).

like

if I = I we have

if ~ = i then k = I.

....

(2.3)

A' ~ A', we have

we have a and b both

/.\. and to look at

and f'--~ u'

when a = b, ~ = i if and

f'--~u';

and s i m i l a r l y

composition

if f - ~ u

the a d j u n e t i o n s and f--~u,

f--~u

~-A

1

"~B

and f'---~u,

be r e s p e c t i v e l y

and under

the a d j u n c t i o n s

88 i A

D- A

B ..........

T h e n by t h e

~B

i

"horizontal"

i A .............

and

part

B--

of Proposition

~

I

2.2,

m A

B

p and v are mutually

inverse. 2.3.

If D:

K ~

in K c l e a r l y applying

L is a 2 - f u n c t o r ,

gives

D to

an a d j u n c t i o n

(2.4)

Propositio..n

f--~ u a n d

2..4.

f'--J u'

Dp a r e m a t e s

when

(2.5)

D~,Da:

p are

and

if D:

Df--~ Du:

2.5.

~: D ~ E b e a 2 - n a t u r a l DA

mates

Df--~Du

in g e n e r a l

~A

Let

DA ~ DB in

und.e.r., t h e

and

L.

By

ad~unctions

Df'--~Du'

true

that

D,E:

K ~

[ be

2-functors

transformation. ,- EA

"

DB ................

Then

in

~ is the

under

Proof.

~

EB

L. D

identity

,

and

let

identity ~A

DB

~

2-cells

~ EA

EB

,

~B

the . adSunctions The

the

DA

~B

above

A ~ B

k is; but we h a v e :

Proposition

are mates

f--~u:

K -* [ is a 2-fu,n,c,,t,o,,r~ , t h e n DI a n d

the a d J u n c t i o n s

it is not

~,¢:

we get:

I_~f t

in K, a n d

under

Of course 2-cell

and

an a d j u n c t i o n

2-naturality

Df--~Du

and Ef--~Eu

of ~ expressed

in

in the

L. form

(I.i)

gives ~A D A ...............

DB -

~

~ EA

DB .............

equals

~ ~B

EB

~A

DA

~EA

DB ~B

~' EB

1

~

EB;

89

pasting the

E~

form

on the r i g h t

of each and

The notion

MONADS

of m o n a d

object

B of K, is a n e n d o m o r p h i s m

i ~ t,

o f the m o n a d ~-tn

A detailed Street

t; = i,

in

~.t~

generality

case

where

o f the

2-category

in K, on the

with

and

2-cells

the m u l t i p l i c equations

= ~-~t. has b e e n

simpler

given

aspects

and

by

then

K = 2-CAT.

of the m o n a d

t above

on an a r r o w

s: A ~ B is a

v is c a l l e d

a t-algebra

v: ts ~ s s a t i s f y i n g = i,

s together

A).

s ~ s'

v-tv

with

A morphism

such

= v-~s. an action

o__fft - a l g e b r a s

(with

common

domain

A)

(with

is a

such that

(3.3)

v'.t~ the

= o-v.

t-algebras

forgetful

functor

~r:

t 2 r ~ tr~

tp:

t r ~ tr'

FA:

K(A,B)

t-algebra

with

t-algebras

A form a category

sending

A ~ B it is c l e a r

is a t - a l g e b r a ; is a m o r p h i s m

and that

(s,v) from

It is f u r t h e r

s, the a c t i o n

v: ts ~ s is a l s o

following tr free

(3.1)

of t - a l g e b r a s .

the a c t i o n proposition,

t-algebras:

~s.

in the

that

from

these

light

tr,

gives

with action

a functor

(3.2)

a morphism

From

with a

O: r ~ r',

This

clear

A/g(A,t),

to s a n d ~ to i t s e l f .

for any

~ A/g(A,t).

ts is t a k e n w i t h the

domain

U A to K ( A , B )

F o r a n y r:

verify

we w a n t

in any

the u s u a l

= i,

some

An arrow

when

what

A monad

the u n i t

satisfy

we r e c a l l

v.~s

Thus

~-~t

to

in 123] ; h e r e

(3.2)

a:

respectively are

sense

t: B ~ B t o g e t h e r

in this

special

makes

K ~ CAT.

of m o n a d s

An a c t i o n

domain

these

where

treatment

to the

2-cell

is that

~: t 2 ~ t c a l l e d

(3.1)

pass

case

(= t r i p l e )

classical

ation

gives

IN A 2 - C A T E G O R Y

K; the

n:

(2.2)

(2.4).

§3 3.1

using

that,

for a

of t - a l g e b r a s ,

remarks

of which

we e a s i l y we

call

the

9O

Proposition

3.1

s: A "~ B be a t - a l ~ e b r a bijection $iven

between

to U A.

a n d r: A -~ B any arrpw.

2-cells

c

In the commonly

of B, a n d A/g(~,t) by B t.

:

In d e t a i l ~

Then

~: r ~ s a n d t - a l ~ e b r a

v-t~,

s: ~ ~ B is then v: ts ~ s,

Identifying in this

course

in the

general

K(A,t)

on the

category

K~A,B) K(A't)

there

let

is a

morphisms

~:

tr ~ s,

with

A, t h e n

with

primary

with

case

case

the

sense

B, we w r i t e

as ut:

the m o n a d

K(A,B),

is a n o t h e r

t ~ t'

such

denotes

ZT:

clearly

(s,v)

(3.6)

and

category

The

adjoint ft:

object

functors

the

U~

Of

a classical

is just

s

denoted

B ~ B t.

t on B i n d u c e s

The

category

is c l a s s i c a l l y

B t ~ B and

Alg(A,t)

~.

corresponding in B.

the

is m o r e

monad

category

monad

on the

same

B, a m a p

of m o n a d s

This

~:

tt ~ t't'.

q' = T.q; If

is a t - a l g e b r a

(S,V')

is a t ' - a l g e b r a

with

where

s ~ s' of t ' - a l g e b r a s

gives

is a

that

v = v'.~s;

a morphism

t-algebras.

the unit

s ~ s' are m o r p h i s m s

~''T 2 = T-H, 2

domain

"t-algebra"

of K ( A , t ) - a l g e b r a s .

(3.5) T

~.nr.

K : CAT the p h r a s e

in this

classical

If t' T:

~:

=

identified

K(~,B)

and F !

2-cell

case

to t h o s e

of t - a l g e b r a s

3.2

a

classical

restricted

t-algebra

and

a dJoint

by

(3.4)

here

F A is left

a functor

Alg(A,t ' )

A/g(A,T)

Alg(A,T)

(3.7)

K(A,B).

is also

a morphism

rendering

of

commutative

~ A/g(A,t)

domain

91

In particular the

the

t'

itself

is a t - a l g e b r a

under

action

(3.8)

8: tt'

a n d T c a n be r e c o v e r e d (3.9)

T: t

since

~''t'n'

given,

when

= i.

t't'

(3.9)

and

with

8.t~'

immediate

that

= f'~'u'

f'--q u'. (3.11)

Proposition 2-cells

v:

3.3 every

is

~ t' be a t - a l g e b r a m o r p h i s m ,

that

is, t h a t

u'

on B w h e r e senerated

is a t ' - a l g e b r a

t'u'

= u'f'u' and

3.2.

with

t'

= u'f'

by t h e

and

adjunction

action

~ u'

if t is a n y m o n a d

between

in K, it is

2-cells

o n B, t h e r e

T: t ~ t'

= u'f'

is by

and

by T = vf'.t~'

T is a m a p

of m o n a d s

if a n d o n l y

if v i s

of t o n u'. P~'pof.

(3.11)

and

If T is a m a p thus

o f t on u',

trivially

of t on t'

for it to arise as in (3.8)

C ~ B is any a d j u n c t i o n

v = u'~'-Tu',

an a c t i o n

t't'

St';

the m o n a d

given

Proposition

~':

t-action

is t h i s m o n a d

(3.12)

action

this

that

~ u',

0: tt' ~ t'

condition

is a m o n a d

2.1 a b i j e c t i o n tu'

action

= ~'.St'

u'~': I f t'

is t h a t

f'--q u':

We c a l l

form

~ t', 8

the

(t',~',~')

Observe

8 in the

sufficient

of m o n a d s

If ~ ' , ~ ' :

u' b y

from

~ t'; ~'

If a n a r b i t r a r y

is t a k e n

(3.10)

~ t't' Tt'

~ tt' t~'

the necessary

from a map

~'

t'-algebra

We monad

v is an a c t i o n

then

satisfies say that

of monads,

~ = vf'

(3.10);

t in K, the n o t i o n

the

by

is an a c t i o n

o f t o n u' by

is a n a c t i o n

whence

K admits

u'~'

(3.9)

o f t o n u'f' T is a m a p

construction

of t - a l g e b r a

(3.6).

I f v is a n = t',

which

of monads.

of a l ~ e b r a s

c a n be

o f t' on

if,

E

for

"internalized"

in

92

the sense that the 2-functor A ~ representable,

Alg(A,t)

from K °p to CAT is

so that

Alg(A,t)

(3.13)

~ K(A,B t)

(2-naturally in A) for some B t in K, called the object of t-algebras. When this is so, the a d j u n c t i o n of P r o p o s i t i o n 3.1 becomes an adjunction K(A,B t) ~ K(A,B) which, because of its 2-naturality in A, arises from an a d j u n c t i o n Bt,ct:

ft-~ ut: B t ~ B.

As the n o t a t i o n

CAT does admit the c o n s t r u c t i o n of algebras, and in that

suggests,

case the Bt,ft,u t are those at the end of §3.1. W h e n K = V-CAT for a symmetric m o n o i d a l closed category

V,

K admits the c o n s t r u c t i o n of algebras p r o v i d e d that V has equalizers (at least of pairs with a common left inverse); in detail by Dubuc [5].

this case is treated

Here again the primary m e a n i n g of t - a l g e b r a

is one w i t h domain ~, which now denotes not the unit category but the unit

V-category;

so in this sense a t-algebra is an object s of the

V-category B with an action of t on it.

The category A/g(~,t)

of

these t-algebras admits a canonical enrichment to a V-category, which is B t.

In particular,

taking V = CAT, this applies to the case

K = 2-CAT. The best general result we know o f - - i t a more general result is c o n t a i n e d in Gray [12] the c o n s t r u c t i o n of algebras 2-cate$ory.

is easy to prove and -- is that K admits

if it is finitely complete as a

In accordance with the general d e f i n i t i o n of complete-

ness for V-categories in [4], this means that K has all finite limits, that these are p r e s e r v e d by the r e p r e s e n t a b l e and that K admits cotensor products [X,B] and each B in K.

functors K(A,-):

K ~ CA~

for each finite category X

It turns out to be sufficient to demand the exist-

ence of the C o t e n s o r product [~,B]

where ~ is the a r r o w category

0 ~ i; the existence of the other [X,B]

then follows.

If we replace

"all finite limits" above by "all pullbacks" we get the representable.

93

2-categories

of Gray [ii]

so a finitely termi n a l

complete

object

~.4

2-category

preserved

We h e n c e f o r t h

algebras.

or of Street's

suppose

morphism

ut~,

check that

and sending

the

ft: B -~ B t, and that

p: A ~ B t r e n d e r i n ~

volume;

one with a

3,.3.

Let

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

of

is a t - a l g e b r a

an arrow p: A -~ B t to the

2-cell

~: p ~ p' to the a l g e b r a

K(A,B t) -~ Agg(A,t).

t: B -~ B arises

the m o n a d

Prooosition

K admits

as follows:

sending

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

the t - a l g e b r a

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

that

(3.13)

ut: B t ~ B; and the functor utp,

in this

by the K(A,-).

We can express

t-algebra

paper [25]

thus

utf t g e n e r a t e d f'---~u':

It is easy to

from the a r r o w by ft__~ u t is t itself.

A -+ B.

Then

the arrows

commutative

(3.14)

P

A

Bt

k/ B

are in b i j e c t i o n with monad-mapsT:

with

t-actions

t ~ u'f'.

If the monad called the c a n o n i c a l be monadic In fact

u'f'

it is easily

and does not d e p e n d

p: B t' ~

arrow;

the a d j u n c t i o n

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

on the choice

B t renderin$

3.4.

f'-~

3.2

is a p r o p e r t y

of the left adjoint

There

commutative

ft'

|u

t'

is

~' is said to

(an equivalence). of u' itself,

f' to u'

3.3 we let t' be a second m o n a d

{u' to be the a d j u n c t i o n Proposition

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

is t itself and ~ : i, the p in (3.14)

seen that m o n a d i c i t y

If in P r o p o s i t i o n take f'

and hence

D

comparison

(weakly monadic)

on u',

on B and

: B t' ~ B, we get:

is a bSject.i.on b e t w e e n

arrows

g4

B t'

P

~

Bt

B and

monad

maps

T:

t

We write it an algebraic clear

that

~

t'.

B T for the p in

map b e t w e e n

(3.15)

the d ~ l

category

K0JB

of objects

provides

a reflexion

duality. particular 3.~ monad

This

For

details

further

in [18]

in 2-CAT on CAT.

It is

the canonical

t

t~u

on B fully comparison

of those

,

in t h e arrow

u in K0/B w h i c h

form of the " s e m a n t i c s - s t r u c t u r e "

on the general

laws,

that

we refer

used the name

theory

of monads,

again to Street

"equational

[23].

doctrine"

Here we use the name d o c t r i n e

and in

for a

(or 2-monad)

for

in 2-CAT. For us,

D: K ~ K with,

then,

a doctrine

for its unit

on a 2-category

K

j: I ~ D and m: D 2 ~ D, s a t i s f y i n g

equations

(with D , j , m for t,~,~).

(3.1) Various

to be doctrines by isomorphisms. isomorphisms

relaxations (of.

[15]

What

are possible.

Some

2-natural on the nose

have

the equalities

Z~berlein

[27]

calls

in (3.1),

The lax monads

D only

a lax 2-functor

and mere m o r p h i s m s

ations

we stick to the

strict

doctrines,

(3.1) not

j and m only

of Bunge in

in

"doctrines"

but have

[3]

(3.1).

and hope

the

things we should

below)

for equalities

transformations.

is a 2-functor

and its m u l t i p l i c a t i o n ,

transformations

natural

to T, and call

B t' and B t.

of monads

subcategory

is one

on d i s t r i b u t i v e

Lawvere

any m o n a d

over B;

adjoints.

objects

3.4 as a s s e r t i n g

of the category

into this

corresponding

(3.7).

Proposition

• ~-~B T embeds

left

(3.15)

the algebra

internalizes

We can r e g a r d

have

D

replaced

only have lax

are w e a k e r To a v o i d

that

there

like

still:

complicis a nice

95

coherence theorem that will allow the results in our papers below to be applied at least to the "pseudo" case, We also take D - a l g e b r a here in the strict sense: an object A of K (or more generally a 2-functor A with codomain K) with an action n: DA ~ A satisfying

(3.2)

considers lax algebras

(with A,n for s,v).

in [25]

below,

However Street

and defines them there;

Kelly considers them in r e l a t i o n to strict algebras below.

When K = CAT, we also use "D-category" For m o r p h i s m s of D-algebras,

are the usual ones in nature,

and [17]

for "D-algebra".

on the other hand, the lax ones

as we said in §1.5.

from the n o m e n c l a t u r e of ~3.1, and define, D - m o r p h i s m F: A ~ A' ~to be a pair

in I16]

and

We therefore depart

for D-algebras A,A', a

(f,f) where f: A ~ A' is an arrow in

K and ~ is a 2-cell n (3.16)

DA

Df

~

~

DA'

A

f

n'

f

A'

~

satisfying the axioms (3.17)

n

D2 A - -mA~

Df

D2flr

D2A ,

D2A

~ A

f

DA'

mA'

n

Dn

DA

f

~

n'

A'

D2f

~ DA

D

D2A '

b

f

f

~ DA' Dn'

A

A' ~

n'

g8

jA (3.18)

n

A

m

DA

=

~- A

identity.

> Df

A'

~

f

DA' ........

jA'

In the the when

case

n'

K : C A Y we a l s o

D-morphism

F strong

~ = i, so that

fn = n ' . D f .

of D - a l g e b r a s "

D-morphism

F : (f,l).

If we r e v e r s e and

F = (f,~)

(3.18),

"D-functor"

of

the

we

The

§3.1;

sense

get w h a t

is a s t r o n g

for " D - m o r p h i s m " .

if ~ is a n i s o m o r p h i s m ;

"morphisms

(3.17)

say

strict

we a l s o

of ~ in

we

call

D-morphisms, write

(3.16),

also

we

call

an op-D-morphism.

D-morphism,

then

(f,~-l)

call

it s t r i c t then,

f for the

and

We

are

the

strict

in the

axioms

Clearly

if

is a s t r o n g

op-D-morphism. D-algebras of vertical with

the

pasting

same

We n o w m a k e For

and

of diagrams

objects

these

D-morphisms

~:

into

f ~

like

by restricting

D-morphisms

be a 2 - c e l l

form a category (3.16).

W e get

to s t r o n g

or to

Df

F,G:

DA

A ~ A' we d e f i n e

~A

I

operation

subcategories strict

D-morphisms.

a D-2-cell

~:

F ~

G to

g satisfying

DA

Dc~

the

2-categories.

n (3 .i9)

under

~---

DA

n

~-A

b

Df'

f,

~

n'

A v

DA '

'~ A' n~

97

In the case

K = CAT we also

"D-2-cell".

With

D-morphisms,

and D-2-cells

write

as D-CAT

(here

the obvious

monadic

sub-2-category

it is this

of algebras)

that it is doctrinal,

or 2-monadic,

in 2-CAT, and not just

substantive

are e x a m i n e d

relations

Examples and m o n o i d a l

functors

symmetric

monoidal

categories,

categories arbitrary

and m o n a d

natural

structures

@ and @

appropriate

is the object

D-morphisms

finite

that we m e a n

some

D-morphisms

natural

bearing

coproducts,

With

categories

functors,

a monad,

with

bearing

and m o n o i -

with

of Street

arbitrary

categories

functors,

transformations;

monoidal

transformations

2-cells.

monoidal

strict m o n o i d a l

and having a d i s t r i b u t i v i t y and

We say of K D

and strict

categories,

categories

transformations;

morphisms

strict

of a l g e b r a s

to go beyond definitions:

symmetric

functor

with a s s i g n e d

only the

sense of §3.3.

and m o n o i d a l

transformations;

functors

here

transformations;

monoidal

monad

in w h i c h

over K (to e m p h a s i s e

monoidal

arbitrary

dal natural

K = Cat.

when

in [16].

of D-CAT are:

natural

w h i c h we also

in CAT).

between

b e l o w by Kelly

that

K D in the

It is not our i n t e n t i o n more

D-Alg,

or as D-Cat

K = CAT

are considered;

for

laws of composltion, D-algebras,

by D - A I ~ the

2-category

transformation"

form a 2-category

in the case

We denote D-morphisms

say "D-natural

the

/23];

functors,

and

two m o n o i d a l

of @ over ~, with

K = CAT 2 the objects

of

D-Alg may be pairs of m o n o i d a l categories with a m o n o i d a l functor between

them;

with

K = CAT IAI

for a category

A, the objects

of

D-Alg may be lax functors A ~ CAT, and the m o r p h i s m s lax n a t u r a l transformations

(cf. Street

On the other hand categories,

CAT.

the category

and of m o r p h i s m s

nose - the i n t e r n a l - h o m Indeed

be no n a t u r a l

~241).

preserving

monoidal

all the structure

closed

on the

as well as @ and I - is not d o c t r i n a l

it is only a category, definition

of symmetric

of 2-cell.

not a 2-category: It is monadic

there

over

over

seems

to

CAT, but the

98

monad

CAT is o n l y a f u n c t o r ,

on

3.6

Because

D-Alg §3.4

= K D,

some

new

doctrine

sense: acted

questions

objects

of

K acted

D.

Write

U:

D-Alg

true

in the

o n by

we

formation

DU

functor

~Jvv~+ U.

G = UE:

sense

that

G: A ~

K with

D-Alg

have

about

ors A

~

the

as w e l l

matter

through

the

action

D-Alg

the

.

n and

2-functor

A ~

well

if we

want

to

need

this

K,

do

stay

as

The 3.4.

A map

gives

a

f in

and

not

D acts

with

E:

an

in

as §3.2

in the

and

only

2-functors

op-lax;

the

when

certainly

to

laxity

the

in our

papers

K Then

is

not

corresponds

course

D-A/~

not

thing an

G only

to

that

a

to a in the

to

and

when we

A

the

do

object

Those 2-funct-

factorize is

to

of K b u t

definitions

ones

can,

there

of D on

right

given

as we

on U:

of

the

2-functor.

op-lax-natural.

A ~

A is

A ~

op-lax-natural-trans-

an a c t i o n G is

primary

2-functors

A ~ D-Alg

of D c o r r e s p o n d

(3.16),

generality

and

rather

therefore,

reduce

D,

that

but

~vx~

to t h o s e

far f r o m

extra

further

actions

to be

so,

§3.5

together

Perhaps,

the

of

DG

D-algebras

~ K f o r the f o r g e t f u l

So a 2 - f u n c t o r

K

D- Alg ,, or

the

o n by

D U ~ U,

A ~

honest

to d e n o t e

sense

transformation

no

2-functor.

arise

D-Alg

2-natural

both

case

using

are

actual

is n o t

weak

a

above. We

it

in the

not

work = ~;

don't

below.

So we

shall

course,

applies

a

equally

but

think

allow

we

we shall

pursue

it

here. same d:

kind

D ~

2-functor

D'

of

observation,

of d o c t r i n e s ,

d- Alg

of

in t h e

sense

of § 3 . 2 ,

= K ~ satisfying d-Alg,

(3.20)

D'-Alg,

~

D-Alg,

to

Proposition

not

only

99

but also an evident

d-Alg s a t i s f y i n g

2-functor

d-Alg D'-Alg

(3.21)

~ D-Alg

K

where

of course

sense.

While,

the d-A/g,

we are u s i n g U in an e x t e n d e d however,

the only P: D'-A/g,

for a 2-natural

d: D ~ D' that

P: D'-Alg ~ D-Alg

many more

op-lax-natural

d: D ~

Again we hope

to avoid

D'

d-Alg,

is 2-complete, 2-1imits

but

: K

products.

~20J,

is so if the category

K : Cat/A, show that

K 0 of K is locally

etc.)

for a locally

for example, a left

to

this

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

1.5.1

4.1

if the of Manes

of r e f l e x i v e

is so when the under-

K ~ Cat, K : C a ~ ,

(e.g.

in their

sense

K 0 the various

We have just not

d-Alg has a left adjoint

If K

not only

has c o e q u a l i z e r s

presentable

presentable

coincide.

preserves

cases.

has a 2 - 1 e f t - a d j o i n t

|91,

and when D' has a rank

of whether

are

the algebraic

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

D'-A/g,

10.3 of G a b r i e l - U l m e r

ions of h a v i n g a rank question

2-functor

d-Alg,

has a left adjoint.

category

there

axioms.

in suitable

It follows

functor

lying

map,

correspond

that

D' , and d-A/g,

underlying

By Satz

with UP : U' are

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

we recall

the

pairs.

they

has a left adjoint

so is D'-A/g, cotensor

level,

of Kelly I 14] that

this

~ D-A/g,

as a r e s t r i c t e d

these.

in (3.20)

also

satisfying

as well

is a d o c t r i n e

with UP ~ U';

While we are at this 2-functor

,

- moreover

reasonable thought

in this

they

definit-

out the

situation;

j-Alg: D-Alg ~ l-Alg is just U: D-Alg ~ K; does it have

adjoint? We have

doctrine

maps

satisfying

in the present

case the n o t i o n

B: d ~ d: D ~ D'; namely

of a m o d i f i c a t i o n

a modification

B: d ~

of

100 (3.22) Here

j'

= ~j, m ' . 6 6

66 d e n o t e s

composite

= 6.m.

neither

the v e r t i c a l

6*6 of m o d i f i c a t i o n s • Dd

(3.23)

~ D~

DD,

~

coincide

frightful

common

D'D'

=

DD

(I.i)

value

and

the h o r i z o n t a l

of D'd

D,D

~6D

~ dD'

by

6-6 n o r

dD

~6D,

Dd

which

the

dD'

"~___~

DD

but

composite

_~

$ D,6 ~'D, _

~

dD

(1.2);

notation

D'd

in a 3 - c a t e g o r y

presents

problems.

If 6 is s u c h a m o d i f i c a t i o n D'-algebra n:

with DA

action

n':

of d o c t r i n e

D ' A ~ A, t h e n A has

~ D ' A ~ A, dA n'

It is e a s i l y

verified

(3.24)

DA

that

maps,

dA

if A is a

two D - a l g e b r a

~: DA _~ D ' A ~ dA n' (l,n'.6A)

and

is a D - m o r p h i s m

structures

A.

(A,~)

~

(A,n):

6-Alg

rendering

n' ~ A

D'A

6A

DA

~ dA

It f o l l o w s

that

6 induces

~ A

D'A

.

n t

a 2-natural

commutative

d-,Alg (3.25)

K

transformation

101

in t h e

sense

that

U.~-Alg

Proposition

= identity.

3.5.

Any

We

2-natural

leave

transformation

D'-Alg~Jt~" ~

B-Algf o r It

there

are

a unique that

equivalent

doctrine

maps

d:

D'-A/g;

D-morphism

D'-Alg,

and

As ®:

A×A

each

D-algebra morphisms,

the

prove:

K rendering

m

equivalent.

@

m ) ~ ®,

A when

D'

~ D. = i;

~:

where we

so t h i s

map

let

I: ~ ~ A;

functors n

set

comes A on

go

e:

•

maps,

even

However

to

D'

give

is b y

be

with

= m I +...+

2 @ = @,

m n.

O I = @;

d.e

sense

~ i and

D'-Alg** where

no m e a n s

the

the

case

the

stron~ that

D-Alg,

A•

with

A with,

for

isomorphisms

A D'-algebra

strict

D-Alg

equivalent•

category

a category

coherent

e°d ~ i

so t h a t

are

only

a monoidal be

that

2-categories

isomorphisms,

and

D-Alg

of

a D'-category

A n ~ A, a n d

are

maps.

in the

equivalent

D-Alg**

it

D,D',

~ D with

here

a D-algebra

let

m

d ~ d of d o c t r i n e

doctrines

sub-2-category

are

a D'-algebra

e.d

strong•

taken.

morphisms

whence

is

are

strict e:

to

D-Alg

6:

modifications

an e x a m p l e •

n ~ 0,

m

gives

is

~ A and

n I @ ( ® ,...,

the

(3.24)

D-morphisms

D ~ D'

of doctrine

since

D-Alg**

where

modification

follows

by modifications and

reader

d-Alg

commutative

is

the

morphisms

A gives

a

go

strict

into

from

a d o c t r i n e m a p d: D ~ D'. A D-algebra 0 1 n n-1 s e t t i n g @ = I, ® = i, @ = @(i• ~ ); again

strict

morphisms,

The

composite

D-Agg

~

the

composite

D'-Alg ~

so t ~ i s

comes

D'-Alg ~ D-Alg D-Agg

~

D'-Alg

from

is is

the

A

a doctrine identity,

clearly

102 isomorphic However

to i, whence

the composite

to i; the original

d.e

~ i, and D and D' are equivalent.

D'-A/g, ~ D-Alg,

~ D'-Alg,

and final D'-algebra

i: A ~ A is a strong, but not a strict,

is not isomorphic

structures on A are such that isomorphism between them.

~,~IOGRAPHZ [ 1]

J. Benabou, Math.

[2]

Introduction

47 (1967),

M.C. Bunge,

13]

M.C. Bunge,

195 (1971), Coherent

(preprint,

[4]

Dubuc,

C. Ehresmann,

pairs,

Lecture Notes

70-122. and relational

Enriched

106 (1969),

Kan extensions

Lecture Notes

[6]

induced adjoint

extensions

B.J. Day and G.M. Kelly,

E.J.

in

algebras

March 1973).

Notes in Math.

[5]

Lecture Notes

1-77.

Bifibration

in Math.

to bicategories,

in Math.

Cat6gories

80 (1963),

functor categories,

178-191.

in enriched

category

theory,

=

145 (1970).

structur6es,

Ann.

Sci. Ecole Norm.

Sup.

349-425.

[7]

C. E h r e s m a n n , Cat6$ories ' et structures

[8]

S. Eilenberg

and G.M. Kelly,

Categorical

Lecture

Al~ebra(La

(Dunod,

Paris,

Closed categories,

Jolla 1965.)

Proc.

(Springer,

1965). Conf.

on

New York,

1966). [9]

P. Gabriel and F. Ulmer,

Lokal pr~sentierbare

Lecture Notes in Math.

[ io]

J.W.

Gray, The categorical in Math.

[ 11]

J.W.

99 (1969),

Gray, Report in Z~rich, 195 (1971),

[ 12]

J.W.

=

221 (1971). comprehension

scheme,

Lecture Notes

242-312.

"The meeting of the Midwest Category Seminar

August

24-30,

1970",

Lecture Notes in Math.

248-255.

Gray, Formal category in Math.

Kategorien,

theory, to appear in Lecture Notes

103

[ 13]

A. Grothendieck,

Cat6gories

fibr6es et descente,

SSminaire

de

G@om@trie Al$@brique , Institut des Hautes Etudes Scientifiques,

[ 14]

G.M. Kelly,

Paris

Adjunction

in Math.

(1961).

for enriched categories,

106 (1969),

166-177.

[ 15]

G.M. Kelly,

On clubs and doctrines

[ 16]

G.M. Kelly,

Coherence

distributive

[ 17]

G.M. Kelly,

[ 18]

F.W. Lawvere,

F.E.J.

E. Manes,

(in this volume).

Ordinal sums and equational

Linton,

and for

(in this volume).

80 (1969),

Autonomous

Journal of Alsebra

[ 20]

(in this volume).

for lax algebras

Doctrinal adjunction

Notes in Math.

[ 19]

theorems

laws

Lecture Notes

2 (1965),

algebras over a triple

Lecture

141-155.

categories

A triple miscellany:

doctrines,

and duality of functors,

315-349.

some aspects of the theory of

(Dissertation,

Wesleyan Universit~

1967).

[ 2l]

P.H. Pa!mquist,

The double category

Lecture Notes in Math.

[ 22]

J.E. Roberts,

[ 23]

8 (1968),

R. Street,

2 (1972),

functors,

Journal

181-193. J. Pure and Applied

149-168.

Two constructions

Topologie

squares,

123-153.

of initial

R. Street, The formal theory of monads, Algebra,

[ 24]

195 (1971),

A characterization

of Algebra

of adjoint

et G@ometrie

on lax functors, Diff@rentielle

Cahiers de

XIII,

3 (1972),

217-264.

[ 25]

R. Street,

Fibrations

and Yoneda's

lemma in a 2-category,

(in this volume).

[ 26]

R. Street, Elementary

[ 27]

V. Z6berlein,

cosmoi,

(in this volume).

Doktrinen auf 2-Kategorien

Inst. der Univ.

ZUrich,

1973).

(Manuscript,

Math.

.FIBRATION$ AND YON.EDA'S ~EMMA IN A 2-CATEGORY by ~o~$ Street

Our purpose is to provide w i t h i n a 2-category a conceptual proof of a set-free version of the Yoneda lemma using the theory of f i b r a t i o n s .

In doing so we carry

many d e f i n i t i o n s of category theory i n t o a 2-category and prove in t h i s more general s e t t i n g results already f a m i l i a r for CAT. The La J o l l a a r t i c l e s of Lawvere [ 5 ] and Gray [ 2 ] have strongly influenced t h i s work.

Both a r t i c l e s are w r i t t e n in styles which allow easy transfer i n t o a 2-

category.

However, they also f r e e l y use the fact that CAT is cartesian closed, a

luxury we do not allow ourselves. The 2-category is required to s a t i s f y an elementary completeness condition amounting to the existence of 2-pullbacks and comma objects.

This relates the

2-category closely to a 2-category of category objects in a category.

Such con-

siderations appear in ~1 and were considered by Gray [ 3 ] . Fibrations over B appear in ~2 as pseudo algebras f o r a 2-monad on the 2category of objects over B. Kock [ 4 ] .

This 2-monad is of a special kind distinguished by

We define lax algebras and lax homomorphisms for general 2-monads and

provide a l t e r n a t i v e descriptions of pseudo algebras and l a x homomorphisms for the special 2-monads.

We are able then to give an equivalent d e f i n i t i o n of f i b r a t i o n

generalizing the s e t t i n g for the Chevalley c r i t e r i o n of Gray [ 2 ] p 56. In order to eliminate the need f o r our 2-category to be cartesian closed in the remainder of our work we are led to introduce an extra v a r i a b l e ; we must consider b i f i b r a t i o n s from A to B rather than f i b r a t i o n s over B.

A p a r t i c u l a r class of spans

from A to B, called covering spans, is introduced in ~3.

As with t h e i r analogue in

topology, covering spans are b i f i b r a t i o n s . Furthermore, any arrow of spans between covering spans is a homomorphism.

In the case of CAT, b i f i b r a t i o n s correspond to

category-valued functors and the l a s t sentence r e f l e c t s the fact that covering spans correspond to those functors which are discrete-category-valued; that i s ,

105

set-valued.

With this interpretation of covering spans as set-valued functors, we

see that Corollary 16 is a generalization of the Yoneda lemma of category theory. The concept of Kan extension of functors is one of the most f r u i t f u l concepts of category theory, and the definition just begs translation into a 2-category. This has already been used to some extent (see [6] and [ 7 ] ) .

But the Kan extensions

of functors which occur in practice are all pointwise (using the terminology of Dubuc [ I ] ) . Using comma objects we define pointwise extensions in a 2-category in @4. Note that, in general, for the 2-category V-Cat, this definition does not agree with Dubuc's; ours is too strong (we hope to remedy this by passing to some related 2category). do not.

For V=Sct and V=2, the definitions do agree; for V=AbGp and V=C~t, they

The closing section gives some applications of the Yoneda lemma and

fibration theory to pointwise extensions i l l u s t r a t i n g their many pleasing properties.

106 11. Representable 2 - c a t e g o r i e s . Let

A

denote a c a t e g o r y .

A span from

A

to

B

in

A

i s a diagram

(uo,S,ul):

When no confusion is l i k e l y , span

(uI,S,Uo)

w i t h the span

Let

is a b b r e v i a t e d to (1,A,u)

f : ( u o , S , u 1)

from

, (u~,S',u~)

SPN(A,B)

we a b b r e v i a t e

A

S*.

to

B to

to

S;

Also we i d e n t i f y

B.

then the reverse

an arrow

u:A---+ B

An arrow o f spans

is a commutative diagram

denote the category of spans from

When A has p u l l b a c k s , a span from

(u0,S,uz)

(uo,S,u 1)

C have a composite span

A to from

B and t h e i r arrows.

A to

(uo~o,ToS,vl~l)

B and a span

from

A to

(v0,T,v 1)

C where the

f o l l o w i n g square i s a pullback. A

Ul

ToS

>" T

S

>-B

Ivo

Ul If

f : S - - - - ~ S'

spans from

is an arrow o f spans from

B to

C then the arrow

A

to

gof:ToS

g:T

13 and ; T'oS'

~ T'

i s an arrow o f

induced on p u l l b a c k s i s an

arrow o f spans, An opspan from

A

to

B

in

A

is a span from

arrows o f opspans are arrows o f diagrams in

Suppose A has pullbacks. following data from A:

A

to

B

in

A°P; however,

A,

A category object

A

in

A

c o n s i s t s of the

107 - an object -

-

Ao;

a span (do,Ai,d I ) arrows of spans

from

Ao to

i:(1,Ao,l)

Ao;

~ (do,Ai,dz),

A

A

c:(dodo,AloAl,dldl)

~ (do,Ai,dl);

such that the f o l l o w i n g diagrams commute 1oi

iol

1oc

AioAI
B

f

r/s

r/s-----~r-)--B

dz

#

are equal, then there exists a unique 2-cell

U l

S~_~r/s

do

S

~- D

such that

f, = do@, q = d1@. In non-elementary terms,

r/s

is defined by a 2-natural isomorphism

K(S,r/s) ~ K(S,r)/K(S,s), where the expression on the r i g h t hand side is the usual comma category of the functors

K ( S , r ) , K(S,s).

The comma object of the i d e n t i t y opspan @A.

I t is defined by a 2-natural isomorphism

(1,A,I)

from

A to

A is denoted by

109 2

K(S,~A) and so is the cotensor in exists f o r each object

~

K of the category

If

, 2

with the object

A.

A and when K has 2-pullbacks we say that

representable 2-category (Gray [ 3 ] Example.

K(S,A)

K

is a

uses "strongly representable").

A has pullbacks then

@comma objects in

When "#A

K = CATIA)

is a representable 2-category. / /

K are comma objects in

both representable and oprepresentable,

~

K°p.

In a 2-category which is

has a l e f t 2 - a d j o i n t

is automatically a 2-1imit in

#

and any l i m i t

which e x i s t s in

K0

Proposition i.

In a representable 2-category each opspan has a comma object.

The formula is

Proof.

K.

r / s : s*otDor.

// In a representable 2-category, an i d e n t i t y 2-cell

arrow

i:A

~ tA,

I A ~ I~A I

corresponds to an

and the composite 2-cell ~Ao~A

~A

tA

o

corresponds to an arrow ~od ~ A ~ A

~Ao~A

c

~>A.

For each arrow

f~ B corresponds to an arrow

~f:~A

f:A

, B, the 2 - c e l l

~ ~B.

dl Proposition 2.

In a representable 2-category the following results hold.

A, the arrows

(a) For each object object

A

in

f:a

enrich

d0,d1:~A

~A

to a category

K 0.

(b) For each arrow arrow

i,c

f:A

..... ~ B,

the pair of arrows

f,tf

constitute a functorial

~ B. f

A ~ . ~ . _ ~ B,

(c) For each 2-cell

f, formation from (d) The assignment

to

f'.

the corresponding arrow

a'A

~ ~B

is a trans-

110

f

~f 7~'B

~ f,

f,

defines a 2 - f u n c t o r from Proof.

K to

(a) For each o b j e c t

functions f o r the category

CATIKo),

X,

IK(X,A)21~

K(X,A);

so

are the source and t a r g e t

Ko(X,@A)-'--T Ko(X,A)

t a r g e t f u n c t i o n s f o r a category, f u n c t o r i a l l y s t r u c t u r e o f a category object in

]K(X,A) I

K0.

in

X.

are the source and

So @ A ~ A

c a r r i e s the

I t is r e a d i l y checked t h a t t h i s s t r u c t u r e

agrees with t h a t o f the p r o p o s i t i o n . (b) For each

X, ( K o ( X , f ) , Ko(X,@f) )

K(X,f): K(X,A) (c) S i m i l a r l y , K(X,~): K(X,f)

corresponds to the f u n c t o r

~ K(X,B). Ko(X,~)

corresponds to the natural transformation

~ K(X,f').

(d) What we have shown is that the composite

K

~ CAT(Ko)

>

[Ko°P,CAT]

is the Yoneda embedding, a well-known 2 - f u n c t o r . a

I t f o l l o w s t h a t the f i r s t

arrow is

2-functor.//

§2. Lax algebras and f i b r a t i o n s Suppose

D is a 2-monad on a 2-category

denote the u n i t and m u l t i p l i c a t i o n . an arrow

c:DE

C and l e t

i:i

A lax D-algebra consists of an object

~ E and 2 - c e l l s E iE

D2E ~

cE

Dc

DE

~

E

~ D, c:DD

DE

c in the 2-category C such that the composites

c

..... )~ DE

~D E,

111 i

DE ......

> DE c

c

~Dc

(1): D E ~ E (2):

@

c

c

1

CDE

CDE

D3E

(3)

• D2E

D3E

~ D2E

DE c

c

A pseudo D-algebra is a lax D-algebra in which

are equal as indicated.

A normalized lax D-algebra has

isomorphisms.

D-algebra is a lax D-algebra with both in

C,

DE with

Kock [ 4] c -~iD

CE:D2E

identities.

~ DE is the free D-algebra on

are

A

Of course, for any E E.

has distinguished those 2-monads D with the property that

in the 2-functor 2-category c.Di D~,~ I

i d e n t i t y modification modification

~,0

~ an i d e n t i t y 2 - c e l l .

~,0

Di D , ~

D2 .

iD isomorphism with inverse

[C,C]

Suppose E is a lax D-algebra such that

~, and consider the composite

E

~ iDE

On the one hand, On the other hand,

Then the

D corresponds under the adjunction to a

DiE D

with i d e n t i t y counit.

c.Dc D2E ~

E . cc E

el E = (CCEiE)(e.Di E) = e.Di E = c.D~. 01E = (eiDE)(C.DC.1 E) = (Cc)(c.DC.~E).

~ is an

112 So we have the e q u a l i t y

DiE D E ~

Dc. Di E

D2E

Dc ~ DE

c ~E

=

(4) c

c) E

D E ' D E I

I

The next proposition generalizes s l i g h t l y some of Kock's r e s u l t s ;

he considers

the normalized case. Proposition 3.

Suppose

D

is a 2-monad with the Kock property and suppose

the 2-cell E

DE is an isomorphism (a) ~

with inverse

c ~

~E

satisfying equality (4).

is the counit for an a~junction

Then:

c ---AiE with unit given by the

composite i

°EC

°2E Dc °E

41E /

iE c. Dc

(b)

the 2-cell

D 2 E C ~0 ~ E

iDE-i E 2-cell

E C I ~

DiE.iE holds;

corresponding under adjunction to the identity

c.c E D2E

is unique with the property that the equality (1)

113 (c)

this 2-cell

Proof.

(a)

gives

0

E,C,~

enriches

Let

with the structure of pseudo D-algebra.

denote the composite 2 - c e l l displayed in (a).

~-c.c~ = 1.

Since the composite

~EiE

is the i d e n t i t y ,

E q u a l i t y (4)

we also have

iE~-.~i E = I. (b)

Let

denote the composite

T

DE

I

~-DE

i~Dc u b ~ D2E___r._...~,DE

c dE

E Then the 2 - c e l l

@ described in (b) is the composite

DiDE D2E

D3E ID2E

iD

Dc

E • D2E

~

Dc

~ DE

c

• E

c

DE The 2 - n a t u r a l i t y

of

i:1

, D implies the e q u a l i t y

which i t e a s i l y f o l l o w s t h a t

% = ~c.

i E~c = D-#.DC.iDE,

Using t h i s and the equations

from

CEiDE = i ,

IDEiDE = 1, we deduce the e q u a l i t y (1). To prove uniqueness, suppose e

8

satisfies

(1).

The 2 - c e l l corresponding to

under adjunction is the composite

iDEi E

(uni t)iDE i E Di E" i ESi DEi E • ~ DiE.iEC.DC.iDEi E ~ DiE.iEccEiDEIE

DiE. i E ( c o u n i t ) DiE.i E. So ( I ) implies t h a t t h i s composite is independent of composite is the i d e n t i t y , 0 (c)

@.

For one such

so the composite is the i d e n t i t y f o r a l l such

@ the 0.

So

is unique. Clearly

@ is an isomorphism, so i t remains to show t h a t

@ satisfies

(2)

114

and (3).

E q u a l i t y (2) f o l l o w s from the equations

c.Dc. DCE.DiDE.t E = c. Dc.t E ,

• = ~c

arrows by t h e i r r i g h t a d j o i n t s " ,

and (4).

CE.Di E = 1, c.Dc.DCE, tDE.DiE =

By the n a t u r a l i t y

e q u a l i t y (3) holds since i d e n t i t y

of " r e p l a c i n g 2 - c e l l s appear

in the squares o f the transformed e q u a l i t y . / /

A lax homomorphism o f l a x D-algebras from f:E

, E'

E

to

E'

c o n s i s t s of an arrow

and a 2 - c e l l c

in

C

~ E

DE. . . . c

~ E'

such t h a t the composites c

DE

~E

(5)

DE

E

I .

D2E

m E i

DE'.----=---~ E'

cE

,L DE

D2f

D2E

=

,I,

)-DE

D2E '

~-DE'

I

Def IDf

Dc"~ DE' are equal as i n d i c a t e d . ef

cE

c

(6)

when

DE

DE'

c A l a x homomorphism f

i s an isomorphism, and i s c a l l e d a

is called a

homomorphism

~, E'

c

pseudo homomorphism

when

8f

i s an i d e n t i t y .

115 Proposition 4.

f:E

~ E'

D is a 2-monad with the Kock property and suppose

Suppose

is an arrow between pseudo D-algebras.

l:Df.iE----~ i E , . f

which corresponds under adjunction to the identity 2-cell unique with the property that equality (5) holds, f

ef:c. Df

Then the 2-cell

Furthermore, this

~ fc is

enriches

9f

with the structure of lax homomorphism.

Suppose 9f

Proof.

is as explained in the proposition.

since both the 2-cells iE,f

1 ~ iE,f

ciE,f

~ f

correspond to the i d e n t i t y 2-cell

under adjunction (recall that

On the other hand, suppose

9f

Equality (5) holds

satisfies

~

(5).

is the counit for Then 9f

c ---~ iE,)-

corresponds under

adjunction to the composite ~.Df.i E iE,Ofi E iE,f~Df.i E ~ iE,c. Df.i E iE,fci E •iE,f which is independent of

@f by (5);

(6) since both the 2-cells i d e n t i t y 2-cell 8:c.Dc

so

c. Dc. D2f

9f

Finally,

Of

i

DiE,.iE, f

(recall that

I:iDEiE-~-* DiE.iE). / /

For convenience we henceforth work in a representable 2-category Proposition 5. counit

~

Suppose

and unit

~.

f:A

counit the identity and unit

~g

Using

g:C ........ >, B,

the arrow

is a right adjoint for

B:I

d0~ = u~.qd0 Proof.

is an arrow with a right adjoint

~ B

For any arrow

corresponding to the 2-cell

satisfies

fcc E correspond under adjunction to the

D2f. iDE,iE = iDE,.iE, f

~ cc E corresponds to

is unique.

,

~ vd I

,

dl:f/g

v:C ~ C

defined by the equations

d1~ = i.

s f . f q = I, we see that the two composite 2-cells

K. u, ~ f/g

with

116 I A

f/g

,

~ C ..... \d \

I

A

~ B

~ f/g v /~' /

f/g

~C dI

~ f/g

f/g

~A

~ C~----~-*-B dI

I are equal;

so there e x i s t s a unique 2 - c e l l

also see t h a t

Corollary 6.

6v = I.

So

For any arrow

6

is a u n i t f o r

p:E

ip with

unit the identity.

with

is the identity 2-cell

h

~

as asserted.

dI ~

~ B, the arrow

Explicitly,p ip

v

Using

ue.nu = 1, we

with i d e n t i t y c o u n i t . / /

do:p/B

~ E

has a left adjoint

is the unique arrow whose composite

E~B. P

Proof.

Since

I:B

> B has a l e f t a d j o i n t , a dual of the proposition y i e l d s the

result.// Corolla~#

dz:f/B

7.

An a r r o w

f:A

, B Foe a right adjoint if and only if the arrow

~ B has a right adjoint.

In this case there is a right adjoint for

dz

with counit the identity. Proof.

If

d I --4 v

f = dlif

--I

dov.

Proposition

we can compose with

i f --~ d o

of C o r o l l a r y 6 to obtain

The converse and the l a s t sentence f o l l o w d i r e c t l y from

5.//

C o r o l l a r y 4 applied to

p = i B gives

i

as l e f t a d j o i n t f o r

.do:@B

, B.

la

The u n i t o f t h i s adjunction is the i d e n t i t y and the c o u n i t

@B,.~@B is the 1 2 - c e l l defined by the equations d o t o = I , d l t o = ~. D u a l l y , d1:@B , B has i I as r i g h t a d j o i n t w i t h c o u n i t the i d e n t i t y and u n i t t B ~ @ B defined by v

d o t l = h, d l t I = 1.

idl Using the 2-pullback property o f the square A

dl

@B

J, B

dl

,

117 loi

we see t h a t

d l t o = 1 = do~ I

i m p l y the e x i s t e n c e o f a u n i q u e 2 - c e l l

@B . ~ @ B o ~ B iol

A

such that

A

dol = t o , dlt = tl,

Proposition 8.

(a)

The composite 2-cell loi

~B ~

c ~, L

L(E,p) = (tBop,dlp , Lf = 1of, L~ = io~.

denote the 2-natural transformations w i t h iol E

The diagrams which say t h a t say t h a t

L

is a 2-monad on

Proposition 8 shows t h a t

L

Let

i:l

L,

(E,p)-components col

~¢Bop ,

~Bo~BoE

~ ~BoE .

(see Proposition 2 ( a ) ) is a category o b j e c t p r e c i s e l y KB with u n i t

i

and m u l t i p l i c a t i o n

c.

Moreover,

has the Kock property so that Propositions 3 and 4

apply. An arrow

p:E

~B

is c a l l e d a O-fibration over

the s t r u c t u r e of pseudo L-algebra.

B when (E,p)

The O - f i b r a t i o n is c a l l e d

supports

split when (E,p)

supports the s t r u c t u r e of an L-algebra. Proposition 9. over

B

(Chevalley criterion].

if and only if the arrow

The arrow

p : ~ E ....~ p / B

p:E

> B is a O-fibration

corresponding

to the 2-cell

119

Pdl

OE

:, B

doI E

>B P

has a l e f t a ~ ' o i n t w i t h u n i t an isomorphism.

Suppose

Proof.

(E,p)

The c o u n i t of C o r o l l a r y 6 is

is a pseudo L-algebra.

r e a d i l y seen to be (id0)ol p/B = ~BoE ~ o o 1 ~

~BoE = p/B

1 t h i s 2 - c e l l corresponds to an arrow p/B

k , {(p/B) i

p/Bq~.~p/B

~c

~ {E.

k:p/B

~ ~(p/B).

One r e a d i l y v e r i f i e s

denote the 2 - c e l l

L( ;

that

Let

£

be the composite

p£ = k(C%E).

i t is an isomorphism.

Let

~E.~__~E 1

Let

~z denote the unique 2 - c e l l s a t i s f y i n g pi = iE). Also

By applying

do(e£.£q) = 1

counit

c

and u n i t

do,d I

(Note t h a t

dos = ~ d 0 , d i s : ( ~ d 1 ) ( c p 1 1 ) .

to

pe.qp

is immediate.

i t is r e a d i l y seen t h a t

l-q

To complete the proof t h a t

q, we must show t h a t

d l ( s Z . £ n ) = I.

pe.qp = I.

~ with

dz(sl.l~)

But

=

(TdlZ)(cplzZ)(cL~). From the c a l c u l a t i o n s A

doP11£

= do11£

= ~l = ~.~c.k

= c~k

= C(~ool)

= C(dool)(1ol)

= do(loc)(1ol),

diPli£ = ~p.dl~i£ = l~(pc)k = Idi k = Idi = di(liol ) = di(dlol)(lol) we deduce t h a t

p l i £ = ( l o c ) ( ~ o l ) = Lc.1E.

= di(loc)(iol),

So, by c o n d i t i o n (4), we have

di(eZ.Zn) = (~c)(c. LC. IE)(CL~) = 1. Conversely, suppose Z~p

with counit

£--~p

~ and

dlei:d1£Pi dz ~ @E ~ E and

with counit

di --[

with c o u n i t

~ d i i = 1.

p/B

~= di~i.

i

~ and isomorphism u n i t

w i t h c o u n i t 1, we have So put

c

di£- ~

n.

Since

pi = i E

equal to the composite

I t is r e a d i l y checked t h a t the composite

120

p/B

E

i

~

B E ~

~ . ~

BE

1

is an isomorphism w i t h inverse the composite 1

Z

E

~ > BE

p/B

~

p/B

~ E

BE

So ~ So

is an isomorphism.

,

> BE

,

1

The e q u a l i t y (4) f o r Proposition 3 f o l l o w s e a s i l y .

E is a pseudo L - a l g e b r a . / / Compare the above p r o p o s i t i o n with Gray [ 2 ] p.56;

so we have r e l a t e d the

d e f i n i t i o n of O - f i b r a t i o n here with the d e f i n i t i o n of o p f i b r a t i o n in K = Cat.

Notice t h a t the u n i t of the adjunction

an isomorphism but an i d e n t i t y . we w i l l

1 --IP

[2]

when

f o r Gray is not j u s t

I t is worth poiHting out the reason f o r t h i s since

need the observation in the next paper.

A O-fibration will

be c a l l e d

normal when there is a normalized pseudo L-algebra s t r u c t u r e on i t .

In Cat every

O - f i b r a t i o n is normal, but in other 2-categories t h i s need not be the case. the proof of the Chevalley c r i t e r i o n ,

for a normal O-fibra~ion,

For any arrow

g:B'

p:BE

if

, p/B

~

is an i d e n t i t y then so is

q.

In So,

h a a l e f t a d j o i ~ t with u~it an ide~J~Cty.

, B, " p u l l i n g back along

g"

is a 2 - f u n c t o r g * : K ~

KB,;

121 f o r each

E in

KB, the diagram A

g

g*E

B' g

is a pullback.

The c o m p o s i t e 2 - c e l l

~/B'

dl B ~

°°I g*E

A

)

BI

g

E

"

P V

induces an arrow of spans gE: A

p/B',

~/

p~;~\

t/o \',, E

B' V

f o r each

E in

KB.

One r e a d i l y checks t h a t

gE' E e KB are the components of a

V

2-natural

transformation

g:

KB

g*

> KB,

t st KB

g*

~ KB, v

Indeed, in the language o f Street [ 6 ] ,

the pair

(g*,g)

is a monad functor from

122 to

(KB,L)

(KB,,L)

Proposition 10.

in the 2-category 2-CAT. g:B' ........> B

Suppose

K.

is an arrow in

For each lax L-algebra

E, the arrow V

gE

Lg*E enriches

f:E

g*E

~ E'

g*(c)

~ g*LE

g*E

>

with the structure of lax L-algebra.

For each lax homomorphism

of lax L-algebras, the 2-cell V

Lg*E

enriches

gE

~ g*LE

g*(c)

Lg*E' v ~g*LE' ~ g*E' gE' g*(c) ~ g*E' with the structure of lax homomorphism.

g*(f):g*E

pseudo L-algebra or an L-algebra then so is or a homomorphism then so is Corollary 11.

~ g*E

g*E.

If

f

If

E

is a

is a pseudo homomorphism

g*(f).//

The pullback of a (split) O-fibration along any arrow is a (split)

O-fibration.//

Let

R:KA

~ KA denote the 2-functor given by:

f (E,q)ZIC(E'

A/f ,q')

'

(A/q'd0)

~

g

A/g

There is a 2-monad structure on K by

Kc°.

An arrow

(A/q',d0).

q:E

R and the theory develops as for

>A

is called a 1-fibration over

A

L;

j u s t replace

when (E,q)

supports the structure of pseudo R-algebra. Note that the category

SPN(A,B)

of spans from

2-category by taking as 2 - c e l l s the 2 - c e l l s

a

of

A to

B becomes a

K as in the diagram

123

where

q'o = lq, p'o = lp.

Let

, SPN(A,B)

M:SPN(A,B)

denote the 2 - f u n c t o r

given by:

f

1ofol

E ~ E '

1

~ @BoEo@A

-- @BoE'o@A.

£io~oI

logol

g

This 2 - f u n c t o r supports the s t r u c t u r e of 2-monad t o o ; multiplication

c:MM

i:1

~ M and

~ M have as components

ioloi

coloc

E

~ ~BoEo~A and

A span

the u n i t

(q,E,p)

for

A

~Bo@BoEo~Ao~A

to

B

, ~BoEo~A .

i s c a l l e d a bifibration from

supports the s t r u c t u r e o f pseudo M-algebra.

A

to

B when i t

A split bifibration is an M-algebra.

Results on L-algebras and R-algebras can be t r a n s f e r r e d to M-algebras via the following result.

The corresponding statement f o r l a x algebras is l e f t

to the

reader.

Proposition I 2. c:¢BoEo~A CL:tBoE

,Suppose

~E

is a span from

A

to

B.

The M-algebra structures

are in bijective correspondence with pairs of arrows of spans

~ E,

structures on

E

CR:EotA E

, E

c L, c R are L-algebra, R-algebra

such that

related by the condition that ME

loc R ~

RE

~ LE

I

~ E cR

commutes;

the bijection is determined by loloi

c L = (~BoE

c

~ ~BoEo~A iolol

c R = (Eo~A

c

~ E) c

~ @BoEo@A-

~ E)

= CL(lOC R) = CR(CLOl).

Furthermore, an arrow of spans is a homomorphism of M-algebras if and only if it is a homomorphism of both the corresponding L-algebras and the corresponding R-algebras.l/

124 Combining this with Corollary I I and the dual for 1 - f i b r a t i o n s we have: qorollary 13.

For any arrows

E from

B

A to

f:A'

~ A,

g:B'

~ B, each (split) bifibration g*oEof

induces a (split) bifibration

A'

from

B'.//

to

There is a more general composition of b i f i b r a t i o n s which we w i l l not need. If

E is a b i f i b r a t i o n from

the b i f i b r a t i o n

F ~ E from

A to

B and

A

C can be defined by the usual "tensor product

to

F a b i f i b r a t i o n from

B to

C then

of bimodules" coequalizer, provided this coequalizer exists and is preserved by certain pullbacks. ..k..3. Yoneda's Lemma within, a 2-category. Again we work in a representable 2-category

K.

A covering span is defined to be a span which is the comma object of some opspan. Theorem 14.

Any covering span is a split bifibration.

Any arrow of spans

between covering spans is a homomorphism. Proof. of

Any comma object

r/s

is a composite

M at the i d e n t i t y span of

D;

so

s*o~Dor.

But

tD

is the value

@D is a free s p l i t b i f i b r a t i o n .

So r/s

is a s p l i t b i f i b r a t i o n by Corollary 13. Suppose f : r / s that

f

~ u/v

is an arrow of spans from

commutes with the M-algebra structures on

12, i t suffices to show that structures separately.

f

r/s

A to

B.

and

u/v.

We must prove By Proposition

commutes with the L-algebra and R-algebra

By d u a l i t y , i t suffices to show that

f

commutes with

j u s t the L-algebra structures. The L-algebra structure

c:@Bo(r/s)

~ r/s

comes from that of

@D via the

commutative square ~so~ @Bo(r/s) . . . . . . . .

@Do@D

Ic r/s Equivalently, note that from

A to

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

~ ~D

@Bo(r/s)

~B (composed with

is the comma object of the opspan (r,D,sdo)

d1:~B

~ B) since we have the pullback

125 ^

dl .~B

r/sdo

i,o

A

do r/s and

c:r/sd o

~ r/s

~B

dl

corresponds to the composite 2 - c e l l r/sd o

^

do The main t r i c k of the proof i s to introduce the 2 - c e l l

r/sdo~

A

defined by u/v.

doa

ldo,

=

The arrow

k(f)

dl~

r/s c

= Xdz;

of c o u r s e , we a l s o have such an

for

is defined by the commutative diagram A

A

do

d1

r/s-

u/v ~

r/sd o

^ do

u/vd 0

~ ~B

^ dl

~ ~B

The c a l c u l a t i o n s

doaL(f ) = ldoL(f) = 1do = doa = dof~ A

d l a L ( f ) = XdzL(f )

= Xd I

= dl~

=

dlf~

show t h a t the following composites are equal A

A

do r / s d 0 ~

f

L(f)

r/s

~-u/v

=

do ulvd0T~.~..~ C

~ulv

r/sdo

C

SO c . L ( f ) = f c , which proves that Let

COV(A,B) denote the f u l l

covering spans. on SPN(A,B);

Let

SPL(A,B)

f

•

is a homomorphism.//

subcategory

SPN(A,B)

whose objects are the

denote the category of algebras f o r the monad M

i t is the category of s p l i t b i f i b r a t i o n s from

A to

B and t h e i r

126

homomorphisms (up to e q u i v a l e n c e ) .

Corollary 1 5.

SPL(A,B)

underlying functor

A

to

B.

~ %PN(A,B)

factors through the

~ SPW(A,B).//

Corollary 16 (Yoneda len~na). span from

COV(A,B)

The inclusion functor

Suppose

f :A

is an arrow and

~ B

Composition with the arrow of spans

bijection between arrows of spans from

to

f/B

E

if:f

F

is a covering yields a

> f/B

and arrows of spans from

f

to

E. Note t h a t

Proof.

f/B

This gives a b i j e c t i o n

is the f r e e M-algebra on the span between arrows o f spans

But by Theorem 14, any arrow of spans Take

Remark.

to

B

K = CAT and

functorially

An arrow

f: ~

>B

is j u s t an o b j e c t

eb

b

, E

A

to

e

from

of

B.

Covering spans

The functor

K(A,B) °p

g A~~ . . ~ B

B

i n t o some category o f sets.

The b i j e c t i o n o f the c o r o l l a r y B(b,-)

>e

I b,

~ SPN(A,B)

given by o/B

i

~

f/B

~ g/B

f

is fully faithful. Proof.

The d e f i n i t i o n

o f comma o b j e c t s gives the b i j e c t i o n f

A

~B h >

A

between 2 - c e l l s bijection §4.

o

g

between such

h

f

• g/B

~B

and arrows o f spans

h.

and arrows o f spans

The Yoneda lemma provides the f/A

~ g/B.//

Pointwise extensions. Recall the d e f i n i t i o n

of left

f/B

E from

The f o l l o w i n g special case o f C o r o l l a r y 16 appears in Gray [ 3 ] .

Corollary 17.

B.

is a homomorphism.//

between n a t u r a l t r a n s f o r m a t i o n s

obtained by e v a l u a t i n g a t

from

~ E and homomorphisms

in C o r o l l a r y 16.

correspond to f u n c t o r s

becomes the usual b i j e c t i o n elements o f

A =~

f/B

f

f

extension in a 2 - c a t e g o r y (see [ 6 ] ) .

and

> E.

127

There is a bijection between 2-cells

Proposition 1 8.

dl

j

j/B

• B

A

>×

>

~ B

f

",,, × /

~

exhibits

k

f obtained by composition with extension of left

f

along

extension o f

Proof.

and arrows o f spans j

f

j/B

The f i r s t


f/k.

Yoneda lemma.

The 2-cell

if and only if the corresponding

aZong

fd o

By d e f i n i t i o n

o f spans

j

ij.

~ ~X

~ X, t h e r e are b i j e c t i o n s


f/l

~ fd 0

-~ I d I


i

(

~ u*

~ f/A

(

I t is readily checked that

n,e

i f and only i f the corresponding An arrow j:A ju C ~ B , jv

f/A > B/u

m • B/u n )f/A

are a unit and counit for an adjunction f m,n are mutually inverse isomorphisms.//

~ B is said to be fully faithful when, given any 2-cell u

there exists a unique 2-cell

Cf ~ o ~ A v

such that

•

is the

I u

129 u

composite

j

C..~A v

, B.

I t is r e a d i l y seen t h a t

i f and only i f the arrow of spans

@A

~ j/j

j

is f u l l y f a i t h f u l

corresponding to

jh

is an

isomorphism.

Proposition 22.

If

j:A

~ B is ~lly ~ i t h ~ l

a ~ if the 2-cell

J A

~B

f

~

k X

exhibits

k as a pointwise left extension of

f along

j, then

o

is

an

isomorphism.

Proof.

Since

k

i s a pointwise l e f t extension and

, j/j

~A

is an isomorphism,

the composite 2 - c e l l

~J/J

X

exhibits

kj

as a l e f t extension of

fd o along

corresponding 2-cel I

~A

f• kj

exhibits

f

as a l e f t extension of as a l e f t extension of

For a O - f i b r a t i o n pullback of

b

along

By Proposition 18, the

i A

exhibits

d I.

p:E p.

f f

kj along along

~ B and arrow

1A. 1A. b:G

But also the i d e n t i t y So a

2-cell

is an isomorphism.//

B, we denote by

Eb

the

130 A

P

Eb

i G

;l

E

b

~B P

Proposition 23.

Suppose in the diagram

P

E

p

that

is a normal O-fibration.

extension of

f

p

along

The 2-cell

o

exhibits

if and only if, for each arrow

A

ob

~ B

A

kb

exhibits

as a left extension of

fb

^ E/b

B~ the 2-cell

b:G

A

along

p.

p'

~ p/b

BE

p

is a normal 0 - f i b r a t i o n ,

(Chevalley c r i t e r i o n ) . adjoint

as a pointwise left

The f o l l o w i n g square is r e a d i l y seen to be a pullback.

Proof.

Since

k

£'

~ p/B

p

has a l e f t a d j o i n t w i t h u n i t an i d e n t i t y

This property is preserved by pullback:

with u n i t an i d e n t i t y .

The arrow

d1:E/b

~ Eb

so

p'

has a l e f t

has a r i g h t a d j o i n t

A

i~:Eb-----~ E/b

(dual of C o r o l l a r y 6).

right adjoint

p'i^b.

Let

q

So the composite

denote the u n i t o f t h i s adjunction.

checks the equations A

A

b = doP'i b , pdi£' = d I dIZ' p / b ~

and

Pdoq = h .

A - j' ~

diZ':p/b

Eb

o

f

P

~G

....~ Eb

has a

One r e a d i l y

131 A

So

fd0n

exhibits

fb

as a l e f t

extension of

fd o

along

I t follows

dlZ'.

that the composite 2-cell p/b

dl

> G

X exhibits

kb

as a l e f t

extension o f

fd o

A

kb

as a l e f t

extension o f

Proposition 24.

K

that

exhibits

dI

i f and only i f

A

qb exhibits

A

p.//

along

Suppose in the diagram

k

J/g

dl

7, C

A

J

~ B

as a pointwise left extension of

composite 2-cell exhibits

Proof.

fb

along

T a k e b:G ......~ C.

kg

f

along

as a pointwise left extension of

j. fd 0

Then the along

d I.

The following square is a pullback. j/gb

dz

>G

J/g

dl

~ C

I f this is mounted on the top of the diagram of the proposition we obtain the diagram j/gb

d~

~ G

gb

and this composite 2-cell does exhibit

dl

(from the pointwise property of

K).

kgb as a l e f t extension of

fd o along

By Proposition 12 and Theorem 14 we

132 have that

d1:j/g

~ C is a normal O - f i b r a t i o n s (indeed, s p l i t ) .

Proposition 23 applies with

p = dz:j/g----~ C to y i e l d the r e s u l t . / /

So

133 Bibl i o~rap.hy. [1]

E.J. Dubuc, Kan extensions in enriched category theory. Lecture Notes in Math. 145 (1970) 1-173.

[2]

J.W. Gray, Fibred and cofibred categories. Proc. Conference on Categorical Alg. at La Jolla (Springer, 1966) 21-83.

[3]

J.W. Gray, Report on the meeting of the Midwest Category Seminar in Zurich. Lecture Notes in Math. 195 (1971) 248-255.

[4]

A. Kock, Monads f o r which structures are a ~ ' o i n t to ~ i t s .

Aarhus University

Preprint Series 35 (1972-73) 1-15. [5]

F.W. Lawvere, The category of categories as a foundation for mathematics. Proc. Conference on Categorical Algebra at La Jolla (Springer, 1966) 1-20.

[6]

R.H. Street, The formal theory o f monads. Journal of Pure and Applied Algebra 2 (1972) 149-168.

[7]

R.H. Street, Two constructions on lax funotors. Cahiers de topologie et g6om6trie d i f f 6 r e n t i e l l e XIII (1972) 217-264.

ELEMENTARY COSMOI I by Ross S~re~,t

The theory of categories enriched in some base closed category V, is couched in set-theory; some of the i n t e r e s t i n g results even require a hierarchy of settheories. nature.

Yet there is a sense in which the results themselves are of an elementary I t seems reasonable then to ask which are the essential elementary results

on which the rest of the theory depends.

In unpublished j o i n t work with R. Walters,

an axiom system was developed which amounts to Theorems 6 and 7 of the present paper restated in terms of the representation arrow. of the desired theory.

We were able to deduce a great deal

One model f o r this system is provided by the 2-category

V-Cat of small V-enriched categories together with the 2-functor P:(V-Cc~t)o°°P÷V-Cat given by PA = lAMP,V] (= the V-enriched category of V-functors from A°p to V) where V is an appropriate small f u l l subcategory of V. In the case V = SET, V = S e t , there is a universal property of the presheaf construction P which is more fundamental than the axioms mentioned above.

With size

considerations aside this universal property amounts t o , f o r each category A, a pseudo-natural equivalence between the category of functors from B to PA and the category of covering spans from A to B.

Generalizing to a representable 2-category

K, we obtain the d e f i n i t i o n of an elementary precosmos as presented in this paper, the adjective "elementary" is dropped f o r b r e v i t y .

( S t r i c t l y the universal pFoperty

only determines P:K°°°P÷Kas a pseudo f u n c t o r , so we f u r t h e r ask that there should be a choice of P on arrows which makes i t a 2-functor.)

A cosmos is a precosmos f o r

which P has a l e f t 2 - a d j o i n t . Our use of the word "cosmos" is presumptuous.

To J. Benabou the word means

"bicomplete symmetric monoidal closed category", such categories V being rich enough so that the theory of categories enriched in V develops to a large extent j u s t as the theory of ordinary categories.

I t is not modifying this meaning much

135 to apply the term to the constuction P of V-valued V-enriched presheaves for such a V, together with whatever structure is needed to make P well defined.

However, we

frankly do not know how a cosmos in this sense in general gives an example of an elementary cosmos in the sense of the present paper. The problem amounts to a well-known one in the theory of V-enriched categories concerning the relationship between comma objects and pointwise kan extensions.

I f we naively take K to be the

2-category V-C~ then the pointwise l e f t kan extensions given by the coend formula (see Dubuc [6]) are not always pointwise l e f t extensions in K in the sense of the previous paper [22]; the comma objects in K are just not right for extension purposes with a general V.

We conjecture that there is some variant of V-Co~twhich

is an elementary cosmos and provides f u l l e r information on V-enriched categories (see Linton [19] p 228). Despite this degree of ignorance, we believe there is good reason for presenting our work in i t s present form. Although we do have proofs for many of our results (we mention in particular Theorem 35) in the j o i n t work with Walters, the proofs of the present paper are shorter and simpler.

Further, our work can be regarded as a

different approach to the elementary theory of the (2-) category of categories emphasising the role of the set-valued presheaf construction (compare Lawvere [14]). Also the (pre-) ordered objects in any elementary topos provide an example of an elementary cosmos; in particular, the 2-category of ordered-set-valued sheaves on a site is a cosmos. This observation contributes to topos theory in that our theory puts the techniques of adjoint arrows, kan extensions, comma objects, completeness, etc, at our disposal to examine the ever-present ordered objects in a topos.

The

2-category of category-valued sheaves on a site is most probably a cosmos. Finally, we repeat the hope that enriched categories can be shown to f i t

into our present

framework and mention that in a forthcoming paper we w i l l show that CoJt-enriched categories ( t h a t i s , 2-categories) do f i t The notations and results of [12] paper.

in by expanding to double categories. and [22] are f r e e l y used throughout this

136

Table of Contents page §1.

Internal attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudo f u n c t o r i a l i t y of

138

SPL; 2-categories endowed with a t t r i b u t e s ; admissi-

b i l i t y and legitimacy; new characterizations of adjunctions and pointwise l e f t extensions.

§2.

Precosmoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L43

D e f i n i t i o n ; hom-statement of the Yoneda lemma; extension and l i f t i n g

properties

of hom-arrows.

§3.

The representation arrow Definition of

PgA - t

147

YA ; relation between covering spans and a t t r i b u t e s ; denseness

and f u l l y faithfulness of adjunction

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

gA; existence of internal r i g h t extension V f ;

the

gPA; the Chevalley-Beck conditions; formula for l e f t extensions

for absolutely cocomplete objects; degeneracy.

§4.

Extension systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

156

D e f i n i t i o n ; special cases; the extension system Pro~; closed and enriched categories of arrows; tensor product of arrows; the extensional bicategory Pro~(G}.

§5.

Cosmoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161

The comprehension scheme; existence of internal l e f t extension

3f; definition

of a cosmos; l e f t extensions which are always pointwise; small objects and kan arrows; r i g h t extensions; more on 3f.

~6.

Universal constructions

166

"Colimits" in 2-categories; examples including pushouts, opcomma objects, k l e i s l i constructions, l o c a l i z a t i o n s ; relation between eilenberg-moore and k l e i s l i

137 page constructions; recognition of objects of the form

~7.

Examples

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

PK.

I72

Ordered objects in a topos; the pre-Spanier construction; categories.

138 1.

Internal a t t r i b u t e s . Let

category

K be a representable 2-category. SPN(A,B)

of spans from

each span S the span

A to

For objects

B supports a monad M which assigns to

MS obtained as the inverse l i m i t of the diagram ~nA

S

h°mB

where, in t h i s paper, we denote the comma o b j e c t c a l l the M-algebras s p l i t b i f i b r a t i o n s from is s t r i c t l y

A, B, recall that the

A/A = ~A by

A to

homA.

We f r e e l y

B, although a s p l i t b i f i b r a t i o n

the underlying span of an M-algebra.

Before giving the next d e f i n i t i o n we must describe a pseudo functor SPL: K c ° ° p x K °p

For each pair of objects from

A

to

B.

SPL(A,B)

A, B,

is the category of s p l i t b i f i b r a t i o n s

For each pair of arrows

f: A' ~

g o-of: SPN(A,B) lifts

~ CAT.

A,

g: B'---* B, the functor

~ SPN(A',B')

in a canonical way (see Corollary 13 of the l a s t paper) to a functor

SPL(f,g) = g*o-of: SPL(A,B) h For 2 - c e l l s

A~.~A,

g B'~T

f

B,

the natural transformation

k

SPL(o,T): SPL(f,g) ----~SPL(h,k) with action

~ SPL(A',B').

is defined as follows.

For each

c: ME ...... ~ E, the component SPL(~,~) E = T*oEo~

SPL(A,B)

E in

is the composite

g*oEof !°iE°~ g*oMEof=(B/g)oEo(f/A) (B/T)°I°(~/A)~(B/k)oEo(h/A):k*oMEo,h~°c°~ k*oEoh. The routine v e r i f i c a t i o n s required to prove that

SPL is a pseudo functor are l e f t

to the reader.

P: Kc°°p

Suppose we have a 2-functor split bifibration

EA from

a t t r i b u t e s when the functors

A

to

PA.

> K and, f o r each object

A

in

These data are said to endow K with

K, a

139

K(B,PA)

{A,BI-}

SPL(A,B)

h B ~ A

I

, (h*o A

k*o A)

k are f u l l y

faithful

and form the components of a pseudo natural transformation.

Pseudo n a t u r a l i t y o f

{A,BI-}

in

(Pu)*oeA~ ~A, OU n a t u r a l l y in o f the form We c a l l

{A,Blh}

is automatic, but in

u: A - - ~ A '.

f o r some h: B

A

i t means t h a t

Split bifibrations

isomorphic to ones

~ PA are c a l l e d a t t r i b u t e s from

A

to

B.

PA the object of attributes of type A.

A p a i r o f arrows

a: X

to a split bifibration Y.

B

~ A,

E from

Then there is an arrow

b: Y - - * B is said to be admissible with respect

A

to

B when b*oEoa

E(a,b): Y

is an a t t r i b u t e from

X

to

~ PX defined uniquely up to isomorphism by

the c o n d i t i o n {X,YIE(a,b)} = b*oEoa . For 2 - c e l l s

~: a'

8: b

~ a,

are admissible with respect to

) b',

where both the pairs

E, we can d e f i n e a 2 - c e l l

so t h a t the c o n d i t i o n of the l a s t sentence becomes n a t u r a l .

a, b

and

a ' , b'

E(a,~): E(a,b) - ~ Then

"E"

E(a',b')

becomes a

pseudo natural t r a n s f o r m a t i o n in the f o l l o w i n g sense. Proposition 1.

If the pair of arrows

respect to a split bifibration ~ Y, the pair

v: K

au,

bv

E

a: X

from

A

~ A,

to

B

b: Y ~

B

is admissible with

then, for all arrows

is admissible with respect to

E

u: H

~ X,

and there is a

natural isomorphism E(au,bv) i~roof.

~

Pu.E(a,b).v .

The f o l l o w i n g isomorphisms are a l l n a t u r a l : (Pu.E(a,b).v)*oC H ~ v*oE(a,b) o(Pu) oEH ~ v*o(E(a,b) OEx)OU v*o(b*oEoa)ou ~ (bv)*oEo(au

We say t h a t an opspan

X f,

A

~ E(au,bv)*oE H . // 9

Y is admissible when the p a i r

f,g

is

140

hOmA;

a d m i s s i b l e w i t h r e s p e c t to from

X

to

Y.

homA(f,g): Y

or in o t h e r words, when

~ PX

f:

ooadmissiblewhen when

homA

i s an a t t r i b u t e

As a s p e c i a l case o f our above n o t a t i o n , we then have an arrow satisfying

the c o n d i t i o n

{X,YIhomA(f,g) ~ ~ Call an arrow

f/g

×

admissible

~A

the opspan

is an a t t r i b u t e .

1A,f

f/g

.

when the opspan

is admissible.

f,l A

is admissible;

Call an o b j e c t

call

legitimate

A

The f o l l o w i n g i s an immediate consequence o f

P r o p o s i t i o n 21 of the l a s t paper. Theorem 2.

f:

B

An arrow

~A

U: A

~ B is a right adjoint for an admissible arrow

if and only if there is an isomorphism

hOmA(f,l) ~ hOmB(l,u). Recall t h a t an o b j e c t

G is said to be

orthogonal

to an arrow

f: A

~B

when the f u n c t o r K(G,f): is an isomorphism. when i t

gonal to

on o b j e c t s and f u l l y

strongly generating f

then

' K(G,B)

This is an elementary c o n d i t i o n since a f u n c t o r i s an isomorphism

is b i j e c t i v e

said to be

K(G,A)

f

When the arrows

faithful.

when, given

f: A

A class ~ B,

if

G of objects of each

G in

j:

A

> B,

f " A ---~X

J

are each a d m i s s i b l e , we have the

X

A )

B

A/I , X

Yoneda
p2A

PA are legitimate then

PA

with counit an isomorphism.

Take f = YA in Theorem 11. ¥YA =

PYA: P2A

By Theorem 8 we have

h°mpA(h°mpA(YA'l)'l)

~ h°mpA(l"l)

:

YPA"

is a left

150 The counit is an isomorphism since Proposition 13.

A f , X J-g B,

YPA is f u l l y f a i t h f u l . ~

For any legitimate object

A and any admissible opspan

there is a 2-cell dl

f/g

~"B

~PA

A

YA which exhibits Proof.

homx(f,g )

as a pointwise left extension of

YAdo along

d I.

Consider the composite 2-cell dl

yA/ hornx ( f , g )

~'" B

YA

/ ~

A

°mx(f'g) ~ PA

By Theorem 10, the lower t r i a n g l e is a pointwise l e f t extension; 24 of the l a s t paper, the composite exhibits sion of f/g ;

YAd0 along

d I.

homx(f,g )

so the r e s u l t follows.#

Chevalley [ 4 ] .

Rf

as a p o i n t ~ i s e l e f t exten-

Using Theorem 8 we have YA/hOmx(f,g) ~ homx(f,g)*oE A

Functors which are simultaneously O- and

R: B°p

so, by Proposition

~ CAT ;

1-fibrations have been considered by

A O-fibration over a category

B corresponds to a pseudo functor

when the original O-fibration is also a 1 - f i b r a t i o n , the functor

has a r i g h t adjoint

and Benabou-Roubaud [ 3 ]

v

Rf

for each arrow

f

in

8.

Chevalley (and l a t e r Beck

in t h e i r study of descent data) considered a compatibility

condition on the f i b r a t i o n which, in terms of

R, amounts to the following:

151 k

P

, B

hI

f o r each p u l l b a c k

RA

I0

A

i

~ C

Rh

~'RP

riCO ~ RB

RC

f

Rf

i s an isomorphism. This statement s t i l l

makes sense v e r b a t i m in the case where

the r e s u l t i s too stror~.. t h i s case.

The f i r s t

is to require that

levels!

i s a 2 - c a t e g o r y , but

There are two g e n e r a l i z a t i o n s which present themselves in

is a category t h i s i s no c o n d i t i o n on a t two d i f f e r e n t

B

f

should be a O - f i b r a t i o n in

f);

B

(if

B

an i n t e r e s t i n g combination o f f i b r a t i o n s

The second i s to replace the p u l l b a c k by a comma o b j e c t

dl f/g

~ B

A

~C f

in

B (if

B is a category this reduces to the pullback again).

The two generali-

zations are closely related and we have both in a precosmos. Theorem 14.

Suppose

f: A

C,

g: B

~ C are arrows between legitimate objecta

Then the 2-cell

Pd0 PA

~ P(f/g)

PC . . . . . . . . . . . Pg corresponding under adjunction to

Proof.

Proposition 13 yields that

YAdo along

dI.

PB

Ph, is an isomosphism whenever

homc(f,g)

~f, Yd I

exist,

is a pointwise l e f t extension of

By Theorem 3, we have an isomorphism

~mc(f,g)/PA ~ homB(dl,l)/homPA(yAd o,1). Now hOmpA(YAd0,1)

~

Pd0.~mmpA(YA,1) ~

Pd0 ;

and a l s o

hOmB(d1,1)

i s admissible

152 when Vd I

exists (Theorem 11).

So the span on the r i g h t hand side of the displayed

isomorphism above is an a t t r i b u t e and we have an isomorphism

~mpA(homc(f,g),1) ~ ~mp(f/g)(hOmB(d1,1),Pdo). Using Proposition 1 and Theorem 11, we obtain the result.# Theorem 15.

Suppose

and

are legitimate.

E, B, G

p: E

> B

is a normal O-fibration,

b: G

~ B is an arrow,

Then the 2-cell

PE

)" PEb

¥p

p

PB

~

PG

Pb

corresponding under adjunction to the identity 2-cell morphism whenever

Proof.

~p, ~

~ P~.Pp, is an iso-

exist.

From Proposition 13, the 2 - c e l l

l e f t extension of

P~.Pb

YE along

p.

×

exhibits

hOmB(P,1) as a pointwise

By Proposition 23 of the l a s t paper t h i s implies

that the composite A

Eb

~, G

P

E

exhibits

hOmB(P,1)b

>B

as a pointwise l e f t extension of

A

yEb along

A

p.

By Theorem 3,

we have an isomorphism A

A

hOmB(P,b)/PE ~ hOmG(P,l)/hOmpE(YEb,1). But

hOmpE(YE~,l)

~

P~ , and

A

hOmG(P,1)

A

is admissible when Yp

exists.

So the

span on the r i g h t hand side of the above displayed isomorphism is an a t t r i b u t e .

This gives:

153

hompE(hOmB(P,b),l) ~ hOmpEb(hOmG(~,l),~). The r e s u l t follows using Proposition I and Theorem I I . #

Remark.

Consider the case where

K = CAT. The isomorphism of Theorem 14 expresses

i n t e r n a l l y the fact that r i g h t extensions of set-valued presheaves are pointwise. Of course, the arrows can be replaced by t h e i r l e f t adjoints to obtain an isomorphism 3do PA
X

The 2 - c e l l e x h i b i t i n g t h i s ex-

is f u l l y f a i t h f u l . YA PA

X

f: A

ool@nit

154

Theorem 16. f: A

Suppose along

~X

A

is legitimate.

The pointwise left extension

exists if and only if, for all admissible arrows

YA

the pointwise left extension

k

composite

of

f

along

hOmB(J,1) Since

exists.

~ PA

j: A

In this case, k

~ B,

is the

~ X.

YA is admissible (Theorem 8), " i f " is clear.

By the pointwise property, for any

of

lexf

B

Proof.

j

lexf

g: C

Suppose l e x f

exist~

> B, the composite 2-cell dl

YA/hom(j,g)

~' C

YA

A

• PA

X exhibits

lexf .hom(j,g)

kg =

YA/hom(j,g)

span

can be replaced by

Proposition 17. lexf

as the l e f t extension of

Suppose

A

lexf

we should have

f'

If

homx(f,1)

leXA: PA so

If

has a left

~ A.

If

f

by Theorem 3;

along

YA'

and this

is the i d e n t i t y arrow of the l e g i t i m a t e object

then

absolutely cocomplete when i t

= PYK ;

is admissible.

hom(YA,1)/hom(f,1) ~ PA/hom(f,1) condition that f ' --4 hom(f,1).~

deserves special a t t e n t i o n ;

adjoint

homx(f,1).

>X

should be a pointwise l e f t extension of

The p a r t i c u l a r case when f

to be

f: A

f'/X ~

is precisely the

But the

exists.

In order that

Proof.

d 1.

(Theorem8).~

is legitimate and

exists then it is a left adjoint for

adjoint then

leXA

j/g

fd o along

hom(f,1) = YA: A

~ PA.

An object

is l e g i t i m a t e and YA: A

A = PK for some l e g i t i m a t e

A is absolutely cocomplete.

A

A

is said

> PA has a l e f t

K then Theorem 12 gives

For absolutely cocomplete objects we

have the most favourable form of a d j o i n t functor theorem.

155

Theorem 18.

Suppose

A

is absolutely cocomplete.

has a right adjoint if and only if lexf

~ flex A If

Proof.

lexf

~X

f: A

exists are the canonical 2-cell

is an isomorphism.

f~

u

then, composing with

u = hom(l,u) ~ h o m ( f , l ) .

flex A ~yA

An admissible arrow

So

YA ' we have

leXA-~

lexf

exists and is isomorphic to

f /exA. Suppose f l e x A ~ l e x f .

u = leXA.hom(f,1) f

Theorem 16 applies to

as the l e f t extension of

preserves this extension.

Remark.

1A along

f

along

f

A

is the terminal object in

cocomplete categories have terminal objects. c o l i m i t s in

X is: (lexf)G

=

X is obtained from

f: A

> X by evaluating at the terminal presheaf.

the i d e n t i t y functor of

I t remains to prove that

f.

(Theorem 16). H

For categories, the c o l i m i t of a functor

l e x f : PA

to y i e l d

fu = f leXA.hom(f,l) ~ lexf .hom(f,1), and the

But

l a t t e r is the l e f t extension of

A

1: A

A.

Recall that the c o l i m i t of I t follows that absolutely

The formula f o r

lexf

in terms of

fa

Ga ® f a . #

Since we have discussed the case where the representation arrow has a l e f t a d j o i n t , we digress b r i e f l y to point out some t r i v i a l i t i e s i t has a r i g h t a d j o i n t .

An object

A

regarding the case where

is called degenerate when i t is l e g i t i m a t e

and YA: A ..... ~ PA has a r i g h t a d j o i n t . Proposition 19.

A degenerate object

A has the following properties:

(a)

YA: A----+PA

(b)

each admissible arrow with source

(c)

each arrow

some

S: B

Proof. yAt - - ~

B ~

RA

A

has a right adjoint;

is isomorphic to one of the form

hOmA(1,s)

for

~ A.

Let I

is an equivalence;

t

be a r i g h t a d j o i n t f o r

is an isomorphism since

YA"

Recall Theorem 10.

YA is dense, and the unit

The counit 1 - - ~ ty A

is an

156 isomorphism since 2 to obtain that

YA is f u l l y f a i t h f u l . u = t hom(f,1)

This proves (a) and (b). K(B,A) ~ K(B,PA);

§4.

So YA: A ~ PA.

Now we can apply Theorem

is a r i g h t a d j o i n t f o r admissible

f: A

~ B.

Using (a) we have an equivalence of categories

(c) follows from t h i s and Theorem 8.#

Extension s£stems. Monoidal (= m u l t i p l i c a t i v e ) categories [ 1 ] , [7] have been generalized by

Benabou

[ 2]

to bicategories.

We now make the corresponding generalization for

closed categories. An e x t e n s i o n system

E

(i)

objects

(ii)

f o r each pair

consists of the f o l l o w i n g data:

A, B, C . . . .

A,B

;

of objects, a category

called arrows and whose arrows are called 2 - c e l l s (iii)

for each object

(iv)

f o r objects

A, an arrow

f o r arrows

whose objects are

;

YA c E(A,A);

A, B, C, a functor [ , ] : E(X,A) °p × E(X,B)

(v)

E(A,B)

f ~ E(X,A),

g c E(X,B),

h Vf,g: [ f , g ] ........ [ [ h , f ] , [ h , g ] ] ,

~ E(A,B) ;

h c E(X,C), 2 - c e l l s

f × : YA

~ [f'f]'

mg: g - - - ~ [ y x , g ]

,

the l a t t e r an isomorphism; such that the following axioms are s a t i s f i e d : ESI.

h f Vf,g, × , ~g

are natural in t h e i r subscripts and extraordinary natural in

t h e i r superscripts; ES2.

the f o l l o w i n g diagrams commute

157 [f,f]

[f,g]

v• [[g,f],[g,f]]

\ / YA'

v ,[[f,f],[f,g]]

[f,g]

v

YX ~ [[yx,f],[yx,g]]

S

[i,~] [YA,[f,h]] h)

[f,g]

[~,i]

[f,[YX,g]]

k [[k,f],[k,g]]

[ l , v g]

[[h,f],[h,g]]

~)[h,k]~

[[k,f],[[h,k],[h,g]]]

[[ [ h , k ] , [ h , f ] ] ,[ [ h , k ] , [ h , g ] ] ]

;

[wh,1]

the composite function

ES3.

[-,g]

E(A,A) (×g,1) E(a,A)([g,g],[f,g])

E(X,A) (f,g)

E(A,A) ( y A , [ f , g ] )

is a bijection. Special cases.

An extension system with precisely one object is a closed

1)

category.

This does not quite agree with the d e f i n i t i o n of closed category appeaHng

in [ 7].

Reference to a category of sets has been eliminated as required for exam~e

by Lawvere [ i ~ X @-

p12. Also, a monoidal category such that each of the functors

has a r i g h t adjoint is closed in our sense (compare t h i s with [7 ] Theorem

5.8 p493).

Note that, for any extension system

E,

E(A,A)

becomes a closed

category. 2)

A bicategory

system with

[f,h]

B in which a l l r i g h t extensions e x i s t yields an extension

taken as the r i g h t extension of

h along

f.

Such a bicategory

we call an extensional bicategory (also called "closed bicategory" by some authors). 3) k: A

Suppose E is an extension system such that, for a l l > C, there exists an arrow

k ~f:

[k ® f , - ] Then

D ~

f: D

~ A,

> C and a natural isomorphism [k,[f,-]].

E becomes an extensional bicategory with composition given by ®.

158

In a precosmos K, suppose the arrow

h: C

X

is admissible.

The

composite 2-cell

d~ f/g

)

B

-~,

hom(h,f) induces an arrow of spans f / g

Provided the source and

~ homx(h,f)/homx(h,g).

target of the l a t t e r arrow are a t t r i b u t e s , we obtain a 2-cell ~,h:

h°mx(f'g)

....... h ° m p c ( h ° m x ( h ' f ) ' h ° m x ( h ' g ) ) "

The next two theorems can be proved using Theorems 6 and 7. Theorem 20. (i)

An extension system

Prof

is defined by the following data:

the objects are the legitimate objects

A

of

K

PA

for which

is

legitimate; (ii)

Prof(A,B) = K(B,PA); g c Prof(X,B), take

f ~ Prof(X,A),

(iii) f~r (iv)

YA is the representation arrow;

(v)

~, X

Theorem 21. data

For each object

hOmA, w, X

VK is

where

K(K,-): L

are as previously defined and of

Prof

enrich the category

Prof(K,K)

~ CAT

K

g © f:

2-cell:

is the isomorphism of Theorem 8.#

and each legitimate object

K(K,A)

lifts to a 2-functor

L

Given arrows

f: A

~ X,

A

of

K, the

with the structure of a VK-category,

with its closed category structure.

Indeed, the 2-functor

........>' V K - CAT.~

Freyd's tensor product of functors [ 8 ] a precosmos.

~

I f , g ] = k~mx(f,g);

g: B

p120 can be carried over to arrows in > PA,

t h e i r tensor product

B - - - ~ X , when i t e x i s t s , is defined as the l e f t extension as e x h i b i t e d by a

159

{A,BIg }

, B

t

FI

A

If

h

A f~x

~X

is an admissible opspan then there is a b i j e c t i o n

~C

{A,BIg}"

{A,BIg}

~ B

I FI A

h

~

~

\

~ X

{A,BI- }

/

{A,Blhom(f,h)}

Using the l e f t extension property of f a i t h f u l n e s s of

g®f

g ® f

on the l e f t hand side and f u l l y

on the r i g h t , we obtain a natural b i j e c t i o n between

2-cells: g®f

g

h Proposition 22. lexf

exists.

Suppose

hom(f,h)

A is legitimate and f : A

Then, for all

g: B---~ PA,

g ® f

> X is an arrow for which exists and there is a natural

isomorphism g ® f

Furthermore, if

~

(lexf)g.

X is legitimate then, for all arrows

h: C

Xj there is a

natural isomorphism

homx(g ® f , h ) Proof.

{glA, B} = YA/g ;

By Theorem 8,

pointwise property of

~

lexf.

hOmpA(g,~mx(f,h)). so the f i r s t

isomorphism follows from the

When X is l e g i t i m a t e the l e f t hand side of the

second isomorphism e x i s t s , and

{B,Clhom(g ® f , h ) } so the arrow

~

hom(g ® f , h )

g ®f/h

~

(lexf)g/h

~

g/hom(f,1)h

has the defining property of

~

g/hom(f,h) ;

hom(g,hom(f,h)).#

160 Let

G denote a class of objects of

G-cocomplete when, f o r each exists.

I f each

X G in

is

G in

G in

An o b j e c t

G and each arrow

X of

f: G

K

> X,

is c a l l e d lexf:

PG

~X

G is l e g i t i m a t e Theorem 16 i m p l i e s :

G-cocomplete i f and o n l y i f ,

G and a l l arrows

K.

f: G

f o r a l l admissible arrows

j:

G

~ X, the pointwise l e f t extension o f

f

~ B with along

j

exists.

Theorem 23. for each

Let

G

objects in

in G

G G.

be a class of objects of The restriction

K is in

K(K,A) becomes

G-cocomplete then

of

such that

PG

is G-cocomplete

of the extension system

Prof

to

is an extensional bicategory with composition given by tensor product,

In the notation of Theorem 21, if

Suppose

Prof(G)

Prof

G

A

and

is legitimate and

a tensored VK-category. ~

G is as in the l a s t theorem o n l y regard i t as a f u l l

K, and l e t

I : Gc°

> Prof(G)

sub-2-category

denote the pseudo functor which is the i d e n t i t y

on objects and which is given on hom-categories by the functors K(A,B)oP the isomorphisms

horn(-,1)

l ( g f ) ~ Ig ® I f

,

; K(B,PA)

=

Prof(A,B) ;

IIA = YA are canonical.

The f o l l o w i n g p r o p o s i t i o n extends a theorem o f Benabou on profunctors to our setting.

Proposition 24. Indeed, if Eroof.

Arrows in the image of

f: A

The 2 - c e l l

hom(f,l) ®hom(l,f) g: B .... i C in

B

is in

G

then

× ' : hom(f,1)

I

have right adjoints in

Prof(G).

homB(1,f)in

Prof(G).

hOmB(f,1 ) --~

~ hom(hom(1,f),#Dm(1,1))

hom(l,1) : YB"

What we must show is t h a t , f o r a l l arrows

Prof(G), the composite 2 - c e l l B

corresponds to a 2-cell

~m(1,f)

y~~~om

C

~ A

(f,~)

161 exhibits

g ®hom(f,1)

as a r i g h t extension of

g ®YB

along

are in an extensional bicategory so the r i g h t extension of hom(l,f)

hom(1,f).

g ®YB = g

But we

along

is

hom(hom(l,f)g)

~

Pf.g

~

g ®horn(f,1),

as required.g Lawvere [18] has viewed non-symmetric metric spaces as enriched categories and found a condition which can be stated in our context and reduces to Cauchy completeness for metric spaces. when each arrow

A--+X

arrow of the form

§5.

An object

X of

Prof(G)

in

homx(f,l)

Prof(G)

is called Cauchy(-G)-complete

with a r i g h t a d j o i n t is isomorphic to an

(compare Proposition 19).

Cosmoi. Another way of expressing the precosmos condition is:

for all objects

A, B,

the composite functor K(B,PA) is f u l l y f a i t h f u l .

{A~BI-}

>

SPL(A,B)

und

,

SPN(A,B)

We now show that r e f l e c t i o n s with respect to t h i s " i n c l u s i o n "

functor are j u s t certain l e f t extensions. Theorem 25. and that hOmA(1,u)

(Comprehension scheme).

u: S

along

>

is coadmissible.

A

v

Suppose

(u,S,v)

is a span from

The left extension

k: B

~ PA

of

exists precisely when the left adjoint of the functor

{A,BI-}: K(B,PA)

,SPN(A,B)

exists at the span (u,S,v). Proof.

A to B

For any arrow

h: B S

"•,

, PA, there are bijections S

pullback property

{A,BIh}

\

/ V*o{A,BIh}

162

A/u do///

~

V

S

dl

~ B

/

horn (

~

h PA k

So we have the bijection between 2-cells

a

and 2-cells

B~

i f and only

$T~A h

f

i f we have the b i j e c t i o n between arrows of spans Proposition 26.

Suppose the span

which is an admissible opspan.

(Lax D o o l i t t l e ) .

COV(A,B)

A

from

A

to

B

T.//

has an opcomma object

is legitimate then the left adjoint in the

(u,S,v).

last theorem exists at

}Woof.

If

(u,S,v)

and 2 - c e l l s

The r e f l e c t i o n for the inclusion functor

>SPN(A,B) is obtained by forming the comma object of the opcomma object.

When A is l e g i t i m a t e , AZ~t(A,B) c COV(A,B). The condition of the theorem ensures that the r e f l e c t i o n lands in Theorem 27. Pf:

PA

Suppose

.....~. PB

extension of

Proop°.

A~Jt(A,B) ~ K(B,PA).#

A is legitimate and

has a left adjoint hOmB(1,f ) along

By Theorem 5,

3f: PA

> PB

YA

exists.

3f

~

hom(1,f)

f: A

~ B is coadmissible.

The arrow

if and only if the pointwise left

In this case, there is an isomorphism

lexhOmB(1,f ).

is admissible and

hom(hom(1,f),l) ~ Pf.

The

r e s u l t now follows from Proposition 17.// Under the conditions of the l a s t theorem, we have an isomorphism

A

which exhibits case where

3f

YA

~PA

as a pointwise l e f t extension of

hom(l,f)

B is legitimate this is the more f a m i l i a r diagram

along

YA"

(In the

163

I

YA

A

~PA

B

>

PB

YB which expresses the pseudo naturality of

y: i ...........> 3.)

Theorem 25 and Proposition 26 provide conditions under which certain l e f t extensions e x i s t .

I f these results are to be used to produce 3 f , we need to know

that the extensions are pointwise (by Theorem 27).

In a cosmos we shall see t h a t ,

with size conditions, l e f t extensions of arrows into objects of the form necessarily pointwise. (Proposition 26).

PK are

At best t h i s approach requires opcomma objects in

K

In order to continue t h i s approach and to present another approach

which does not depend on opcomma objects (but does s u f f e r from size problems in the non-uniform case) we require

K to be a cosmos.

A cosmos is a precosmos for which the 2-functor

adjoint

P*: K

P: Kc ° ° p

>K

has a l e f t 2-

This means of course that there is a 2-natural isomorph~m

.~. ~ , o o p

K(A,P*B) °p ~ K(B,PA).

We denote the composite of this isomorphism with the functor {A,BI-}*: K(A,P*B) °p and the attribute

{A,B]IA}*

split bifibration

E from A to

B to

A in

K°°, so that

from A to

P*, ~

B in

{A,BI- }

by

.........~ SPL(A,B);

B when A = P*B is denoted by ~B" K becomes a s p l i t b i f i b r a t i o n

endow Ko°

with attributes.

A

E* from

Indeed, Ko°

is

also a cosmos. All previous precosmos theory dualizes. I t is consistent with our notation to write corresponding to

E(a,b): Y

E*(a,b): X

.>. P*Y for the arrow

~ PX in the situation of Proposition 1.

In particular,

for an admissible opspan X a A b y, we have hom*(a,b): X ... ~ P*Y defined by {x,YJhom*(a,b)}*

~

a/b.

164 An arrow j : A ~ B will be called lax-fibre small when A and B are legitimate and there exists a strongly generating class G of legitimate objects such that, for all arrows g: G i B with G in

hOmB(1,g):B Theorem 28,

~ P*G and hom~(1,do): A If

j: A

G, j/g

, P*(j/g)

is legitimate and both

are coadmissible. j

, B is lax-fibre small then any left extension along

of the form

J

A

~ B

PK

is pointwise. Proof.

Take

G as above.

We w i l l

applying Theorems 11 and 14 in

K° °

prove t h a t (c) o f Theorem 3 is s a t i s f i e d .

By

we o b t a i n an isomorphism

P*d 1

P*C ....

>- P*(J/g)

P'B,

where ¥ g, ¥*d o are the l e f t a d j o i n t s P'g, P*d o, r e s p e c t i v e l y .

r

p*j

Now apply

(in

K(K,-) °p

>

P*A

K; t h a t is r i g h t a d j o i n t s in and use the adjunction

Kc ° ) of

P* --~

P to

obtain an isomorphism

K(G,PK)

K()PK)

K(d~,l) ~ K(j/g ,PK)

K(j ,i)

~-~K(A}PK)

where the vertical arrows are the right adjoints of

K(g,1), K(do,1). The diagram

obtained by replacing the arrows by their left adjoints (when defined) also commutes up to isomorphism. But the value of the left adjoint to the value of the left adjoint to the left extension of

K(dl,1)

fdo along dl. ~

at

K(j,I)

at

fd o is isomorphic to

f kg.

is

k.

So

So kg is

165 An object f: A

A is called small when YA: A

~ B is called kan when A

object of the span (f,A,y A) Corollary

29.

adjoint

f

Proof.

If

f: A

B to

PA exists and is an admissible opspan.

~ B is a kan arrow then

Pf: PB

~ PA has a left

~f.

PA is l e g i t i m a t e ,

exists by Theorem i i .

An arrow

is small, B is l e g i t i m a t e , and, the opcomma

from

and a right adjoint

Since

~ PA is l a x - f i b r e small.

bomB(f,1): B

~ PA is admissible;

so ~ f

By Proposition 26, the l e f t a d j o i n t of the functor

{B,PA]-}: K(PA,PB) ----~SPN(B,PA) exists at

(f,A,YA).

along YA exists.

By Theorem 25, the left extension

k: PA

By Theorem 28, this extension is pointwise.

~ PB of

hOmB(1,f)

So, by Theorem 27,

k = 3 f.# Left extensions in a precosmos have been discussed. cosmos we obtain results about r i g h t extensions. of arrows into objects of the form

P*K.

r i g h t extensions of arrows into

PK.

Suppose

j: A

The results about l e f t extensions

PK dualize to results about r i g h t extensions of

arrows into objects o f the form

Theorem 30.

From d u a l i t y present in a

More s u r p r i s i n g l y , we can prove results about

is an arrow with

~B

is coadmissible then each arrow

f: A .........~ PK

served by any arrow of the form

Pg: PK - - ~ PK'.

B

legitimate.

If

has a right extension along (If

~P

w

hOmB(1,j) j

pre-

is representable then

the converse of the last sentence holds.)

Proof.

By the dual o f Theorem 11,

existence of a l e f t a d j o i n t P*--~

¥*j

hom~(1,j)

to

coadmissible is equivalent to the

P ' j : P*B .............> P*A. From the 2-adjunction

P we have a commutative square

K(j ,I) K(B,PK) . . . . . . .

K(K,P*B)°P

~ K(A,P,K)

,,

~ K (K,p*A)°P

K(1,P*j) °p

166

But

K ( I , P * j ) °p ---4 K ( I , ~ * j ) °p, so

f : A---~PK

K(j,I)

has a r i g h t extension along

the n a t u r a l i t y of the above square in Corollary 31. hOm~A(1,Pf)

Suppose

j. K.

Then

So each arrow

The preservation property follows from We leave the converse to the reader.#

is an arrow such that

f: A --~ B

is coadnissible.

has a r i g h t a d j o i n t .

Pf: PA --+ PB

is legitimate and

PA

has a left adjoint

3f: PA--+PB.

Proof. along by

Apply the theorem to the arrow Pf

Pf

gives an arrow

we have

3f ~

3f:

The r i g h t extension of

I : PB - ~ P B

PA--~PB, and since this r i g h t extension is preserved

Pf.# f : A--~B

Corollary 32.

Suppose

Pf: PB ---~PA

has a left adjoint

§6.

Pf.

is an arrow in a uniform cosmos. 3f

and a right adjoint

Then

~f.#

Universal constructions. Whilst 2-CAT with

PA =

[A °p, Cat]

is not a cosmos in our sense, we do have

pointwise l e f t extensions in the sense of Dubuc ( [ 6 ] with tion arrows.

V = CAT) and representa-

So we can carry over the d e f i n i t i o n of " c o l i m i t " given in ~3 to 2-

categories themselves.

Since ~

extension of an arrow into

is a strong generator f o r

2-CAT, any pointwise l e f t

K along a representation arrow should be constructible

from the tensor product of arrows Fe7 :~---~[A°P,coat] we i d e n t i f y 2-functors out of

~

with arrows F:

A---~K.

If

with objects of the target 2-category, the

d e f i n i t i o n of tensor product (see ~4) amounts to the f o l l o w i n g . Given 2-functors

o:

A°p --+Ca~t, r: A - - ~ K , the object

@~ ?

of

K is

defined up to isomorphism by an isomorphism K(e 0 r ,K) ~ natural in K.

[A°P,cowt](e,Er-,K])

Note t h a t we could also ask f o r 2 - n a t u r a l i t y in

K in which case we

167 say

O ©F

is 2-enriched; 1)

Examples.

If

@© F = lim F •

A

t h i s i s automatic when

i s a category and

1 ~

Take

A

object

Let

9: A u\v

3)

fall

F: A

> K.

~

then

So pushouts

as s p e c i a l cases.

0: A°p

> £a~t denote the opspan ~ 6o ~ - ~ - -~i - ~ in Ccut. A u v i s a span A ~ S > B in K and @ ® ? i s the opcomma

>K

o f t h i s span.

Let

one object

A denote the simplicial category.

*

monad (A,s) @: A°p

@ i s the constant f u n c t o r at

t o be the t h r e e o b j e c t category w i t h two n o n - i d e n t i t y arrows thus:

2----~I '

functor

is representable.

j u s t the usual c o l i m i t o f the f u n c t o r

c o e q u a l i z e r s , coproducts a l l 2)

K

and A(*,*) = A.

A 2-functor

in the 2-category

> Co~t is a monad in

K Oxt.

Take A to be the 2-category with

F: A

; K can be i d e n t i f i e d with a

(using the language of [21]).

So a 2-functor

We take this monad @ to be the monad called

AA by Lawvere in [15] pp150-1. Then [A°P,cxt](@,[F-,K]) So @®F in

K°P) 4)

~

[A°P,ccut](@,(K(A,K),K(s,1)))

is the construction

As

of k l e i s l i algebras

for the monad (A,s)

in

K (again see [21]).

Take

~

K(A,K) K ( s ' l )

~

(= construction of algebras

A

to be the 2-category w i t h two o b j e c t s and n o n - i d e n t i t y arrows and f 0~1. A 2 - f u n c t o r £: A ~ K is just a 2-cell A~o~'~B

2 - c e l l s thus:

v

in

K.

K(As,K).

We leave i t

to the reader t o f i n d the 2 - f u n c t o r

(that is, a natural transformation)

such t h a t

g

9: A° p - - - * C a t

@ ® F = B[o - I ]

w i t h the f o l l o w i n g

universal property: - t h e r e i s an arrow and, i f

B

u n i q u e l y as We c a l l 5)

~K B

B

~ B[a - I ]

such t h a t the 2 - c e l l

i s an arrow such t h a t > B[a - z ]

B[a - 1 ]

ka

p@ i s an isomorphism,

i s an isomorphism then

k

factors

~ K.

the l o c a l i z a t i o n

of

B at

~,

There i s a dual n o t i o n o f cotensor which g e n e r a l i z e s " l i m i t " .

As examples

we o b t a i n p u l l b a c k s , e q u a l i z e r s , p r o d u c t s , comma o b j e c t s , e i l e n b e r g - m o o r e c o n s t r u c t i o n s and o p l o c a l i z a t i o n s w i t h i n a 2-category.~

168

Theorem 33.

A representaSle 2-category with a 2-terminal object admits the follow-

ing constructions: (a) finite limits; (b) the construction of algebras; (c) oplocalization. These constructions are all 2-enriched. Proof.

I t is well known that pullbacks and a terminal object imply a l l f i n i t e

limit~.

Part (b) is intimated by Gray ~ 0 ] and a proof w i l l appear in his f o r t h -

coming book [11];

we also discussed i t in [12]. f A~ B .

For part (c), take a 2-cell structure on

isoB

......... ,

hom B

homB such that, for a l l

K, the f u l l image of the composite

K(K,iSOB)

........ K(K,hOmB)

is precisely the f u l l subcategory of which are isomorphisms.

One r e a d i l y checks that oe

Using l i m i t s and the category object

g (= ~B in [22] Proposition 2), one r e a d i l y constructs an arrow

~

K(K,B) z

K(K,B) ~ consisting of the arrows

K

> B

Then form the pullback

e

i

~ ii°B

A ......

~

homB

is the universal arrow into

A with the property that

is an isomorphism.# Familiar techniques prove the following.

T~orem 34.

Suppose @: A°p

exists. If ~: K

~ Cat,

F:

A

~ K are 2-fun~tors such that 9 ® F

> 11 is a 2-functor with a right 2-adjoint then o@(~r)

~

~(O@F).t/

169 As promised in the i n t r o d u c t i o n of [21] we shall prove the r e s u l t which relates the eilenberg-moore construction to the i n t e r n a l sheaves of a c e r t a i n type on the k l e i s l i construction.

Recall that

eilenberg-moore object

E (= As )

([12]§3.3), f o r a monad (A,s)

in

K, the

is defined by the condition that there is a diagram 1

u

E

flu

A

• A

s such t h a t

(u,y)

is the universal s - a l g e b r a .

Dually, the k l e i s l i o b j e c t

K (= As)

is defined by the c o n d i t i o n t h a t t h e r e is a diagram A

s

~A

K

J,K 1

such t h a t

(j,,))

is the universal s - o p a l g e b r a .

K is the co-eilenberg-moore object in Theorem 34 y i e l d s that (PA,Ps)

in

K;

K°°°p.

Note t h a t So i f

(A,s)

is a comonad and

P has a l e f t 2 - a d j o i n t ,

PK is the co-eilenberg-moore object for the comonad

that is: PK ,,

1

)-PK

PA

>PA Ps

is the universal Theorem 35. that

(A,s)

P: Kc°°p

Ps-coalgebra.

Suppose >K

K

is a representable 2-category endowed with attributes such

has a left 2-adjoint.

Suppose

is a monad for which the kleisli object

the eilenberg-moore object for

(A,s)

K

A

is legitimate and that

exists in

Ko

An object

if and only if there is a pullback

E

is

170 E

, PK

Pj

A i°PO0~,

For any arrow

a: X - - - ~ A

YA

~PA

we have b i j e c t i o n s A/a

~X

X

el

Yoneda a


X

~,,

PA

Ps

One r e a d i l y sees t h a t coalgebra; (Pj,/~) x: X

(a,~)

is an s - a l g e b r a i f and only i f

(YAa,O)

is a Ps-

the diagrams j u s t t r a n s l a t e n a t u r a l l y through the b i j e c t i o n s .

But

is the universal Ps-coalgebra, so in t h i s case t h e r e e x i s t s a unique arrow > PK such t h a t

g j . x = yA a

and

~.x

= O.

In o t h e r words, we have a natural

bijection

X

a

~A

A

X

x

A

~PK

~PA YA

between such s-algebras and such commutative squares. U The above theorem represents o n l y one amongst many ways in which the various "limits"

are r e l a t e d .

One would l i k e to o b t a i n the k l e i s l i

eilenberg-moore since the l a t t e r

does not r e q u i r e c o l i m i t s .

c o n s t r u c t i o n from the There are two approache~

171 The f i r s t

is to take the l e f t a d j o i n t

A J >K k

E where

k

The monad (A,s)

eilenberg-moore object t e l l us that object f o r

j

u

and factor i t

is " b i j e c t i v e on objects";

but

The second is a general one which applies to any

gives a comonad (PA,Ps)

and we can form the

X f o r this comonad. Now we need a "recognition theorem" to

X is equivalent to (A,s).

~ E to

is f u l l y f a i t h f u l and

when do such f a c t o r i z a t i o n s exist? "colimit".

f: A

PK f o r some K.

Then K should be the k l e i s l i

The f o l l o w i n g "recognition theorem" does not seem good enough.

One would hope to be able to generalize the work of Mikkelsen on complete atomic boolean algebras in a topos to improve the r e s u l t .

For enriched categories the next

theorem appears in [ 5] p189, Theorem 36.

In a precosmos suppose

K is legitimate and suppose Z: K - ~ lexz

an admissible fully faithful~ dmnse arrow such that by

homx(Z,1).

Proof.

Then

X

is equivalent to

X is

exists and is preserved

PK.

We have the f o l l o w i n g l e f t extensions.

K

z

-X

K

YK

~PK

/

YK

By Proposition 17, sion. z

But

is 1.

z

f ~

lexz

lexz --4 hom(z,1) and so lexz preserves the f i r s t l e f t exten-

lexz.y K~ z So

~"~/hom

and

z

is dense so the l e f t extension of

lexz.y K along

lexz.hom(z,1) ~ 1. A s i m i l a r argument proves hom(z,1).lexz ~ i.~

172 ~7. I)

Examples

Ordered objects i n a topos A span (u,R,v)

given arrows f o r the f u l l

from

A

to

w,x:C --~ R, i f

B

in a category

uw = ux

subcategory o f

and

A

vw = vx

an ordered s e t .

An ordered object in

span (do,A1,d I)

is a r e l a t i o n ;

for

A

then

w = x.

A

A

to

t h a t an arrow f: A ~B

is a category o b j e c t

A

(d1,Az,do).

fo = f .

Since t h i s

f

Ao.

(I,A,1).

Given ordered o b j e c t s

is uniquely determined by

is

f o r which the Objects Write

A,B,

f

A

A°p

we say

is order preserving when there e x i s t s a f u n c t o r i a l

f:Ao --~ Bo

with

There i s a t most

the span i s then c a l l e d an order on

w i t h the reverse o r d e r

RcZ(A,B)

B; t h a t i s , RcZ(A,B)

are regarded as ordered o b j e c t s v i a the d i s c r e t e o r d e r

Ao

Write

SP~V(A,B) c o n s i s t i n g o f the r e l a t i o n s .

one arrow o f spans between any two r e l a t i o n s from

of

is c a l l e d a r e l a t i o n when,

arrow

we o f t e n w r i t e

f : A ---~ B. The f o l l o w i n g d e f i n i t i o n

is e q u i v a l e n t to t h a t of Lawvere-Tierney [ 1 7 ] .

An

(elementary) topos is a category

E which has f i n i t e limits and, for each object

has an o b j e c t

EA from A to

PA and a r e l a t i o n

A,

PA satisfying the following

"power-object" condition: -given a such t h a t

relation

R from

A

to

B, there e x i s t s a unique arrow

R ~ h*~EA.

Given an arrow

f: A-~B,

define

In t h i s way we o b t a i n a f u n c t o r

f: PB --~?A P: E°p--~ E.

P A x PA.

by the condition

The composite span

PA also comes equipped with a p r o j e c t i o n i n t o to

h: B ~ P A

A

~B°f = (Pf)*°EA. * EAOEA

from

PA

and so leads to a r e l a t i o n from

A

This r e l a t i o n corresponds under the p o w e r - o b j e c t c o n d i t i o n t o an arrow

^: P A x p A - - + p A .

The e q u a l i z e r (d° 1

dl

CA ~

defines an order

(do,CA,dl)

^ ; P A x PA ........... ~ PA proj l -

on

PA.

Henceforth we s h a l l w r i t e

PA

for this order-

ed object. Let

to

K denote the f u l l sub-2-category of

CAT(E) consisting of the ordered

173

objects.

Each of the categories

K(A,B) is an ordered set.

The essential

property of the order on PA is that r B

PA i f and only i f

A\

R~ S

s

where

r,s

correspond to the r e l a t i o n s

condition.

Note that

particular,

~A is

K AI

R,S

from

A to

B

under the power-object

is a representable 2-category with f i n i t e 2-1imits. with the order

In

(~1,AIoAI,~I).

There is a kind of l a x 2 - 1 i m i t which we did not mention in ~6 but which can be constructed in any representable 2-category with f i n i t e 2-1imits.

Given any ordered

pair of arrows f , g : A~ ~B, t h e i r s~equalizer (Lambek [13]) is a universal diagram of the form

k E

~A

A

,B

,

f A construction

for the subequalizer

E

is t h e p u l l b a c k

5

~ #B

A

BxB

~

We now wish to extend our functor For

A

in

K, l e t

.

P: E°P--~K

PA be the subequalizer of

to a 2-functor

P: Kc°°P--*K.

Pdl, Pdo:PA0 --~PAI.

inc PA

~ PA0

inc I

~

PAO

IPdo ~ PA1

.

Pd1 The 2-functor structure o f Suppose A,B t i o n from

A

to

P is induced using the enriched " l i m i t " property.

are objects of

K.

An ideal from

B which is a r e l a t i o n .

of

SPW(A,B) consisting of the ideals.

Bo

in

E, there is a unique order

Let

A to

IdX(A,B)

B

is a s p l i t b i f i b r a -

denote the f u l l subcategory

Given a r e l a t i o n (u, R0,v)

( d o , R l , d l ) on

Ro

from

A0

such that (u,R,v) is a

to

174

relation from A to

Proposition#7.

B in

K.

So Id£(A,B)

is a subcategory of

ReZ(Ao ,Bo)-

The oomposi~ f~ctor

K(B,PA) K ( I ' S n ~ K(B,PAo)c

~ E(B0,PAo)~ RcZ(Ao,Bo)

induces a~ equivalence of categories

K(B,PA) ~ ~dZ(A,~).

Proof. Each functor in the composite is clearly f u l l y faithful so i t remains to show that the composite is surjective up to isomorphism onto the ideals. Proposition 12 of the last paper, a relation A to

R from Ao to

By

Bo is an ideal from

B precisely when BIoR ~ R and R°AI ~ R (the extra conditions are diagrams

in the ordered set RP-Z(Ao,Bo) and hence automatically commute). Let correspond to

R under

E(B0,PAo) ~ ReZ(Ao,Bo).

translate to the following conditons on ~Bo

> PAo

Bo

1r

~

B0

~A 1 ~ R

r

BL

[

BIoR ~ R,

r.

dI

do

The conditions

r:Bo--~PAo

rl

@A0

,

PAo

r

I Pd0

~I

~ PAl Pd:

The f i r s t of these says precisely that second says precisely that

r: B --+PAo

is order preserving and the

r: B--+PA 0 factors uniquely through inc to y i e l d

r: B --~ PA.~ An arrow of spans between ideals from of s p l i t b i f i b r a t i o n s from an ordered set.

B is automatically a homomorphism

B since the homomorphism axioms are diagrams in

So Id~(A,B) is a f u l l subcategory of

is a sub-pseudo-functor of of

A to

A to

Id~, the inclusion

SPL(A,B).

SPL (see early @1). With t h i s pseudo f u n c t o r i a l i t y

Id~(A,B) --~R~(Ao,B0)

becomes pseudo natural in

composite functor of Proposition 37 is c l e a r l y pseudo natural in that the equivalence

Theorem 38.

K(B,PA) ~ ~cL~(A,B)

The 2-categoz~

Furthermore, I ~

K

is pseudo natural in

of ordered objects in a topos

the attributes are precisely the ideals.

A,B.

A,B.

The

I t follows

A,B.

E

is a cosmos in which

175 Proof.

We have already seen that the f u l l y f a i t h f u l functor

is pseudo natural in ideal

EA from

A to

A,B.

Set B = PA and evaluate at the i d e n t i t y to obtain an

PA which consequently endows K with a t t r i b u t e s .

We have

immediately that the attributes are the ideals so the precosmos condition is s a t i s fied.

One r e a d i l y v e r i f i e s that

P has a l e f t 2-adjoint

P*

given by

p.A = (pAoP)°P.# For an ordered object

A, the s p l i t b i f i b r a t i o n

r e l a t i o n and hence an ideal. Corollary39.

@A from

A to

A is a

So K is a uniform cosmos.

For each order-preserving arrow

f : A --+B, the order-preserving arrow

Pf: PB --*FA has both a left and a right a~oint.#

2)

The pre-Sp~nier constuction as a

P.

The 2-category Simp and 2-functor

P presented here are taken from unpublished

work of Day-Kelly on categories l i k e categories of topological spaces. A function

f : X--+Y

between sets

through the one-point set i . functor

I I: A-+Set

X,Y

A simple category is a category

SC2. the image of

I la, b

SC3. there is an object

is in

a, a' in A, we write

A(a,a').

A together with a

satisfying:

SC1. I la,b: A(a,b) - - ~ S e t ( l a l , l b l )

For objects

is called constant when i t factors

A functor

contains a l l the constant functions;

a of a ~ a'

f : A--+B

is an inclusion of sets;

A with when lal

lal ~ o. =

la'I

and

Iial:

la l---~la'l

between simple categories is simple when the

following diagram commutes. f A

Let

~B

Simp denote the 2-category whose objects are simple categories, whose arrows

are simple functors, and whose 2-cells are natural transformations. A,B

in

Simp, the category Simp(A,B)

Then, f o r each

is an ordered set; indeed, there is a

176 f natural transformation

A~'+-~ B

p r e c i s e l y when

fa ~ ga

for all

a

in

A.

g Pullback in Simp is t h a t of CAT and Simp is a representable 2-category. simple functors

A--~r D,

BS

D,

The objects are p a i r s (a,b) with arrows are pairs

the simple category a

in

A, b

(m,B): (a,b) --~ ( a ' , b ' )

in

where

r/s

B and

For

is defined as f o l l o w s . ra ~ sb

~: a - - ~ a ' ,

in

D.

~:b - - ~ b '

The are in

A ,B

r e s p e c t i v e l y , and the f o l l o w i n g square commutes. ra --

IIa

]+ sb

ra' I-T-~_, sb'

[al

Let

A

An o b j e c t

x

f o r each

a

be a simple category. of in

A is a set

FA

indeed a 2-functor 4.3

(4.6)

forgo,

sending

di:

4. 3

(A,£)J(B,A)

~ A'SB

SB

~a'gB

gB

~B

to check that ~ respects

vertical

of 2-cells in CATf*T and CATfT; thus J

of CATf*T x CAT~T

CAT#*T

(3.14),

onto its factors,

give new forgetful

and is

and

composed

2-functors

x CATfT ~ CAT ,

(B,A)) to A and B respectively.

clear from (4.12) Proposition

~ AB k

2-functors

forg1:

((A,£),

aB

AB

of the form (4.9).

The projections

with the forgetful

k

~,FA

It is very easy from (4.15) composition

~ (~,T)#(S,~)

is

~A

horizontal

transformation

The following is

(4.15):

The functors do:

(A,F)J(B,A)

~ B are the components

~ A and

of 2-natural transformations

do: J ~ forg O and d1: # ~ forg I. D

5. 5.1

THE 2-FUNCTOR

o AND THE EMBEDDING

Form the composite

CATf*T x

CAT~*CAT ~ [CAT,CAT]

2-functor

T---~----~

CAT[*T

×

CATfT

i × r • where r m

(I,~B~),

: Y ~ CAT~T etc.

is the 2-functor

Our interest

write the above 2-functor (5.1)

o: CATf*CAT

~ CAT, #

(3.13)

is in the special

as

x CAT ~ CAT.

sending B E Y to case Y = CAT; we then

201

We usually

abbreviate

value on morphisms

its value

(A,F)oB on objects

or 2-cells we write

except that we are sometimes

interested

to AoB ; for its

(T,T)oS and (q,8)o~ in full; in the restriction

CAT/CAT 0 × CAT, where T and 8 are identities,

of o to

and then we write simply

ToS and qo~. An object of AoB is a pair

(A,X) where A C A and X is a functor

FA ~ B; we write this object as A[X]. pair

A morphism A[X] ~ A'[X']

is a

(a,x) of the form

FA

A

(5.2)

a

ra

A' here a is a morphism the morphism

8;

FA'

S

in A and x is a natural

transformation.

B~

B', the functor

We write

(5.2) as

A[ X]

(5.3)

l

a[ xl

A'[X'] If (T,T): (T,~)oS:

AoB

~

(A,F) ~ (A',F') A'oB'

and S:

sends, as a special

case of (4.12),

the morphism

(5.2) to (5.4)

TA

Ta I TA'

F'Ta I

TA ~Ta

~-~FA ~

~B'

F'TA'

(T,T)oS ~ (T,T)~S has,

-component

X

!i"~~XX B

TA , (T,T) ~ (T,~) and ~: S ~ S, the natural

If (q,0): (q,O)o~:

F 'TA --

transformation

as a special case of (4.15),

the A[X]-

202

(5.5)

TA

F'TA S

HA

r'r~ A

FA

~-B~ B '

...... X

F 'TA

TA

We have

(5.6)

do:

sending ent

as

(5.2)

in

where

fOrgo:

5.2

The

gives

there

K ~

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

4.3

this

is the

compon-

° ~ forg O ,

CAT[*CAT

in p a r t i c u l a r

functor

transformation

x CAT

3-category

2-functors

the p r o j e c t i o n

to a: A ~ A'.

do:

TA

AoB ~ A

of a 2 - n a t u r a l

(5.7)

§4.1

-

~

sends

CAT

of 2 - c a t e g o r i e s

is a b i j e c t i o n

[L,M]

Under

.

this

((A,r),B)

to A, etc.

is of c o u r s e

between bijection

cartesian

2-functors the

closed;

K × L ~ M and

2-functor

o of

(5.2)

a 2-functor

(5.8)

CAT~*CAT -~ [ C A T , C A T ] ,

@:

to wit

(5.9) Our

¢(A,F) first

Theorem

main

5.1

result The

Proof.

Take

components

GB:

unique

= Ao-,

= (T,T)o-,

¢(n,8)

= (~,8)o-.

is:

2-functor

a 2-natural AoB ~ A'oB.

} is 2 - f u l l y - f a i t h f u !. transformation

G:

We are

that

to

show

(A,F)o-

~

(A',F')o-,

G = (T,T)o-

with

for a

(T,T). The u n i q u e n e s s

follows

(5.1o) and

¢(T,T)

that

from

(5.4)

of

is i m m e d i a t e ;

that

Gri(A[irA]) GrA , sends

(T,T)

= TA[TA],

the m o r p h i s m

for

if G is

(T,~)o-

it

203

(5.11)

A

FA

al

Fa

rA'"

A' of AorA'

FA'

to the morphism TA

(5.12)

TA

F 'TA

r'Ta

Tal TA' of A'°FA';

(5.13)

~FA

]I

v

F'TA'

Ta

[

ra

~

U i

rA'

~FA'

TA ,

that is to say,

GrA ,(a[lra]) = Ta[~a ].

So by (5.10) and (5.13) G uniquely Now let any 2-natural that by the naturality

determines

G be given,

T and T.

let AtX]

of G we have commutativity

E AoB, and observe in

GFA

(5,14)

AorA

~ A' rA

A'oX

AoB

B.

If we define TA,T A by (5.10) and evaluate object A[IFA]

(5.15)

we get

a~(AEx]) = TAEX.TA].

In particular, this said,

GFA , (At Fal) -- TAIFa.'rA],

the domain of (5.12).

it now makes sense to define Ta,T a

Next, replace A,X by A',X' the morphism a[ira] (5.16)

both legs of (5.14) at the

of (5.11).

With

by (5.13).

in (5.14) and evaluate both legs at

By (5.13) this gives

GB(a[ Ix, .ra ] ) -- Ta[X' .T a] .

204

Now let

a[x] be the morphism (5.2) of AoB.

By the 2-natural-

ity of G we have commutativity in G AorA

(5.17)

AoX

rA

~-A oFA

Aox

A'orA'

A'oX

AorA'

/

~

A'oX'

~ A ' oB

GB

Taking the A[IrA]-component of each leg we get, using (5.10) and (5.5), (5.18)

GB(1A[ xl ) -- 1TA[X.~ A] .

But the a[x] of (5.2) is the composite A[X]

~-

A[X' .ral

1A[ x]

~- A ' [ X ' I

,

a[ i x , . r a ]

and G B is a functor; so GB(a[x]) is, by (5.16) and (5.18), the composite TA[X.T A]

~-~ TAIX'.Fa.T A] ~ TA'IX'.TA,] ; ITA[X.T A] Ta[X'.T a]

that is, (5.19) GB(a[x])

is

TA

Ta

F'TA

TA

~FA

I I I r'Ta

TA'

F'TA'

~FA'

TA ,

Using the fact that G B is a functor, we now easily get that T is a functor and ~ a lax n.t.

by applying GFA to IAIllr A| and by applying

205

GFA,, to the composite A[ra'.ra]

~A'[ra']~A"[IFA,,].

a[ i] Then by comparing

It remains let

(T,T),

(~,~):

modification

a'[ I]

(5.19) with (5.4) we see that G B is indeed (T,T)~8. to prove that ¢ is fully faithful

(A,F) ~ (A',r')

with components

YB:

that ¥ = (n,8)o- for a unique

and let y:

component

(5.20)

(T,T)o- ~ (T,Y)o-

(T,T)oB ~ (T,~)oB.

So

be a

We are to show

(0,0).

If ¥ is indeed of this form, A[IFA]

on 2-cells.

it follows

from (5.5) that the

of ¥FA is

(~rA)A[irAl:nA[e A],

which proves the uniqueness

of n and 9.

If now for any ¥ we define

and 9 A by (5.20), we have because y is a m o d i f i c a t i o n

commutativity

HA in

(T,T)oFA

AorA~YrA

A' rA

A ToX

Ao X

(T,~)oB Ao B

YB

oB,

(T,~)oB and calculating

(5.21)

the A[iFA]-component

(yB)ALX]

in agreement

with

: nA[XeA] , (5.5).

indeed a modification, YFA'

Proof

That n is indeed natural,

5.2

CAT/T

and that 0 is

follow at once when we express

for the particular m o r p h i s m

Corollary

of both legs gives

and

By the remarks

CAT/*T

(5.11) of A-FA' are indeed

the naturality

of

D

2-catesories.

in §3.1 and §4.2, with the above theorem.

U

206

5.3

We d e v o t e

this

full

embedding

First, note

that

I is the

F = rn~

for

some

discrete

image

the

of ~.

with

were

in the

that

this

[n,-]

sending

part

I have because,

while

h o w to fix image

clear

that

[n,-]

for

D is not

both

DB

form

D need

not

that

form.

[n,-] .

D3 = 38.)

be in the

explicitly,

for

DI is at any rate to [n,B]

I think

image

Now

of p a t h -

D2 = 24 and

equivalent

in the

n.

2 is the

of the

can be g i v e n

of this

of s o m e t h i n g

where

not

it is c l e a r

to m a k e

some

m is the n u m b e r

with

which

There

is e n o u g h

D is a q u o t i e n t n is ~

where

of a d o c t r i n e

the d o c t r i n e ,

of ~; but

be

is c e r t a i n l y

no m is c o n s i s t e n t

I; this

A ~ I, so that

[[B,2],B],

B to

B is d i s c r e t e ,

2-functor

image

where

see this,

CAT ~ CAT we have:

D:

then

it is also

DI ~ I m u s t

2 objects,

when

Consider

to

To

~ [n,B];

a category-with-coequalizers. equivalent

on

A a DI.

But

of ~ w i t h

of n; and

Even

CAT].

of [ C A T

if D ~ (A,F)o-

n.

CAT ~ CAT

= [m,B]

-components

then

category

category

[n,Bl

the w h o l e

informal,

2-categories.

So for a 2 - f u n c t o r

~ I and

image

D:

~ [CAT,CAT]of

is not

category.

(l,rn~)oB

2-functor

comments,largely

~ A,

if DI

so any D in the

to some

clearly

unit

that

(5.24)

(For

image

If D ~ (A,F)o-

follows

the

section

@: C A T f * C A T

(A,F)oI

(5.23) It

its

we have

(5.22) where

this

if D

however

of @, to wit

of

.

no real

idea

I can p i c k

F in terms

how big

out

the

image

the p u t a t i v e

of D, and

hence

have

of ~ in fact

A by

(5.23),

no test

is;

chiefly

I don't

see

for D to be in the

of ~. We are

composition products

and

going

to show

of 2 - f u n c t o r s . coproducts.

in What

§7 that else

the

image

is it c l o s e d

of @ is c l o s e d under?

under

Certainly

2O7

It suffices

to i l l u s t r a t e

CAT~*CAT of (A,F) and and

A',A)oB ~

(A ×

(A,F)

and

(A',F')

(A + A',a)oB ~

But

(A,F)oB is

it would

AoB

as

The product A(A,A')

coproduct

in

= FA + F'A';

CATf*CAT

in

AIA = F and AIA'

has

of

= F'; and

my r o u g h

constructed

a 2-natural

AoB.

pointwise,

(A,F)~(I, r B ~ ) ,

B~, where

adjoint,

G: D ~ ~(A,F)

such a G B would

that

that

seems

false.)

preserved.

modulo

size

: (A,F) o-, with

4.1 and the d e f i n i t i o n

correspond

B~ is the

which

they are p r o b a b l y

a left

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

indicate

them it w o u l d mean

on the other hand,

Consider

chiefly

to our m a i n purpose.

calculations

(If @ p r e s e r v e d

were

equalizers:

not r e l e v a n t

seem to have

FP ~

The

CATf*CAT

@ would

GB: DB ~

er with ~:

where

whether

them.

If it has coequalizers,

components

where

x (A',F')oB.

equalizers,

of lax n . t . ' s

considerations.

(A × A',A)

be a long b u s i n e s s

wou l d not preserve

For then

ones.

+ (A',F')oB.

checked

if it does have

equalizers

is

(A + A',A)

(A,F)oB

I haven't because

(A',F')

by b i n a r y

of

to P: DB ~ A togeth-

composite

DB ~ I ~ CAT,

rB~ that

is, the

bijection selves in

constant

with m o r p h i s m s

would

CAT~*CAT

(5.25)

of

in

is, by

seen to be 5.4

in

other hand

in the

CATf*CAT

already

(A,F), (5.24)

(B,A)on. We shall

CAT,

take

not n e c e s s a r i l y

The

G's them-

~B(DB,B~) ~

(A,F)

that

lies

~ is as far as it can be from

that

any D E [CAT,CAT]

in the

latter.

For

with

a

if

~ CATf*CAT((A,F),(B,a))

and Yoneda,

always

CAT~*CAT.

in

exists.

sense

(A,F)

So D ~

in

the G B are

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

coend

adjoint;

In other words

-~ (A,F)

it seems

[CAT,CAT]((A,r)o-,D)

2-naturally (5.25)

(DB,B!)

if the i n d i c a t e d

a right

2-reflexion

at B.

then be in b i j e c t i o n

On the having

functor

: (I, rr~). Dn;

while

Then the the right

left side

side of is easily

(B,A)o-.

henceforth

use T to denote

full on 1-cells,

but always

a sub-2-category taken

to be full

208

on 2-cells. We have an obvious

(5.26) which

inclusion

CATf*T ~ CATf*CAT composed

(5.27) It is clear precisely

with

¢ gives

¢T:

CATf*T

~

that

(5.26)

is full on 2-cells,

when T ~ C A T

6.

6.1 closed

in [ C A T , C A T ]

Ao(Bo-)

~

CAT

~

is;

so that

goal

under

CAT).

(AoB)o-;

[CAT,CAT].

here

composition

We shall AoB

of A and

B. The c o n s i d e r a t i o n s

n = £A of A E A is a kind

~

of objects

CAT

of B.

If these A: B ~ CAT,

is the composite

suggest

of A.

that

CAT[*CAT is

is, of

that

so denoted

in 55, but now

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

that

An object

objects

the a u g m e n t a t i o n A[X]

of B also have arities,

then A[X]

AX: n ~ B ~ C A T ,

should

of A o B

and where

n(m)

generalizes

In fact

ization

construction

is p r e c i s e l y

the G r o t h e n d i e c k

n and a functor

m: n ~ C A T

over n, which we may

call n(m).

6.2

define

We therefore

8:

the

the G r o t h e n d i e c k

CAT,CAT ~ CAT

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

corresponding

2-functor

of

provided

have an arity n(m),

n(ml,... , m n) = m I +...+ m n of §1.2.

(6.1)

(5.27).

of A and X: £A = n ~ B, so that X is a kind of n-ad

augmentation

category

of

(that

showing

derived

of 51.2 above of "arity"

is true

in §7 b e l o w

do this by

AoB

and is full on 1-cells

CONSTRUCTION

is the category

with an a u g m e n t a t i o n

objects

same

is the proof

provided

consists

the

THE G R O T H E N D I E C K

Our next main

2-functors

,

fibred

by an

where m the generalto a category

209

by setting (6.2)

@ = rill-,

where o f c o u r s e /

is

the 2-functor

We a l s o u s e , where c o n f u s i o n

of

(4.9)

is unlikely,

with

T put equal to

the "parenthesis"

CAT.

notation

suggested above: namely (6.3)

@(A,£)

= A(£),

An object (that domain

@(T,T)

of A(F)

= T(T),

We write

= q(e).

is a pair A,Y where A E A and Y: I ~ FA

is, Y E FA; but it is useful ~).

8(q,8)

this object

to write

as A(Y).

Y as a funetor A morphism

A(Y)

with ~ A'(Y')

is

a pair a,y of the form (6.4)

A

aI

which

I

Fa

is a fancy way of saying that y is a morphism

We write this morphism If (T,T): functor T(T)

sends

(6.5)

TA

ra.Y ~ Y' in rA'.

as a(y) .

(A,£) ~ (A',r')

in CAT/CAT,

then by (4.12)

the

(6.4) to TA ?r'TA

Ta

~a TA'

r'Ta ~F'TA'.

TA , If (q,e):

(T,T) ~ (T,T)

has by (4,15)

(6.6)

its A(Y)

in CATfCAT, component

the natural

given by

~

TA I

~A

transformation

y

~

F'TA

rA

F'~A

q(8)

210 We have as in §4.1 the p r o j e c t i o n functor

(6.7) sending

d1: A(r) ~ A (6.4) to a: A ~ A'.

As in P r o p o s i t i o n 4.3 this is the

component of a 2-natural t r a n s f o r m a t i o n

(6.8)

d1: @ ~ forg,

where forg: A, etc.

CATfCAT ~ CAT is the forgetful

2-functor sending

(A,F) to

W r i t i n g ~ for the arrow category 0 ~ i, we can regard 9,

forg, and d I in (6.8) as c o n s t i t u t i n g a 2-functor @: CATfCAT ~

(6.9) sending

[~,CAT]

(A,F) to dl: A(F) ~ A, etc.

We can call O the a u g m g n t e d

G r o t h e n d i e c k 2-functor. 6.~

The results we need about the G r o t h e n d i e c k c o n s t r u c t i o n are

most easily derived from the following theorem, which is a slight extension of Gray's "Yoneda-Like Lemma" on page 290 of [5]. T h e o r e m 6.1 Proof.

CAT,CAT ~ [~,CAT]

The 2-functor 9:

is 2-fully-faithful.

Let

A(r)~

(6.10)

~ ~

A

A'(r,)

~

A'

T be

a morphism

If

in

M is

[~,CAT].

indeed

T(~)

We h a v e

we

to

show

conclude

(6.11)

M(A(Y) ) = TA(TAY) ,

(6.12)

M(IA(Y) ) = ITA (TAy) ,

which fixes T A on objects and morphisms.

that

from

M = T(T)

(6.5)

with

for

a

a

unique

= 1 A that

M o r e o v e r if we define T A by

211

(6.11) and composite

(6.12), the fact that M is a functor and the fact that the of iA

AtX'

¢1''" X'¢n] aiX,1...X,n,~ 1 A'EX' l ' ' "

X~].

The reader will have no trouble expressing (9.24) and (9.26) in this expanded notation. Since the special case CAY/~ has been treated in detail in [5] and [6], and since there is not much to change, I shall give only an outline in this section.

There are some mild notational differences

from [5] and [6]; there ¢ was always an isomorphism and (10.2) was replaced by xi: X

~

X'i, which is impossible here.

That, however,

had the effect of making (10.6) conformable with the convention that X' stands for IX,; here we have to sacrifice something and we choose it to be this.

244

10.2

What makes

ness

of the

say b e l o w most

images

would

practical Let

CAT/~ ".

the

FA u n d e r

apply case

case

the

equally

that

the a b o v e

well

with

T[S1,...,

S n] ~ T ( S 1 , . . . ,

S n)

for

A club club; are

it m a y

clubs

of the

CAT/~

above

starting

is that,

that

Let

construct

are

forms

free

automatically inductively

the

objects

not

the basis

free

club

discreteof what

stick

we

to the

"club

"multiplication"

same

identity

(Q,F)

a discrete

(Of course further

consequence

the

objects

is false

that

we do in this

be given.

We

for

before

to

- which

construct

and

section.

It is easy

it g e n e r a t e s

and

CAT/cato,

that

it g e n e r a t e s .

there

K from generators

settle

~; and

CAT/~.

in

is c a l l e d

The

in

if K is a club,

same

SET/~.

in

and

SET/~

club

the

a club

of all

of

means

and the

discrete).

It is this

discrete

we

that

as a c a t e g o r y

IKI first

(P,F)

much

IKI ~ K is a club m a p

in c o n s t r u c t i n g

any o b j e c t

the

it f o l l o w s

as a o - m o n o i d

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

10.3

is also

the

set

Q

by

(10.8)

& • Q; if P E p w i t h FP

(10.9)

P(QI''" We e m b e d

agree

that

over

S),

until and not

for any

if Q I ' ' ' ' '

Qn E Q t h e n

qn ) E Q.

FP(Q 1 ... qn)

augmentations

extension

: n and

P in Q by i d e n t i f y i n g

F~ = i,

that

is d i s c r e t e

that

but

"club"

of c o u r s e

inclusion

we can c o n s t r u c t

it is this

by

the

be r e g a r d e d

in

relations,

which

section

discreteness IKI,

that

is the

in fact

CAT~set,

to

for this

set of o b j e c t s

also

simple

operations.

so is its

it f o l l o w s

specially

augmentation;

of f i n i t a r y

us a g r e e

From

present

further

notice

a general

L and

to a club m a p

P(~

... ~),

= n ( F Q 1 , ... , FQn).

are d e n o t e d club

P with

one

"club

map"

To

Let

by F if no o t h e r

name

any m o r p h i s m ~ ~

L.

P ~

save

means

in CATf*~.

and e x t e n d

repetition,

one

in

us also

there

let

CAT/~

(i.e.

agree

that

is given.

L over ~,

F to Q

It

us

all

is c l e a r

is a u n i q u e

245

Now and

let

consider

clear

that

these

congruence and

discrete

then

A model : P ~

is of c o u r s e

always

ITI =

ISI

for the

A; here

~(~,I)

identity 10.4

~

or

objects

into

obtained

by

operation We

can

implies

implies

club m a p s

Q ~S

first

call

S

rT = rS,

MT = MS.

enlarging

the

It is

~ L,where

T ( S I , . . . , S n)

where over

A together

n = FP. ~;

This

the

S is

p to a

of s u b s t i t u t -

discrete

with,

club

for e a c h

is c l e a r l y

the

endo-internal-hom

a club;

and

{ A , A }, or a g a i n

with

P ~ P,

same

thing

( A , A}

so a m o d e l

of P is the

same

as an a c t i o n

Q oA ~ A;

so it

Q-category.

of P,O

is such

a model

TpS.

A of P s a t i s f y i n g

It is t h e r e f o r e

an S - c a t e g o r y

S. example

~ and

is that

I with

p identifying

of a strict

F@ = 2 and

~(~,~)

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

with

discrete

monoidal

FI = 0, and

~(~,

club

®),

category

Q

is s u b j e c -

~(I,~)

with

S is ~ itself,

~, and

with

augmentation. a discrete

and a r r o w s

and let

us be g i v e n as w e l l

graph-morphisms a club-map.

- not

that

now replace L

I assert any

call

over that

such M

the

S, w h e t h e r

a graph of [5]

a morphism

D ~

so that

L; we can

club

let us be g i v e n

objects

M ~

Q~

obvious

or d i r e c t l y ,

M,

ToS

canonically

The most

Given

We m i g h t

A,

club

with

that

is a c a t e g o r y

discrete

ted to r e l a t i o n s

L such

ToS

p.

: An ~ A whenever

P has

that

to the

{ A , A}

a Q-al~ebra A model

club

P

as a c l u b - m a p

is just

such

S = Q/~.

An ~

as a m o r p h i s m

thing

of

relations of

on Q

uniquely

respect

and

IPI

: Q ~

club

setting

P

a functor

M

factorize

~ with

generators

a relation

club-maps

the q u o t i e n t

ion,

0 be

D (in the u s u a l and

of g r a p h s

S by

IDI.

~ such there

by g e n e r a t o r s

that

[6]) w i t h r

factors

club

~enerated

of g r a p h

object-set

for a club

composite

is a u n i v e r s a l

D ~ L

sense

: D ~ ~ extending

Consider the

and r e l a t i o n s

through

IDI = S, r

: S ~ ~.

L those

IDI ~ D ~

L is

such map

D ~ M into

a unique

club-map

by S and

D.

-

In p a r t i c u l a r

a

246

a model f

of

(S,D)

means

an S - c a t e g o r y

: T ~ S in D, a n a t u r a l

in the the

sense

club

of

(9.20).

objects;

formal

first

~ = Ff,

formal

f($

... ~),

but

: ITI ~

A is just

by e n l a r g i n g

an i n s t a n c e

we also

i.e.

so the

extend

IsI of type

rf,

an M - c a t e g o r y

for

D twice,

of an f

bigger

P E

graph

to w h i c h

but

a partial

without

changing

: T ~ S in D to be a

...

O",

instance

g

...

We

pass

that

any m o r p h i s m

IDI ~

L is a c l u b - m a p ,

diagrams

in D'''

g

... R%n)

so d e n o t e d .

etc. are

of c o u r s e

We i d e n t i f y

graph

they

...

g itself

the

f with

D' e x t e n d i n g

D, to

r.

The

of

o -monoid

O ~

L over

extends

...

T

~)

with

~

P($

~(g),

category

~ into

generated

D'''

a club

is not

S

...

~)

a still

sent

There into

by the yet

on it;

L, w h o s e

to a f u n c t o r

on D'''

by

...

and n o w have

can be d e f i n e d

are n e c e s s a r i l y

an e x p a n s i o n

F.

category

uniquely

are t y p i f i e d

define

f, to be an e x p r e s s i o n

extend free

club-structure

which

: T ~ S in D',

of some

we a g a i n

extend

structure

~L;

IDI

: P($

to D''',

we a g a i n

the p a r t i a l

l)

identify

to w h i c h

respects

T(R~I

~ S ( R I ... Rn,)

F.

clear

by D,,,

the o b j e c t s

for an i n s t a n c e

IOl.

We t h e n D",

... R~n)

instances form a b i g g e r

an e x p a n d e d P(1

some

T(R~I

are the o b j e c t s of

and

(i0.ii)

for

we d e f i n e

and w h e r e

We next, of g,

a model

M we b e g i n

f(R I ... Rn,):

not

which

So such

Ifl

for e a c h

expression

(i0.I0) where

transformation

with,

M. To c o n s t r u c t

its

A together

O'''

graph

a club,

and

it is

restriction ~ L

are two

commutative

which

sorts

of

diagrams

247

T(l...h...i...i) (lO.12)

T(SI...S~...Sj...S n)

T(SI...Si...Sj...S n)

T(1...1...g...l)

T(l...l...g...l 1

T(SI...S~...S~...S n)

T(Si...Si...S~...S n) T(l...h...l...1)

and by

(10.13)

T(S¢I . . . . . . . . .

S$n )

f(Si...Si...Sn,) .............

T'(SI...Si...Sn,)

T(1...k...k...l)

~T'(sl...s~...sn,).

T(S¢~ ......... Sen) f(S1"''S!'''Sn')m Here (10.13) needs some explanation.

The left vertical arrow stands for

either leg of a diagram like (10.12); or the corresponding thing when there are more than two maps k in it; and these k's occur at all those indices j for which Sj = i.

The diagram (10.12) says in effect that

T is "functorial", and (10.13) that f is "natural".

It suffices in

fact to impose these diagrams not for all f,g,h,k E D''' but for f E D", g,h,k ~ D'; the more general cases then follow automatically. All this is rather more complicated than in the CAT/~ case of [61. If we write Funat for the class of diagrams of the form (10.12) and Nat for those Qf the form (10.13), and D'''/(Funct + Nat) for the quotient category with the same objects got by imposing these relations, it turns out that this last is the desired club M; for it indeed admits a

club-structure in which, in analogy with (10.7), f(gl...gn ) is

defined to be the composite

248

(10.14)

T ( S I . . . S n)

~ T(S'~I...S'~n)

~

T ( g l . . . g n) 10.5

Now

suppose

diagrams

f,g

: T ~ S in M w i t h

a set

that,

o of such diagrams

morphisms

D ~

L over

be a c l u b - m a p ,

f(S'l...S'n,)

besides

the above,

Ff = Fg;

in D''';

~ into

a club

for which, in the

induced

say K, a q u o t i e n t - c a t e g o r y

of M , a n d

M by i m p o s i n g

expanded

instances

for the of the

in fine

call

P, w i t h r e l a t i o n s

generators

D, w i t h

A model

imposed

K the

A of

usual,

graph-

IDI ~ D ~ L, the

L to

images

of

such

club-map

relations is o b t a i n e d

that

K ~

all

any

L.

the

such

In fact expanded

as

consists

of all

the

a.

club

p, and

the r e l a t i o n s

o of

more

those

L or M ~

a club,

relations,

relations

generators

club

as e x t r a

+ Nat + Imp),

We m a y

the

itself

K = D'''/(Funct

satisfying

~

a set

~here is again a universal such L,

a; or b e t t e r ,

Imp, s t a n d i n g

only

requiring

N'''

of the r e l a t i o n s

(10.15) where

from

given

is p e r h a p s

consider

as R ~ M ~ K ~ L for a u n i q u e

K is o b t a i n e d instances

~lo.

we are

what

L~ still

T ~S

factorizes

or,

a n d we

f and g coincide for each f,g:

map

T'(S'I-.-~'n,)-

~enerated

by the

function-

the n a t u r a l - t r a n s f o r m a t i o n -

o.

(P,p,D,o) is of c o u r s e a m o d e l of (P,@,D)

the r e l a t i o n s

o; it

is the

same

thing

as a K - c a t e g o r y

A for

K. A typical

example

is that

of

symmetric

monoidal

categories~ where

P consists of ~ and I with F@ = 2 and FI = O; where p is vacuous; where D consists of a

:

e(~,~)

~

~(~,e),

a

: ~(~,~)

r

:

@(i,I)

~

i,

~

: i

c

:

® ~ @,

with

identity axioms, a,~

Fa,

permutation together

as i n v e r s e s

with

F~, of

Fr,

~

~

®(i,I),

F~ i d e n t i t i e s

2; and w h e r e

~(~,~), and

and w i t h

~ consists

Fc the n o n -

of the u s u a l

coherence

aa = i, aa = i, r> = i, ~r = i, e s t a b l i s h i n g

of a and r.

Here

the

club

K is in fact

in CAT/S;

249

other e x a m p l e s 10.6

leading

to clubs

It should now be clear that

"coherence explicit already

problem"

determination

of the club

a word-problem,

: T ~ S in D'''

that

is, of k n o w i n g

of course diagram,

in terms

f and g.

: ® ~ @

with

to the

consists

To find the objects

easy,

and trivial

in the of K is

if P is vacuous.

that

is usually

much harder.

and it is a q u e s t i o n

of k n o w i n g

when

coincide "which

in K because

diagrams

components

in the

of the relations

commute".

is the c o n d i t i o n

of generic

For example,

in §1.4 above.

solution

(P,p,D,q)

is a w o r d - p r o b l e m

Ff = rg; this

suggested

a complete

K.

but usually

We know the generators, f,g

are

for such a structure

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

c,l

CAT/~

in

A necessary

in (10.15); condition

for the w r i t a b i l i t y of the "natural

of a

transformations"

symmetric-monoidal-category

Fc ~ rl; and the generic

is

case,

we have

components

(10.16) C ~ B A®B

~ do not where

form a closed

A

In

closed diagram,

the

in the club

diagr a m s

commute,

10.7 (P,p,D,~)

The

situation

K.

If one wants

one has only

considerations

calculation

to conclude

case

is that

F is faithful.

: A ® A ~ A ® A form a

in a t y p i c a l model,

of §§10.3

described

even when

(P',p',D',~'). for example

KoA on A,

when we know K. - 10.5

there

that

and makes

such s p e c i a l i z e d

to look at the free model

of the word,

relations

c,l

to know w h i c h

we also know c o m p l e t e l y

sense

commute"

for f = g, i.e. where

specia ! components

may be r e c o g n i z e d

generato~and

"all diagrams

case.

~ L of the kind

in our current

The

but this does not commute

no sense

which of course

B

sufficient

a rare

(10.16)

®

diagram.

Ff = Fg is also

It is of course

@ A

~

leads

assert

to a club map

is to say, L itself

that a "map"

in

CAT/~.

K ~ L This

is given only by

Thus we are able without

the existence

of a club map

K ~ L

250 over

~ when

symmetric

monoidal

monoidal

club-maps may

also

when

together

very

observe

in

is in c o n n e c t i o n

by

the m o r p h i s m s itself

So an o b j e c t of the

we

that

as

observation

in

law

monoidal

(i.i)

in §9.5 lie

to leave

for

symmetric

symmetric

law,

we h a v e

i that

above. that

not

CAT/~.

in

(lax)

of the

considerations

(~,~)

doctrine

in the

2-category

which

are

lax n . t . ' s

but

should

get

sense

K o- for a club

FUN the not

of

: A ~ A' b e t w e e n

D-morphisms,

D is the

8

A'

a 2-cell

is like

op-lax

op-K-morphisms

We

K-categories

K in

[~,

CAT/~.

CAT[

except

if we t o o k

of K - m o r p h i s m s . is O , O ' , ~

~ B'

8,

where

¥

: O ~ [ and y'

: O' ~ ['

satisfy

(Io.18)

~ ' ~

A'

B

~

B'

-~-

A

A'

y

- 10.5

of [ 1 2 ] § 3 . 5

n.t.'s;

instead

§§10.3

',- B

is y,¥'

A

all

KoL ~ LoK b e t w e e n clubs that

form A

This

CAT/~ for C A T f * ~

FUN is a f u n c t o r ~ : A ~ A', and a m o r p h i s m

of

(i0.17)

while

below

K-morphisms

for

where

Denote

for two

for

K,L in CAT/~ t h e m s e l v e s

application

with

is,

in this volume,

of the

K is that

and

CAT/~.

A further

CAT~

a distributive

a distributive

10.8

[~,

with

in [8]

lie

that

is that

in v i e w

categories

or w h e n

L

K ~ L for clubs

themselves

that

for m o n o i d a l

and

handy,

considering

A and A';

club

categories;

categories

structures is o f t e n

K is the

]~..,

O' @'

~-

B

~

B'

251 There (10.19)

CAT/~

~ :

sending (To@,

is an e v i d e n t

K, ¢ to Ko@,

To8 ' , To~),

(@,e',~)

to

notationally

~ from

It is f i n a l l y

clear

n

So

(e,

: T ~ T and

in the

we have

no n e e d

clear

sense

e',~)

(y,y')

there is r e a l l y

It is m o r e o v e r

FUN,

that

: K ~ K' and

that

: @ ~ @, to

: (8,

@',~)

~

to d i s t i n g u i s h (10.19)

provides

(KoL) ~ ~ ~ K ~ ( L g ¢ ) ,

that

etc.

a 2-adjunction

FUN(K~¢,~) m CAT/~ (K,< ¢,@ >),

(10.20) where

o.

on

T

sending

qoy,).

of CAT/~

an a c t i o n

× FUN ~ FUN sending

and

(noy,

2-functor



denotes

the c o m m a

(10.21)

= { 6 , 1 } / { i , ~ }

in

as in

CAT/S,

object

dI (10.22)

~

{A,B}

~

{A',B'} .........

~-{A,B' }

{¢,I} Thus

FUN is e x h i b i t e d An a c t i o n

is c l e a r l y

@':KoA'

the

(@,0',~):

same

~ A', m a k i n g

- category.

K?@ ~ @ of the

club

thing,

by

§3.5 of [12],

A and

A'

into

KoA

(10.23)

CAT/S

as a t e n s o r e d

K in CAT/S

as a c t i o n s

K-categories,

on @ E FUN

@ : K o A ~ A and

together

with

an ~,

~ A

Ko~

KoA'

~ A'

,

@, making

of

club-map

(~,~)

a K-functor

K ~

A ~ A'

over ~ w h e r e



Such has

an a c t i o n its

then

evident

is a

club-structure.

252 To g i v e a w h e n 8,6' whose composites

are a l r e a d y

given

is to g i v e s u c h a K ~

w i t h d o and d I are the g i v e n c l u b - m a p s

K ~ {A',A'}

and K ~ {A,A}. S u c h an ~ has c o m p o n e n t s

(lo.24) aT[AI...An] w h i c h are to be n a t u r a l (3.18)

of [12]

: T ( ¢ A I . . . ¢ A n) ~

¢ T ( A 1 . . . A n)

in T and in the Ai, and the a x i o m s

(3.17),

become

(zo.25) aT(Sl...Sn)[A1...Am I

~T[ SI(A 1

• . .

Am1 ) . . ..Sn( .

T ( ~ S I [ A i • . .Aml ] "''a S n [ • . .Am] )",

.Am)]

(i0.26)

If now K is g i v e n by g e n e r a t o r s description

of the s t r u c t u r e

give ~ in t e r m s only to give natural

=

in the A i.

To a r e l a t i o n

as a c l u b - m a p

of t h e s e g e n e r a t o r s

aT

~T[A1...An]

K-functor

in the strict m o n o i d a l identifications axioms case,

expressing etc.)

@(@,~)

a T by

(10.25)

a T = a S.

of a T in T o n l y

one has

and

to be (10.26).

It r e m a i n s

to

for m o r p h i s m s

category = @(~,@) in

that,

when K

or for strict m o n o i d a l

functor

naturality

Similarly

First,

is f i n i s h e d .

categories

is a m o n o i d a l

shows h o w to

for T E P, and it is of c o u r s e

In this way one sees for i n s t a n c e for m o n o i d a l

(P~p,D,~), the



and r e l a t i o n s .

i m p o s e the a x i o m

the n a t u r a l i t y

f : T ~ S in 9, and one

either

K ~

One t h e n gets the g e n e r a l

TpS one m u s t

i m p o s e as a x i o m s

and r e l a t i o n s

in the u s u a l case;

sense

the a x i o m s

for s y m m e t r i c

@(@,~)

a

f r o m the

~ @(~,®)

monoidal

categories,

(not a s t r i c t one

in this case are the a:

is the club

same as the in the n o n - s t r i c t

functors,

monad

functors,

253

and o t h e r w e l l - k n o w n In the q:

(¢,~) ~

cases.

same way we see that

(@,B):

transformation

A ~ A',

B:

¢ ~ ~

we need

as we commonly

~ T ( A 1 . . . A n)

q T ( A 1 . . . A n)

.....~---~T(A1...An) ~T[A1...A n]

(3.19)

of [12],

do for m o n o i d a l

natural

only

We conclude

by n o t i c i n g

some other

special

to the case

of clubs

in

CAT~q,

K then be a club

in

CAT/~.

a K-category each

A such that,

i, the functor

for each choice gory,

monoidal

is a c a t e g o r y

closed

A bearing

variance kind of I left

T in P;

that

seem to be

CAT~set O.

or perhaps

By a closed

K-category

FT = n say,

a monad,

were

symmetric

cartesian

and for adjoint

closed

category,

cate-

etc.

is when a K - c a t e g o r y

or a comonad.

when K actually

whether

lies

for a club

in p a r t i c u l a r

the q u e s t i o n

closed

such a thing

the algebras

§1.2 above;

monoidal

we m e a n

in

CAT/~,

the

L of the m i x e d -

they are monadic

the canonical

over CAT.

K ~ L in this

faithful. This

announced of Day

category,

in [7 ] that,

open there

case was

Thus

like to check what

K-categories

properties

for each T @ K, with

of the A. E A. $

I showed

objects

T ( A 1 . . . A i _ 1- Ai+l...~) : A ~ A has a right

blclosed

The reader may

for those

transformations.

10.9

Let

on the natural

~T[ A I ... A n]

T(qA1...nAn) i T(@AI...~A n) the a x i o m

impose

the axiom

(lO.27)T(%A1..-¢An)

representing

for a K - n a t u r a l - t r a n s f o r m a t i o n

question

in [ii].

in [i]

can be a n s w e r e d

In fact,

by using

it is to clubs

seems to extend.

a Yoneda

K in

CAT/~

embedding, that

as

the work

254

For

for a K in CAT/~, we can e x t e n d

if A is a K - c a t e g o r y

K-structure within

on A to one

isomorphism)

on B = IA°P,setl , and

if we ask

indeed

B to be a c l o s e d

ITI:

B n ~ B of T E K m u s t

be the

its r e a l i z a t i o n

ITI:

A n ~ A.

the a b o v e

lies

in C A T / ~ ;

strict

L-map

K; the

strict

since

they

K-map

K ~

the

embedding

that

arbitrary always

is so.

above

doctrine

associativity

for

pseudo-K-algebra

above

extension

for a club

below;

does

extension

question

when

the

of the

Yoneda

that

work

not

~ of

since

K ~ K o I is

for an

K in say CAT~cat O. need

K

embedding,

L is f a i t h f u l

not

of

is a u n i q u e

Yoneda

of c o u r s e

~ Set

loose,

in Set c a u s e s either

with

For

I is

But Set,

be.

for a club

since

the

K in CAT~set,

account

- I have

not

thought

of strict

to be only

of this

as we shall

Set by an e q u i v a l e n t

colimits

lack

B = [A°P,get}

we t a k e

algebras,

or we r e p l a c e

associative

K ~

a K-category

is a l i t t l e

in g e n e r a l ;

do in I81

with

for then

Kan

there

(to

r: K ~ set.

colimits

pseudo-algebras

of w h e t h e r

Yoneda

and yet {l°P,set}

comparing

strictly

on ~ E K).

is a c l u b - m a p the

coincides

i.e.

left

under

on !; w h e n c e

sat, is c e r t a i n l y

Of c o u r s e

image

(We are u s i n g

D, or even

a D-category,

there

effect

on I,

the

or at any r a t e since

~ E L to the

L ~ B then

same

free K - c a t e g o r y Note

B to be {K°P,set}

if we t a k e

L ~ B taking

have

the Y o n e d a the

for

solves

uniquely

K-category;

the r e a l i z a t i o n

This

the

by

to some category

out

the

a

extent admitting

question

set has these. REFERENCES

[11

B.J.

Day,

On c l o s e d 137

12}

S. E i l e n b e r g

(1970), and

G.M.

calculus, I3}

J.W.

Gray,

The

categories

Kelly,

Jour.

99

Lecture

Notes

in Math.

1-38. A generalization

of A l s e b r a

categorical

in Math.

of f u n c t o r s ,

3 (1966),

comprehension

(1969),

242-312.

of the

functorial

366-375.

scheme,

Lecture

Notes

255

[4]

J.R. Isbell, Math. Reviews

[ 5]

G.M. Kelly, Many-variable Notes

[6]

G.M. Kelly, An abstract Math.

[7]

in Math.

281 (1972),

calculus.

I., Lecture

66-I05.~

approach to coherence,

Lecture Notes in

106-147.

G.M. Kelly, A cut-elimination

theorem,

Lecture Notes in Math.

196-213.

G.M. Kelly, Coherence theorems distributive

[9]

functorial

281 (1972),

281 (1972), [8]

44 #278 (1972).

laws,

G.M. Kelly and S. Mac Lane,

for lax algebras

and for

in this volume. Coherence

in closed categories,

Jour. Put e and Applied A l~ebra i (1971), [i0]

G.M. Kelly and S. Mac Lane, Closed coherence transformation,

[ii]

for a natural

Lecture Notes in Math.

G.M. Kelly and R. Street

(Editors),

97-140.

Abstracts

281 (1972),

of the Sydney

Categgr~ ' Theory Seminar 1972, mimeographed by School of Mathematics, 1972;

second printing

editor at his present [12]

G.M. Kelly and R. Street,

1-28.

(originally

Univ. of New South Wales,

(1973) available

from either

address).

Review of the elements of 2-categories,

in this volume. [13]

A. Kock, Limit monads Preprint

[14]

M.L. Laplaza, Math.

[15]

M.L. Laplaza,

in categories,

Series 1967/68 No. 6 (1967).

Coherence

for distributivity,

281 (1972),

F.W. Lawvere,

A new result of coherence

Ordinal

[17]

G. Lewis,

Coherence

in

for distributivity,

281 (1972),

sums and equational

Notes in Math.

Lecture Notes

29-65.

Lecture Notes in Math. [16]

Aarhus Univ. Mat. Inst.

80 (1969),

doctrines,

Lecture

141-155.

for a closed functor,

281 (1972), 148-195.

214-235.

Lecture Notes in Math.

256

[18]

G. Lewis,

Coherence

for a closed functor,

New South Wales, [19]

1974.

S. Mac Lane, Natural associativity Rice Univ.

[20] V. Z~berlein,

Ph.D. Thesis,

Studies

and commutativity,

49 (1963),

28-46.

Doktrinen auf 2-Kategorien,

(Math. Inst. der Univ.

Z~rich,

manuscript

1973).

Univ. of

DOCTRINAL

ADJUNCTION by

G, M.

This

paper deals

with

are categories with structure; isolated

observations

Kelly

adjunctions

its purpose

in the

n~:

f

I u:

is to unify

literature,

A ~ A'where A,A'

and

simplify

various

at the same time e x t e n d i n g

them

widely. (a)

First,

left adjoint (b) if f--~u that

~ I.

tensor

others,

the forgetful

of which from

§§5.1 and

sufficient

5.2),

condition

and A' be symmetric merely (i.e.

in CAT but commute

class

morphisms

the best

(Ab, 0) to

(Set,

w h i c h was not

that a

in Sym Mon CAT,

with the c a n o n i c a l

f(X)

in natural

of c a r t e s i a n

known

tensor

that

the

case

~ fX @ fY

~ X + X, so examples

f

closed

but there

are

is that where

u is

of mine

to this

products;

the a d j u n c t i o n

and finally symmetric

and

×).

itself d i r e c t e d

closed,

closed,

f(X~'Y)

of an old t h e o r e m

for f to preserve monoidal

that

of topoi,

perhaps

One can dig out from the proof (131

result

in general

and let

h o w often

a whole

by geometric

functor

sense

let A ~ A' ~ Set,

products:

perhaps

it is not

in the

Yet it is s t r i k i n g

is p r o v i d e d

innum e r a b l e

categories,

in CAT,

products"

instance,

= A × A.

is the c l a s s i c a l

colimits.

an a d j u n c t i o n tensor

For

preserve

examples

there

If A and A' are m o n o i d a l

is m e r e l y

that u(A) does

f preserves

f "preserves

and fl'

and trivially,

question,

namely f--~u

a

that A lie not

that u be normal

monoidal

functors

A ~ Set

and A' ~ Set). (c) with

Two t h e o r e m s

structure:

Theorem

of Street 9 of ~SJ

deal w i t h a d j u n c t i o n s asserts

for c a t e g o r i e s

that the left adjoint

of a

258

monad-functor

is an o p - m o n a d - f u n c t o r ,

and T h e o r e m i of [9] asserts

that the left adJoint of a lax natural t r a n s f o r m a t i o n is an op-lax natural transformation. (d)

Let V be a symmetric m o n o i d a l

be a m o n o i d a l m o n a d equalizers,

(= c o m m u t a t i v e monad)

the c a t e g o r y

"internal-hom"

closed category and let T on V.

Then p r o v i d e d V admits

VT of T - a l g e b r a s is closed,

sense of [2]

in the original

(cf. Kock |71); it has been shown to have a

tensor product, m a k i n g it m o n o i d a l closed,

only when it is cocomplete;

and c o c o m p l e t e n e s s has only been d e m o n s t r a t e d under highly r e s t r i c t i v e h y p o t h e s e s on V and T. certain adjunctions

In this context Wolff [ii]

VT ~ VT'

has looked at

(arising in fact from a d i s t r i b u t i v e law),

and shown them by direct calculations to be closed adjunctions. the right a p p r o a c h here is to change universes, tensor product,

r e c o v e r i n g the m i s s i n g

and then to use the easier "monoidal" methods;

might also be of value to have a simple criterion, of internal-hom, (e) conditions

Perhaps

but it

d i r e c t l y in terms

for an a d j u n c t i o n to be closed.

Finally there is Day's result Ill giving sufficient for a full r e f l e c t i v e

subcategory A of a b i c l o s e d m o n o i d a l

category A' to admit itself a b i c l o s e d m o n o i d a l

structure,

in such a

way that the a d j ~ n c t i o n f~-~u, where u is now the inclusion,

becomes

a monoidal adjunction. It turns out that there are some simple general results that illuminate all of the above situations.

We start in §i with a doctrine

D, which could be on any 2-category at all, but which we take to be a doctrine on

CAT,

purely because the n o m e n c l a t u r e is there more vivid.

(For what we need here about d o c t r i n e s we refer to [6] above in this volume, e s p e c i a l l y

53.5.)

We suppose that A and A' are D-categories,

and that we are given an a d j u n c t i o n q,s:

f--~u:

A ~ A' in CAT.

Our

first result is that there is a b i j e c t i o n between enrichments of u to a D-functor

(u,~) and enrichments of f to an o p - D - f u n c t o r

(f,f').

259

Our second result is that if the a d j u n c t i o n f--~u admits enrichment to an a d j u n c t i o n above ~', so that

(f,~)~(u,~) (f,~) and

enrichment ~ is given, ~' is an isomorphism,

in D-CAT,

then ~ is the inverse of the

(f,f') are strong; m o r e o v e r that if the

(u,~) has a left adjoint

in D-CAT p r e c i s e l y when

the left adjoint then being

(f,~) where ~ and f'

are inverse; and finally that if the enrichment ~ is given,

(f,f) has a

right adjolnt in D - C A T precisely when it is strong, the right adjoint being

(u,~) where ~ corresponds by our first result to the inverse 7'

of f. The first result encompasses

the two theorems of Street in (c)

above - for the second of these the relevant a suitable CAT/A.

2-category is not CAT but

The second result gives a proper answer to the

q u e s t i o n raised in (b) above: is to say that it is strong,

to say that

f preserves tensor p r o d u c t s

so that this happens precisely when the

a d j u n c t i o n is one in Mon CAT, which can also be expressed as a condition on u, here r e p r e s e n t e d by its generators u: UO :

uA@'uB ~ u(A@B) and

I' ~ ul (cf. §10.8 of [4] above in this volume)

This shows how

wide of the m a r k the sufficient c o n d i t i o n given in (b) above really is: that A and A' should be closed,

or even symmetric,

is totally irrele-

vant; as for the n o r m a l i t y of u, we shall see that it is a consequence of the a d j u n c t i o n ' s lying in Mon CAT, and not an independent c o n d i t i o n at all.

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

(a) that left adjoints preserve colimits can

also be seen as a trivial case of this second result,

by taking D as

the d o c t r i n e whose algebras are categories-with-colimits: functor gives a c o m p a r i s o n of colimits, D-functor,

any

and is hence canonically a

so that the a d j u n c t i o n is a l w a y s in D-CAT and f is always

strong, which here means c o l i m i t - p r e s e r v i n g . In §2 we look at the case where D is the d o c t r i n e whose a l g e b r a s are m o n o i d a l categories, but supposing A,A' to be a c t u a l l y m o n o i d a l closed.

Then the giving of G:

giving of a certain u:

u[A,B]

uA®'uB ~ u(A®B) ~ [uA,uB]'

is equivalent to the

where the square brackets

260

are the internal-homs;

~

and the conditions

on u, u

0

for the a d j u n c t i o n

A

to be m o n o i d a l

translate

into conditions

equivalent

to the n o r m a l i t y

not);

on ~ is i n d e p e n d e n t

that

"sufficient

condition"

on f, fo (namely on f and fo. on the

that

Now,

conditions

level,

products,

in (d) above.

(Recall

closed

natural

transformations.)

precise

conditions.

here

to be a closed

functor

interest

conditions entirely the

even in the absence

"simple

criterion"

categories,

functor,

proo~ would

and

desired

monoidal

similarly

for

still

be in terms

of n o n - m o n o i d a l

closed

and leave

of §i is m e r e l y

to suggest

and

categories

it at that

the

of

of these has not been developed,

ad hoc proof

results

into

is that they are indeed

an ideal

the theory

the c o n d i t i o n s

they are still

closed

or

is the

are e x p r e s s e d

one,

the

= closed

The answer

an independent

of the general

conditions

for m o n o i d a l

I suppose

in v i e w of the m a r g i n a l

translate

we can ask whether

that

= monoidal

but

the

so as to provide

functor

give

Similarly

on u ° is

are closed

a g a i n how m i x e d - u p

(b) above).

that

for the a d j u n c t i o n

pro-D-structures;

(showing

the c a t e g o r i e s

they be isomorphisms)

of the tensor

merely

of u (whether

g i v e n in

however,

internal-hom

on u and u °. That

I

- the role

the right

conditions. Finally

in §3 we r e t u r n

turn to the a n a l o g u e

of Day's

A to be a full r e f l e c t i v e the counit

~:

D-structure

to A too, Our results

shows

the enrichment condition;

By "roughly" structure

I mean

we show that

satisfied

D-structure

it is possible

f must

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

D and

suppose we suppose

be strong;

unique

to give a

the a d j u n c t i o n

to one

this at once

if it exists,

the c o n d i t i o n

only

So we now

or e q u i v a l e n t l y

enriching

in §i show that

doctrine

Now only A' is ~ i v e n as a

under w h i ch

that we may have

on A (axioms

get an honest

of A',

at the same time

in D-CAT.

of a general

in (e) above.

fu ~ I to be the identity.

we seek conditions

a necessary

result

case

subcategory

D-category;

that

to the

is r o u g h l y

and gives sufficient.

to make do with a p s e u d o - D to w i t h i n

in two cases:

isomorphism).

for any D,

We can

if we a c t u a l l y

261

ask f to be strict condition); "the

and for c e r t a i n

same thing"

"flexible"; monoidal reduces

(accepting

one.

then,

our c o n d i t i o n

This

shows

is irrelevant;

D, where

The

for strict m o n o i d a l

to Day's.

bielosedness

correspondingly

"flexible"

as an honest

that

case,

the

since u must

by our

imposes

of [i],

upon h i m s e l f

I.I

is in fact no r e s t r i c t i o n

We refer h e a v i l y

results

on a d j u n c t i o n We suppose

Everything

CAT.

were n: n,c:

f--~u:

and

above

for all

A ~ A'

DA

that

pairs

terminology

of the

of the

We further

of natural

on some

from [6]

u, which

we write

suppose

explained.

2-category.

2-category

in question,

as if the

2-category

with actions

given

transformations

~

A'

Df

,

u,f' n

an a d j u n c t i o n

as in ~

A

5'

DA' ........

n'

We record

Secondly,

introduction

otherwise

DA

u

([6]

one;

both for its

A and A' are D-categories,

u

which are mates

to normal

not

D = (D,m,j)

A

DA' .......

a "monoidal"

in CAT.

n

Du

First,

if A' is.

in this volume,

simplicity,

DA' ~ A'.

We c o n s i d e r (i.i)

to [6]

for n o t a t i o n a l

DA ~ A and n':

result.

and it

The m a i n results

given a doctrine

We suppose

paragraph

is

at all.

we say is quite i n d e p e n d e n t

and so, m e r e l y

or b i c l o s e d

is

In the

sufficient,

Day's

§2, the r e s t r i c t i o n

in the first

i.

and

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

that A is closed

Day

about

categories

is not.

is n e c e s s a r y

we just get as a bonus be normal

D for m o n o i d a l

two things

necessary

a pseudo-D-algebra

categories

the result

stronger

f

~

A'

•

n v

§2.2)

under

Proposition

the a d j u n c t i o n s

2.1 the value

Df--~Du

and

of f' in terms

f--4u. of u, to wit

262

n (1.2)

I

DA

~

DA'

~

A

~

DA'

A

---- A'

1

n'

Denote by I, II the axioms for (u,[) to be a D-functor, (3.17) and

(3.18) of [6] with f r e p l a c e d by u.

Denote by I', II' the

c o r r e s p o n d i n g axioms for (f,~') to be an o p - D - f u n e t o r ~ t h e and

(3.18) of [6]

namely

same (3.17)

with } r e v e r s e d in sense and r e - n a m e d }', and with

A,A' interchanged. Le~ma i.i

Axiom I is equivalent

Proof.

Proposition

2.5 of [6}

to I' and II t_~o II' shows that the mates of the left

squares of the left sides of I and II are identities; P r o p o s i t i o n 2.4 of [6} is D}';

shows that the mate of the left square of the right side of I P r o p o s i t i o n 2.2 of [61

T h e o r e m 1.2

There is a b ijection b e t w e e n enrichments of u to a D-

functor U = (u,[): F'

completes the proof.

= (f,}'):

A ~ A' and enrichments

A' ~ A.

of f to an o p - D - f u n c t o r

T h i s b i j e c t i o n is given by takin$ u and f' too

be mates" under the adjunct ions Df--JDu and f 1.2

Now

{u.

consider a further natural t r a n s f o r m a t i o n f as in DA'

(1.3)

Df

n T

~

A

~

DA

f

~ n

A.

2~

L e t us are

call

I",

satisfied,

marion

s:

II" the

the

condi$ion

F U ~ i is,

(1.4)

axioms

by

DA

DA'

for

(3.19)

A'

=

(f,~)

c:

to be a D - f u n c t o r

F.

If t h e s e

fu ~ i to be a D - n a t u r a l

transfor-

o f [6],

~--- A

,n......

n'

for

~

n

DA

I

DA'

~A

i

Df DA We get Du,

an equivalent

which

to the

Df---~Du a n d identity

thing

natural

process

to say t h a t

i--~i.

A

DA

condition b y p a s t i n g

is an i n v e r t i b l e

same

~

n

The mate

by

we p a s s

D4 to e a c h

(2.1)

to m a t e s

of t h e r i g h t

transformation

and

of n. Df;

side

side a l o n g

(2.2) under of

the m a t e

n

of [6]:

(1.4)

n

-

,,

~-

of the

A

~

f DA

which

in v i e w o f

(1.2)

n

above

(1.5)

c a n be r e w r i t t e n

as

DA

DA' ......

n'

--

A'

f

edge

it c o m e s

is t h e n t h e

Df DA

the

the a d j u n c t i o n s

DA'

"

~A.

~- A.

left

side

is

264

We conclude that,

if F = (f,~) is a D-functor,

and only if f'.f = i, the identity of n. Df.

shows that n:

An exactly

if

similar argument

i ~ UF is D-natural if and only if f.f' = i.

Ii" are equivalent

Proposition

then ~ is D-natural

Since i",

to I',II' when ~ and ~' are inverse, we have:

1.3

Given ~ and ~', and hence the m a t e u of the latter,

n and ~ constitute an a d ~ u n g t i o n in D-CAT between D-functors

U = (u,~)

and F = (f,~) if and only if (i)

(u,~) is a D-functor;

(ii)

(f,f) is a D-functor;

(iii)

~ and ~' are m u t u a l l y inverse.

Moreover (i')

and

(i) is e q u i v a l e n t to

(f,~')

is an op-D-functor;

and (i') and (ii) are equivalent

in the presence of (iii).

Immediate consequences are:

Theorem 1.4

In order that a D - f u n c t o r U = (u,~):

A ~ A' have

a left adjoint F = (f,~) i~n D-CAT, it is necessary and sufficient that f i_~n CAT, and that the ~' $iven by

u have the left ad~oint --

isomorphism.

T h e o r e m 1.5

Then f = f,-1 , and F is n e c e s s a r i l y strong.

In order that a D - f u n c t o r F = (f,~):

a right adjoint U = (u,u) i~n D-CAT,

D

A' ~ A have

it is n e c e s s a r y and sufficient that

f have t h e risht ad$oint u in CAT, and that F be strong. mate of f' = ~-i , in the sense of (i.I).

1.3

(1.2) be an

w

Then u is the

D

We leave the reader to formulate the obvious dual theorems

o b t a i n e d by r e p l a c i n g the doctrine D by the opposite doctrine D*, where D*A = (DA°P)°P; he will get in

this way theorems about adjunction in the

2-category of D-categories and op-D-functors; adjoint that must be strong; and so on.

here it is the right

265

2.

2.1

The m o n o i d a l and closed cases

Now let D be either the d o c t r i n e whose algebras are m o n o i d a l

categories,

or else that

categories.

whose algebras are symmetric m o n o i d a l

These d o c t r i n e s arise from clubs in

CAT/~,

and it follows

from §10.8 of [4] above in this volume that a D-functor is the same thing as a m o n o i d a l case may be. u:

functor or a symmetric m o n o i d a l functor,

In both cases u is determined by its components

uA @' uB ~ u(A@B) and u°:

([2] page 473 in the m o n o i d a l

I' ~ ul, subject to the usual axioms case, with the extra axiom of [2] page

513 in the symmetric m o n o i d a l case). of [4]

as the

It further follows from §10.8

that u is an i s o m o r p h i s m precisely when u,u ° are isomorphisms.

Similarly a D-natural t r a n s f o r m a t i o n is, in both cases, a m o n o i d a l natural transformation. Theorems 1.4 and 1.5 t h e r e f o r e apply to an a d j u n e t i o n ( f , ~ , f o ) _ ~ (u,~,u o) either in Mon CAT or in intermediary of an o p - m o n o i d a l

functor

Sym Mon CAT,

(f,~, fo,).

T h e o r e m 1.4 is now that f' and fo, be isomorphisms;

via the

The c o n d i t i o n in that in T h e o r e m 1.5

is now that f and fo be isomorphisms.

There is no d i f f e r e n c e between

the n o n - s y m m e t r i c and symmetric cases;

it is just automatic

that u

satisfies the extra axiom in the symmetric case if and only if f or f' does.

We get the explicit e x p r e s s i o n s

for fo,:

for f':

f(X@'Y) ~ fX @ fY and

fl' ~ I in terms of u and u ° from (1.2), giving D its value We find for fo, and f' r e s p e c t i v e l y

in terms of the a p p r o p r i a t e club. the composites (2.1)

fl'

~

ful

~

I;

fu °

(2.2)

f(x~,~)

~

f(~)

f(ufX~,ufY)

fX@fY.

fu(fX@fY)

f~

S

266

Given a m o n o i d a l functor U = (u,u,u°): A(I,-):

A ~ Set and V' for A'(I',-):

monoidal-functor

structures

([2]

A ~ A', write V for

A' ~ Set with their canonical

page 504).

There is a canonical

monoidal natural t r a n s f o r m a t i o n V ~ V'U with A - c o m p o n e n t

A(I,A)

~

A'(ul,ui)

UlA

A'(I',uA), A'(u°,I)

as shown in [2] page 510; we call U normal if this is a natural i s o m o r p h i s m V ~ V ' U (we do not require it, as we did in [2], to be actually an equality).

P r o p o s i t i o n 2.1

If we use the a d j u n c t i o n f ~ u

U is normal p r e c i s e l y when

we find at once:

(2.1) is an isomorphism.

In particular normality of U is a n e c e s s a r y c o n d i t i o n for the existence inn Mon CAT.

°f an a d j u n c t i o n F ~ U

One can similarly t r a n s f o r m the c o n d i t i o n that

(2.2) be an isomorphism,

but I see no special m e a n i n g in the t h u s - t r a n s f o r m e d condition,

and omit

it.

2.2

Now suppose that the m o n o i d a l

categories A and A' are closed,

so that -OB and -@'Y have right adjoints of giving u:

[B,-]

and [Y,-]'

Then instead

UA @' uB ~ u(A~B) we may equally well give its mate under

the adjunctions

- ® B - - ~ [ B , - ] a n d -@'uB--~[uB,-]';

w h i c h is a natural

transformation ^

(2.3)

u:

u[B,C]

.......

~-

[uB,uC] '

It is shown on page 487 of [2] that the m o n o i d a l - f u n c t o r axioms for u,u ° on page 473 of [2] translate into the c l o s e d - f u n c t o r axioms for ~,u ° on page 434 of [2]; and similarly for m o n o i d a l natural t r a n s f o r m a t ions and closed natural transformations. We leave the reader to verify that the c o n d i t i o n that an i s o m o r p h i s m translates (2.4)

u[fY,C]

(2.2) be

into the condition that

......

[ufY,uC] ' -

~

In,l]

[Y,uC] '

267

be an isomorphism.

Moreover,

if f is given, and c o r r e s p o n d s to f,

then the c o n d i t i o n that f be an i s o m o r p h i s m translates into the condition that

(2.5)

[Y,uC],

uf[Y,uC]'

............^

n be an isomorphism.

~

u[fY,fuC]

uf ^

O

So ( u , u , u )

~

u[fY,C]

u[l,~]

has a left adjoint when u does if and

only if (2.1) and (2.4) are isomorphlsms;

and

(f,~,fo) has a right

adJoint when f does if and only if fo and

(2.5) are Isomorphisms.

We omit the proofs of these r e d u c t i o n s p r e c i s e l y because we are now going to give direct proofs of the last two statements that apply even to non-monoidal closed categories. given in the introduction.

Our motive for doing this was

As the reader will see, the proofs that

follow largely parallel those of §I, and would doubtless be best seen in the context of p r o - D - s t r u c t u r e s ,

if the theory of these had been

worked out.

2.~

Our d e f i n i t i o n of (non-monoidal)

tial m o d i f i c a t i o n of that of [2]; [ , ]: L:

closed category is an inessen-

a category A, a functor

A °p x A ~ A, an object I of A, and natural t r a n s f o r m a t i o n s

[B,C]

~ |[A,B],[A,C]],

being an isomorphism;

j:

I ~ [A,A],

i:

A ~ [I,A], this last

s u b j e c t e d to the axioms CCI - CC4 of [2] page

429, together with the axiom that the A(A,B) ~ A(I,[A,B])

induced by j

be an isomorphism. We suppose given as before an a d j u n c t i o n ~,e:

f--~u:

A ~ A'

in CAT, but now we suppose A and A' to be closed, and not n e c e s s a r i l y monoidal.

It is still the case that there is a b i j e c t i o n b e t w e e n

m o r p h i s m s u°:

I' ~ ul and m o r p h i s m s

in terms of u ° by (2.1): f--~u.

fo,:

fl' ~ I, where fo, is given

this is of course immediate by the a d j u n c t i o n

But in place of the b i J e c t i o n b e t w e e n natural t r a n s f o r m a t i o n s

u and natural t r a n s f o r m a t i o n s

f' that we had in the m o n o i d a l case, we

get s o m e t h i n g more complicated.

Precisely,

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

^

u as in (2.3) are in b i j e c t i o n with natural t r a n s f o r m a t i o n s

268

u1:

u[fY,C]

---------~" [ Y , u C ] ' , ^

u I being given i n terms of u as the composite ( 2 . 4 ) ; and n a t u r a l ^

transformations

f:

flY,Z]' ~ [fY,fZ]

are in b i j e c t i o n w i t h natural

transformations f1:

[Y,uC]'--~u[fY,C],

fl being given in terms of } as the composite

(2.5).

To see that

^

u~-~u I and f F-~f I are indeed bijections, and

(2.5) in 2-dimenslenal

we have only to write

(2.4)

form as, respectively,

(2.6) A°PxA

..... [..x. ]

>

op A'°PxA

A

LU ~-A'°P×A

i

~ A'°PxA ' l×u

~

A',

[ , ]'

(2.7) [

A'°PxA '

A'°PxA

, ]'

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

>

i

>

A'

~-- A'

A'°PxA --------a,, A°PxA

I

[ ,1

f°Pxl ^

e x h i b i t i n g u I as the m a t e of u under certain adjunctions, and fl as the mate of f under others, in the sense of [6]

§2.2.

A closed functor is of course said to have an adjoint one in the 2-category of closed categories, natural transformations.

The m a i n result of this section,

and e x t e n d i n g the assertions

T h e o r e m 2.2

closed functors,

of §2.2 above,

if it has and closed suggested by

is:

A closed functor U = ( u , ^u , uo) : A

~ A' has a left

adjoint F = (f,~,fo) if and only if u has the left adjoint f and m o r e o v e r fo, and Ul, given by (2.1) and

(2.4) respectively,

are

269 ^

isomorphisms. t__~ofl = Ul

Then fo is the inverse of fo, and f corresponds by (2.5)

-i ^

A closed functor F = (f,f,fo):

A' ~ A has a right adjoint

^

U = (u,u,u °) if and only if f has the right ad~oint u and m o r e o v e r fo a n d fl' the latter ~iven by (2.5), are isomorphisms.

Then u

O

^

c o r r e s p o n d s by (2.1) t o fo, = fo-l, and u corresponds by (2.4) t__o_o u I = fl -I . ^

Proof

^

Take all the data u , u ° , f , f ° as given, with the mates

ul,f°',fl of the first three, without supposing yet that the closedfunctor axioms CF1-CF3 of E2] page 434 are n e c e s s a r i l y satisfied by U or by F.

This last in no way prevents us from

the c o m p o s i t e s FU = ~ = (¢,

m e c h a n i c a l l y forming

,¢o) and UF = ~ = (@,~,~o) as on t2] page

434. Our first o b s e r v a t i o n is that q:

i ~ ~ and s:

¢ ~ 1 satisfy

the axioms CN1 and CN2 for closed natural transformations,

on page

441 of 121, if and only if fo is inverse to fo, and fl to u I.

In

fact s satisfies CN2 if and only if flu1 = I; n satisfies CN2 if and only if ulf I = l; s satisfies CNI if and only if fo,fo = I; and n satisfies CN1 if and only if fofo,

= 1.

We prove only the first of these assertions, to the reader.

Express

flul = i by pasting

leaving the others

(2.6) on top of (2.7).

In

^

the r e s u l t i n g d i a g r a m the 2-cells f and u have a common edge, and the result of pasting them along this edge is just ~. 2-cells are the triangles n °p x I, q, i x s. of these by pasting on an s °p x i

The other three

Get rid of the first two

and an s to cancel them out, as in

(2.1) and (2.2) of ~61; the e x p r e s s i o n of flu1 = i is now that ~ with a I x s pasted on is equal to the identity of uE s °p × I and an s pasted on in suitable places. c in 2 - d i m e n s i o n a l form.

, ](fop x i) with an But this is just CN2 for

27O

This establishes the "only if" part of both the a s s e r t i o n s of the theorem.

For the ~if" part of the first assertion, we use the

p r e s c r i p t i o n of the t h e o r e m to define f and fo; and it remains to show ^

that F = (f,f,fo)

satisfies the c l o s e d - f u n c t o r axioms;

similarly for

the "if" part of the second assertion. We shall show here that F (resp. U) satisfies the a x i o m CF3 of [2] page 434 when U (resp. F) does, and when the hypothesis

is

verified that u I and fl are inverse; we leave the easier axioms CFI, CF2 to the reader.

pl,p2:

Write the two legs of CF3, say for ~, as

~[Y,Z]

~

[~[X,Yi,[~X,~Z]];

we are now for simplicity w r i t i n g [Y,Z]

for [Y,Z]'

We similarly use

pl,P2 for the two legs of CF3 for U or for F. We do not, a priori, n:

know CF3 for ~; but we do know that

i ~ Y satisfies CN2, and by an easy d i a g r a m - f i l l i n g - i n argument

we get that (2.8)

[Y,Z]

=

~[Y,Z]

n is independent of i.

~

[~[X,Y],

[~X,¢Z]]

Pi Now give @ its value uf, and express the Pi

for ~ in terms of those for U and for F, as is done in the big d i a g r a m on page 435 of [2], which we shall call A. First suppose we know CF3 for U. (2.9)

~

Then

(2.8) and A give that

uPi

[Y,Z] ~ uf[Y,Z] --~ u[ f[ X,Y] ,[ fX,fZ]]

[uf[X,Y],

ulfX,fZ]]--* [uf[X,Yl,[ufX,ufZ]]

ll,;l is independent of i.

But since

(2.4) is the i s o m o r p h i s m Ul, the

^

m o r p h l s m s u and ¿l,u] cancelled:

in (2.9) are b o t h c o r e t r a e t i o n s and can be

the composite of the first two m o r p h i s m s in (2.9) is

already independent of i. diagram

This

implies that the top leg of the

271

fuP i

fq .............

flY,Z]

~

fu[ f{ x,Y] ,I fx, fz] ]

fuf[ Y,Z]

[ fiX,Y]

f[ Y,Z]

'Pi is independent triangle

of i.

But the square

commutes

by one of the adjunction-equations;

of i, which

by naturality, hencePi

and the

is independent

is CF3 for F.

Now suppose We conclude

,[fX,fZ]]

instead

that we know CF3 for F and seek it for U.

from (2.8) and A that

(2.10) [Y,Z]

n ; uf[Y,Z]

uf >

Pi u[fY,fZ]~

[u[ fX,fY] ,[ufX,ufZ]] I [ uf,l] luf[ X,Y] ,l ufX,ufZ] ]

is independent

of i.

Write this for Z = uC, compose

[l,[l,uc]]

: [uf[X,Y] , [ufX,ufuC]]

naturality

to move this last morphism

on the end with

~ [uf[X,Y] ,[ufX,uC]] , and use ^

in the process

it becomes

first three morphisms isomorphism (2.11)

u[l',e]

:

back through u[ fY,fuC]

in the thus-transformed

fl of (2.5) and can be removed.

u[fY,C]~

[uf,l]

and Pi;

~ u[fY,C] ; so that the composite

(2.10)

form the

What we now have is that

[u[fX,fY],lufX,uC]]

>- [uflX,Y],[ufX,uC]] ^

Pi is independent ule,l] l~,l]

: u[B,C]

of i.

[ uf, i] Now set Y = uB, compose with

-+ u[fuB,C]

: |uf[X,uB],|ufX,uC]]

naturality

on the front,

-* [|X,uB],lufX,uC]]on

to move the first morphism

that the last three morphisms [f1,1],

we have that

and with the end.

through the second,

then give an instance

Using and observing

of the isomorphism

272

(2.12)

u[B,C]

[ u[ fX,B] ,[ ufX,uC] ]

~Pi

is i n d e p e n d e n t

of i.

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

The

"extraordinary"

naturality

of Pi in A gives

of Do

u[ B,C]

t

...........

[ u[ A,BI ,[ uA,uC] I

[ u[ fuA,B] ,[ u f u A , u C ] ]

~

[ u[ A,BI ,[ u f u A , u C l 1 ;

[l,[u¢,l]] since by but

(2.12)

the top leg is i n d e p e n d e n t

[l,[u¢,ljjis

retraction

a coretraction

by one of the a d j u n c t i o n

is in d e p e n d e n t

of i, which

3.

~.i

cancellable,

equations.

of

Reflective

§I to g e n e r a l i z e

So once 2-category

special (3.1)

again we are on w h i c h

ua b e i n g a

the Pi on the left

subqatq$ories

Day's

result

D, and use the

in [i] .

given an a d j u n c t i o n

D acts,

doctrine

n,e:

but now we suppose

f--~u:

A ~ A' in

that we are

in the

case where fu = I

The results this were

and

being

CAY;

~ = !:

fu ~ i.

independent

then

subcategory

adjunction

equations f~ = i:

of the

2-category,

(3.1) may be e x p r e s s e d

reflective

(3.2)

Thus

leg;

is CF 3 for U.

We return n o w to the case of an a r b i t r a r y

results

the

and hence

of i, so is the b o t t o m

of A', with

((2.1)

and

f ~ fuf;

by c a l l i n g

inclusion

(2.2)

of [6]

~u = i:

we again w r i t e A a full

u and r e f l e x l o n above)

u ~ ufu.

as if

f.

now become

The

273

We suppose a c t i o n n': giving

A', but not A, to be g i v e n with a D-structure,

DA' ~ A'.

By an e n r i c h m e n t

of a D - s t r u c t u r e

enriching way that

to A, with a c t i o n

of u and f to D-functors n,~ provide

By T h e o r e m

an a d j u n c t i o n

1.5 to give

DA'

n:

U = (u,u) F--~U

situation

DA ~ A say,

we m e a n the and the

and F = (f,~)

in such a

in D-CAT.

an enrichment

to A and to give an i s o m o r p h i s m (3.3)

of this

with

is to give

a D-structure

f as in

n'

A'

f Df

f

DA

~

.....

A

n

that m a k e s at all, n.Df

a strong

it is u n i q u e

~ fn',

determined (3.4)

of (f,~)

while

D-functor.

If such an e n r i c h m e n t

to w i t h i n a suitable

(3.1)

gives

Df.

isomorphism;

for

exists

(3.3) gives

Du = i, so that n is e f f e c t i v e l y

by

n ~ f.n'.Du. Now consider

the c o m p o s i t e

(3.5)

DA'

....~.

DA'

i If an enrichment which

enrichment

to exist

D is the d o c t r i n e single

exists,

is the i d e n t i t y

condition

~

A'

~

n'

by

(3.5)

is isomorphic

(3.2).

is that

Thus

(3.5)

for m o n o i d a l

by

(3.3)

a necessary

this

to n. Df.D~,

condition

be an isomorphism.

categories,

A.

f

In the

clearly

for an case where

reduces

to the

274

(3.6)

f(n®'n):

f(X®'Y) ~ f ( u f X ® ' u f Y )

and Day [I] has shown that in this case

is an isomorphism; (3.6) is also sufficient.

H o w e v e r the i n v e r t i b i l i t y of (3.5) can hardly be sufficient for a general D; if D were the d o c t r i n e

for strict m o n o i d a l categories,

the

i n v e r t i b i l i t y of (3.5) would still reduce to (3.6), but the m o n o i d a l structure c o n s t r u c t e d by Day on A, given by A@B = f(uA@'uB), would not in general be strict even though that on A' was strict. the Introduction, "flexibility"

As we said in

the doctrine for m o n o i d a l categories has a certain

- an ability to absorb

isomorphisms

- which the doctrine

for strict m o n o i d a ! categories lacks. It turns out that the i n v e r t i b i ! i t y of (3.5) is n e c e s s a r y even for the existence of a p s e u d o - e n r i c h m e n t ; sufficient.

To be precise,

and that for this it is also

let D' be the d o c t r i n e whose a l g e b r a s are

the n o r m a l i z e d pseudo D - a l ~ e b r a s in the sense of Street [i0] volume.

§2 in this

That is to say, a D'-category differs from a D-category in

that the a s s o c i a t i v i t y axiom for the action is satisfied only to within a (prescribed) axiom h o w e v e r nose.

isomorphism,

subjected to suitable axioms;

(this is what "normalized" means)

T h e n a D ' - f u n c t o r |resp.

calls a lax h o m o m o r p h i s m [resp. algebras.

is what Street

strict homomorphism]

every strict D - f u n c t o r is a f o r t i o r i d o c t r i n e - m a p p:

is satisfied on the

strict D'-functor]

Since every D-category is a f o r t i o r i

the unitary

of p s e u d o - D -

a D'-category,

a strict D'-functor,

and

there is a

D' ~ D; and it is shown in my paper [5] b e l o w in this

volume, which also Justifies the other remarks above, that p is a retraction. ~.2 T~orem

We can now give the m a i n result of this section. ~.!

Let the r e f l e x i o n ~,~:f--~u:

A ~ A' be ~ i v e n as above,

and let A' be a D-category, hence also a D'-category.

Then the

r e f l e x i o n admits a D ' - e n r i c h m e n t if and only if (3.5) is an isomorphism.

275

Proof pA':

First, the necessity. D'A' ~ DA'.

Consider the composite of (3.5) with

By the 2-naturality of p, the m o r p h i s m pA' can be

m o v e d past the triangle Dn in (3.5), turning the latter into D'n. But then pA' composed w i t h n': A'.

DA' ~ A' is just the action of D' on

Thus the composite of (3.5) with pA' is just the analogue of (3.5)

with D' r e p l a c i n g D. a D'-enrichment

We already know that this must be invertible

is to exist.

But then (3.5) itself must be invertible,

since pA' is a retraction. We turn to the sufficiency. n:

Guided by (3.4), we define

DA ~ A by

(3.7)

n = f.n'.Du.

We can then write (3.8)

¢ = f.n'.Dn

(3.5) as an i s o m o r p h i s m :

fn' ~ n.Df,

and we define the ~ of (3.3) to be the inverse of ~. We check that n satisfies the unitary law for an action. have n.jA = f.n'.Du.JA = f.n'.jA'.u

by (3.7) by the n a t u r a l i t y of j

=

fu

by the unitary law for n'

=

1

by (3.1).

As regards the associative law, we have n.mA = f . n ' . D u . m A

by (3.7)

= f.n,.mA,.D2u

by the n a t u r a l i t y of m

= f.n,.Dn,.D2u

by the a s s o c i a t i v e law for n';

while n.Dn : n . D f . D n ' . D 2 u Therefore the i s o m o r p h i s m

if

by (3.7).

We

276

(3.9)

U = ¢.Dn'.D2u:

f.n'.Dn'.D2u ~ n.Df.Dn'.D2u

is an i s o m o r p h i s m (3.10) ~:

n.mA = n.Dn,

and we define v to be

-i

It remains to verify that n and v constitute a n o r m a l i z e d p s e u d o - a c t i o n of D on A, and that f and ~ c o n s t i t u t e a "lax h o m o m o r p h i s m of p s e u d o - D - a l g e b r a s " in Street's language. we have a D ' - a c t i o n on A and a strong D ' - f u n c t o r whenqe the desired D ' - e n r i c h m e n t

rather different:

(5),

(f,f):

A' ~ A,

follows by T h e o r e m 1.5.

The axioms to be v e r i f i e d are the axioms §2 for v, and the axioms

Then in terms of D',

(6) of [I0]

(i),

§2 for f.

(2),

The notations are

Street's i is our j, his c is both our m and our n,

as well as our n'; his ~ is i in our case

(normality);

his e is our v,

and also our v' which is 1 (A' being a honest D-category); is our f. inverses,

(3) of [I0]

and his ¢f

In v e r i f y i n g these axioms we replace v and ~ by their ~ and % respectively,

Axiom (i) reads

inverting the arrows accordingly.

~.jDA = i.

~.jDA

But

=

¢. Dn'.D2u.

=

%. jA'. n'.Du

This will therefore follow from axiom

¢.jA'

Axiom (2) reads p.DjA p.DjA

jDA

by (3.9) by the n a t u r a l i t y of j.

(5), which reads ¢.jA'

: i.

But

=

f.n'. D~,JA'

by (3.8)

=

f.n'.jA'.n

by the 2-naturality of j

=

fn

by the unitary axiom for n'

=

i

by (3.2).

=

i,

But

=

¢.Dn' .D2u.DJA

by (3.9)

=

¢.Dn' .DJA' .Du

by the n a t u r a l i t y of j

=

¢.Du

by the unitary axiom for n'

=

f.n' .D~.Du

by (3.8)

277

=

i

since nu = I by (3.2).

This leaves us with axioms reduces

to the second,

the vertical vertical

composite

composite

via (3.9).

¢.mA'.

(6).

The first of these

For the left side of axiom

of n.D~ with ~.DmA,

of ~.D2n with ~.mDA.

and the right

Now using

=

n.D¢.D2n'.D3u;

u.DmA

=

¢.Dn'.D2u.DmA

=

¢.Dn,.DmA,.D3u

by the naturality

=

¢.Dn,.D2n,.D3u

by the associativity

~.D2n

=

~.D2f.D2n,.D3u

by (3.7);

~.mDA

=

¢.Dn'.D2u.mDA

=

¢.mA,.D2n,.D3 u

and the right So axiom

n.D¢

side is the vertical

(3) follows

It remains

by the naturality

(6) is the vertical

from axiom

then to verify

axiom

by (3.7) and

¢.Dn'

=

f.n'.D~.Dn'

by (3.8);

~.D2f

=

f.n'.Dn.Dn'.D2u.D2f

by (3.9) and

¢.mA'

=

f.n'.D~.mA'

by (3.8)

f.n,.mA,.D2n

by the 2-naturality

f.n,.Dn,.D2n

by the assoclativity

composite

with D2n'.D3u.

Now

f.n'.Du.Df.Dn'.D2~

The vertical

of n.D¢ with

of ~.D2f with

=

=

axiom for n';

of m.

(6) on composing (6).

side is the

of m

composite

composite

(3) is

(3.9)

n.D~

But the left side of axiom ¢.Dn',

(3)and

of the first two of these

(3.8);

(3.8);

of m axiom for n'.

is indeed equal to that

of the second two, both being D2A

DA

~" DA' i

Dn'

~- D A ' ~ i

A' n'

= f

A.

278 This

completes

3.7

the proof.

As we have

we can a c t u a l l y

said in the Introduction,

get a D-enrichment.

not m e r e l y

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

F = (f,f)

to be not m e r e l y

strict.

This may

in nature, extremely

useful

Theorem

~.2

strong

[5]

first

an a r t i f i c i a l

be by T h e o r e m

case;

show,

(unpublished)

is that where

and

~,s:

f--~u:A

Then the r e f l e x i o n

w i t h F strict

(3.5)

Proof must

if and o n l y ~ i f f.n'.Du.Df

For the necessity, have

f.n'

fn = i by

= n. Df.

I have

found

it

related

to

~ A' be g i v e n as above, admits

is the i d e n t i t y

a D-enrichment

(which includes

we are to have

(3.3)

is n. Df.Dn,

with f = i, so that we

which

is the identity

since

is c o n t a i n e d

case we have

in the proof of T h e o r e m

¢ = 1 and ~ = i, or equally

3.1,

since

f = 1 and

D We

said towards

is a retraction; D = D'

that

the end of is, there

with pq = 1.

doctrine-map.

§3.1 that

the d o c t r i n e - m a p

is a 2-natural

In general

q to be a

q with pq = l, we call the

doctr i n e

D the doctrine

D' is flexible;

and we also

show that D is

flexible

if D is of the form K °-, where

K is a club

in

in the or that

sense

whose

discrete

of [4]

monoidal

below,

we show that

JKJ of objects

§10.3 above.

for symmetric

functor-operations

club

[5]

suppose

D flexible.

CAT/set)

In the paper

transformation

we cannot

If we can find a d o c t r i n e - m a p

p: D' ~ D

doctr i n e

in

the

= f.n').

So (3.5)

sufficiency

in the present

q:

1.5) but

(3.2).

The

v=l.

to a s k i n g

it formally:

Let the r e f l e x i o n

that

is

it o f t e n arises

considerations

and let A' be a D-category.

assertion

but

in w h i c h

(3.5)

which corresponds

(as it must

b e l o w will

in various

To state

The

there are two cases

but an identity;

seem rather

as the paper

coherence.

0

Thus

the club

categories,

are g e n e r a t e d

for any

CAT/~

(or even

is a free d i s c r e t e for m o n o i d a l

is flexible,

freely by @ and I; the

since club

club

categories, the for strict

279 m o n o i d a l categories axioms like ®(®,~)

escapes the result

since ® and I are subjected to

= ®(~,®).

By a p p l y i n g the theorem b e l o w in the case where D is replaced by D', we see that T h e o r e m 3.1 remains true even when A' is originally given as a D ' - c a t e g o r y rather than a D-category.

T h e o r e m ~.~

Let the r e f l e x i o n q,c:

above~ and let A' be a D-cate~ory~ r e f l e x i o n admits a D-enrichment

f--~u:

A ~ A' be ~iven as

where D is flexible.

Then the

if and only if (3.5) is an

isomorphism. Proof

The necessity was shown in §3.1.

For the sufficiency,

use T h e o r e m 3.1 to get a D'-enrichment. q-CAT:

D'-CAT ~ D-CAT

(cf. [6]

first

Then apply the 2-functor

§3.6 above)

to get a D-enrichment.

The only point at issue is whether the D-structure A' now has is that it started with;

this is ensured by pq = i. D

REFERENCES

l 1]

B.Day

(=B.J.Day), A r e f l e c t i o n t h e o r e m for closed categories, Jour.

[2]

Pure and Applied A l g e b r a 2 (1972), i-ii.

S. E i l e n b e r g and G.M. Kelly, Categorical Algebra

I3]

Closed categories,

(La Jolla 1965),

G.M. Kelly, On clubs and doctrines,

[5]

G.M. Kelly, Coherence theorems distributive laws,

in this volume.

for lax algebras and for

in this volume.

G.M. Kelly and R. Street, Review of the elements of 2-categories,

[7]

Lecture Notes in

106 (1969), 166-177.

[4]

[6]

Conf. on

S p r i n g e r - V e r l a g 1966.

G.M. Kelly, A d j u n c t i o n for enriched categories, Math.

Proc.

in this volume.

A. Kock, Monads on symmetric m o n o i d a l closed categories, Arch. Math.

21 (1970),

i-I0.

280

[8]

R. Street, The formal theory of monads, Jour. Pure and Applied Algebra 2 (1972), 149-168.

[9]

R. Street, Two constructions Top. et G$om.

[ io]

on lax functors,

Diff. XIII 3 (1972),

Cahiers de

217-264.

R. Street, Fibrations and Yoneda's lemma in a 2-category, in this volume.

[ 11]

H. Wolff, Commutative distributive laws, to appear in Jour. Australian Math. Soc.

COHERENCE

THEOREMS

FOR LAX ALGEBRAS

AND FOR DISTRIBUTIVE

LAWS

by

G.M.

i.i

We p r o v e

two m a i n

o n Cat.

(For t h e m o s t

Kelly

results.

basic

First,

notions

about

let D be a d o c t r i n e , doctrines,

see [13]

say

above,

,

familiarity (honest)

with which

algebras

are

we a s s u m e . )

Let D

the lax a l g e b r a s

be t h a t

for D,

doctrine

in t h e

sense

whose of [24]

,

above.

Since

a D-algebra

is a f o r t i o r i

a D -algebra,

there

is a

,

doctrlne-map

[Cat,Cat]

s:

D

~ D.

of e n d o - 2 - f u n c t o r s ,

doctrine-map)

such that

sB = 1 a n d n h = I.

Cat, "full by

the

existence,

of a m o r p h i s m

h:

D ~ D

sh = I, a n d o f a 2 - c e l l

Borrowing

the

terminology

in the

n:

2-category

(not a

1 ~ hs

usual

s u c h that

in the

2-category W

we m a y

say t h a t

reflective

"pseudo",

this

the

same

case

this

result,

along with

For law between for

natural

second

ones

examples

hopelessly

Moreover, true,

rare

[i0]

result,

at the d o c t r i n e for c a t e g o r i e s , with

actions

D-actlon

is g i v e n

the

structure

some

doctrine

D

onto

"lax"

the

above

n o w n is a n i s o m o r p h i s m ;

of e n d o - 2 - f u n c t o r s .

let p:

original

provided

for

if we r e p l a c e

we

shall also

D ' D ~ DD'

or r a t h e r

in the

of D

Part

prove,

so

of

has

above.

category

algebra

but

some r e f i n e m e n t s

two d o c t r i n e s

strict

s as a " r e f l e x l o n "

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

in m y p a p e r our

D.

is s t i l l

in t h i s

been applied

exhibits

subobject"

that

law;

We prove

.

sense

level while

both

a pseudo-distributive-

of B e c k

(just

[2[

are r a r e

in

as A @ B = B ~ A is

A @ B a B @ A is c o m m o n ) .

of D a n d o f D',

of a ( n o n - s t r i c t ! ) Were

be a d i s t r i b u t i v e

and

for w h i c h

D'-morphlsm,

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

A the

is a n

law a strict

one,

282

DD' would

itself be a doctrine,

with a D - a c t i o n D'-morphism; case there

and a D'-action,

this

pseudo-distributive rate a D -algebra; whose

reflective

subobject

monoidal

again,

monads

strictly

ones are easily monoidal

distinguished, closed

two symmetric

is a special for a more formulate principles

results

- - now more

One

let us now call

doctrine,

monoidal

there

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

of

(although

the e n r i c h e d

those

similarity

~

that he called

this

D'

the

monads.

Once

for symmetric if D is any

[19],

In p a r t i c u l a r

D'.

In that

who considers

@ and @ together

it applies

case

our

a category

with

with a

(A@B)~(A@C).

lax

the present

very

general,

situation. proofs

have

in §2 and §3 below,

of this

on a

but p s e u d q - c o m m u t a t i v e

Then

applies.

particular

case of s o m e t h i n g

collected

are rare,

of our two m a i n results

lax and a less

monads

is a p s e u d o - d i s t r i b u t i v e - l a w

result

structures

A®(B@C)

on coherence

diagrams.

onto a full

often called m o n o i d a l

that of Laplaza

as we have

strict

i = ii ~ DD'.

to be a strong

such is the d o c t r i n e

The rest comme n t s

of D

found.

this:

but is at any

is jj':

doctrines

"distributivity" The

among

category,

and the above

contains

In our

as the unique

it is a c t u a l l y

the D - a c t i o n

when D too is taken to be this result

i ~ D

In that

D'-morphism.

categories;

D'D ~ DD',

and that

~ DD'.

~ DD'

commutative

pseudo-commutative p:

D

a doctrine,

D

:

be categories

of Beck { 2 }.

s is a r e f l e x i o n

if we require

still not a strict)

commutative

with j

that

DD';

s:

would

being a strict

s:

is no longer

so we can define

is once again

symmetric

DD'

composition

Kock [14]

situation

be a d o c t r i n e - m a p

case,

Our result

endo-2-functors

the D - a c t i o n

is the classical

would clearly

D -morphism

and its algebras

introduction

problems,

relating

that each

the d o c t r i n e s

I have not a t t e m p t e d in common only

distinct.

to general

the r e l a t i o n

about

to

some general

and are otherwise

is devoted

including

stated and results

suggests

between

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

such of

2~

1,2

To be told,

for some d o c t r i n e and the 2-cells implicitly. that

D

situations,

(along with the

strict m o r p h i s m s

between

the

strict m o r p h i s m s )

By the "coherence

of d e t e r m i n i n g

soluti o n

as in the above

of this

I understand

D

problem"

explicitly

is o f t e n beyond

a complete

are the a l g e b r a s of these,

is to be given D

for D* I u n d e r s t a n d

from this

information.

our powers.

or partial

what

solution

primarily

A complete

By a "coherence of this problem:

theorem" a

,

result

w h i c h tells

us

something

of Cat.

For instance,

functor

D; or even that

as a full r e f l e c t i v e

that

at least

about

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

it contains

subobject.

D

~ua endo-2-functor

to some k n o w n

endo-2-

a k n ow n D, in some d e f i n i t e

Results

about

certain

diagrams

commuting

m a y be part of such a theorem,

or may be among its

important

consequences;

can,

seen as c o n s t i t u t i n g

but

the essence

I shall return First,

I want

simplified (a)

to observe

in various

The word

"monad".

the case

(b)

such as the

on Cat.

A-indexed

family

with

monoidal

like

internal-hom,

An e q u a t i o n a l

Cat/A.

2-cells,

themselves

point.

to be r e p l a c e d

structural

categories,

structure

closed

V.

structure

are the

functors

are all

subjected

functors

if one considers may be borne are a l g e b r a s

as in of m i x e d

only

by a for a d o c t r i n e

of d o c t r i n e s

But then too there

an E E LCat,CatJ the

by

involved.

and these

i ~ E satisfying

diagrams.

for a mere m o n a d when,

are

be

has been over-

may have

It is easy to conceive

"lax doctrines";

E 2 ~ E and unit certain

the

closed

of categories,

2-category

of c o m m u t i n g

some e q u a t i o n a l

on Cat when

no longer

theorem.

of its polemical

(= 2-monad)

V-Cat for a symmetric m o n o i d a l things

sake

One takes m u c h too n a r r o w a view

doctrines

on the

for the

endowed

I believe,

the last p a r a g r a p h

but they are the a l g e b r a s

of symmetric

variance,

of a c o h e r e n c e

to the m a t t e r

that

ways

for a d o c t r i n e

covariant;

later

"doctrine"

Categories

algebras

such results

way,

"doctrine

on are

with m u l t i p l i c a t i o n axioms"

to so-called

only to within

"coherence

axioms";

2~

such an E is an a l g e b r a

~Cat,CatJ.

Clearly

a mere monad) reasonable

(c) term

some d o c t r i n e

we should

take D

on an a r b i t r a r y

to suppose

such as being

for

locally

2-category

K; a l t h o u g h

(or p o s s i b l y

it m a y be

some of the good p r o p e r t i e s

of

Cat,

presentable.

It is r e a s o n a b l e "coherence

on the

to be a d o c t r i n e

2-category

that K has

D

theorem"

- and p r o b a b l y for a result

very

which,

common having

- to use the found out

,

something

about

these a l g e b r a s Mac Lane's

D

from a k n o w l e d g e

and deduces

original

something

coherence

saying that the d o c t r i n e - m a p functors,

where

and D that

D

gives a complete which

further

determination

consequence

that

symmetric-monoidally although takes

it for granted

case,

and in one other,

"Kategorien

1973").I

equiv a l e n c e

s:

to a D - a l g e b r a pursue would

D

has not

people's

D

symmetric

monoidal

to a strict

to refine

that

one.

statement

But

in fact

it has

category IThis

is

in this

It is a special

is true,

has been

and he

(at the O b e r w o l f a c h

every

conference

of such an

D -algebra

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

been discussed;

nor do we

out for investigation,

of what a c o h e r e n c e

known

Beck 13]

the e q u i v a l e n c e

than

P

the

is well

in print.

the existence

- but it cries

categories

but this a s s e r t i o n

more

as

of endo-2-

This

of fractions.

in ~71,

instance,

is Po- for a club

result).

to my k n o w l e d g e

ideal

categories.

(since

in the proof

~ D implies

For

monoidal

Lane's

How g e n e r a l l y

it in this paper be m a n y

of D

it asserts

error

them.

may be i n t e r p r e t e d

for symmetric

any precise

of Isbell

about

goes back to

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

via c a t e g o r i e s

by its author; a radical

every

E23]

monoidal

and proceeds

case of the a s s e r t i o n

discovered

D

by Mac

equivalent

I do not recall

withdrawn

s:

symmetric

is wholly d e t e r m i n e d

useful

theorem

is the d o c t r i n e

for strict

of its algebras,

theorem

and

should be.

285

1,3

We return now to the m a t t e r of c o m m u t i n g diagrams.

In some

cases we know at once from the d e s c r i p t i o n of the D -algebras that D is Co- for some club C, and that C is given by specified generators and relations; given there.

see §I and §I0 of [9]

above, and the further references

The w o r d - p r o b l e m involved in finding the objects of

C

is t y p i c a l l y easy - often indeed they are freely generated - while that involved in finding the m o r p h i s m s is typically m u c h harder. It is a matter of finding the category g e n e r a t e d by a certain graph subject to certain relations - a g e n e r a l i z a t i o n of the w o r d - p r o b l e m for monoids,

inasmuch as a category is a "monoid w i t h many objects".

Deciding when two words in the g e n e r a t o r s represent the same m o r p h i s m of C is d e c i d i n g which diagrams commute in C.

This is the most

common sense in which solving a coherence problem,

completely or

partially, may involve proving that certain diagrams commute.

The

typical example is that of symmetric m o n o i d a l closed categories,

the

club for which was d e t e r m i n e d in part by Kelly-Mac Lane [ii] , and then more fully by Voreadu

([27],[28],[29]),

but is still not known

completely. (Other examples where the club is partially but not completely d e t e r m i n e d are: V-categories,

a symmetric m o n o i d a l closed category V, two

two V-functors,

and a V-natural t r a n s f o r m a t i o n

(Kelly-

Mac Lane [12] ); two symmetric m o n o i d a l closed categories and a symmetric m o n o i d a l functor

(Lewis [ 21] ,[ 22] ); a category with two

symmetric m o n o i d a l structures ~ and @ along with a "distributivity" A®(B@C) ~ (A®B)@(A@C)

that is not required to be an isomorphism

(Laplaza [ 18] and [19]). d e t e r m i n e d are:

Some examples where the club is completely

a m o n o i d a l or symmetric m o n o l d a l category

(Mac Lane

[231); a category w i t h a t e n s o r - p r o d u c t and a n o n - i s o m o r p h i c a:

(A@B)®C ~ A®(B@C)

(Lapiaza [17]); a category with a monad

(Lawvere [20]; Lambek [16])

- the doctrine is just ~x-; two symmetric

m o n o i d a l categories and a symmetric m o n o i d a l functor

(Lewis [21]

and

286

[ 22]).

I do not

of clubs

suggest

that

by their authors;

last-mentioned

result

but this

of Lewis

for it sets out to determine augmentation meant

r:

the above results is what

they amount

is so expressed,

C completely

C ~ Y is not

are expressed

faithful.

to.

in terms The

and indeed must

in a case where Finally,

this

be:

the

list

is not

to be exhaustive.) ,

One could of d e t e r m i n i n g

D*A for every

free D -algebras. who

also regard

This

sets up generators

the d e t e r m i n a t i o n

object

A;

that

is,

of D

as the p r o b l e m

of d e t e r m i n i n g

is the view taken by Lambek

([15]

and [161),

for D *A , and who attacks

and relations

the

the

,

problem

of which diagrams

category.

in D A commute

To this end he has

Gentzen's

in place

knowledge

of C i m m e d i a t e l y

well at the level

of C.

"equi-generality

D A, and

Lambek's

studying

(cf.

When,

idea of setting

commutativity

for a club C, of

- for any

knowledge

of CoA,

and

techniques

those results

commutativity"

of

turn out

and some of them are

however,

up generators

of diagrams

of

the c o n s i d e r a t i o n

In fact,

implies

[8 ]§6).

is Co-

the c u t - e l i m i n a t i o n

of " r: C ~ Y is faithful";

seen to be false

adaptation

complication

corresponding while

of the form

from a club,

gives

or not;

to be r e - p h r a s i n g s then

by Lambek,

of C is an u n n e c e s s a r y

this for any A, discrete work p e r f e c t l y

a brilliant

Yet when D

considered

D A = CoA

Lambek

introduced

work on cut-elimination.

as it is in the examples

when A is a discrete

at this

D

does not

come

and r e l a t i o n s level,

for

m a y well

,

be the best way of getting D -algebra

is a c a r t e s i a n

by Szabo

(]251

1.4

Thus

establishing

at D

.

closed

An example

category,

such results

w h i c h has been

investigated

and [ 26]). proving

diagrams

a coherence

commutative

theorem,

stated

in §I.i.

imply theorems

may be a tool

it; but it

some grip on D , as is shown by

On the other hand about

in

or even a way of stating

need not be the only way of g e t t i n g our results

is that where a

diagrams

I now point

commuting,

out that

at least

when

287

D,D'

come from clubs:

and imply them wholesale,

clubs are arbitrary. argument

In this way I believe

for calling

them

"coherence

P

CAT/~,

for a club ~ in

, and that

s,h,n all arise

one clinches

it is easy to see that

then D* is P * o from things

for another

change n o t a t i o n

and write

sh : i, s~ = I, n h = i. necessary

condition

in general ~S:

it always

P

If f,g:

~

T ~ S is a d i a g r a m

for it implies

It is however

is in the

"pseudo"

when S is in the image

of h, i.e.

in D

is sf = sg.

, a

This

is not

only that nsf = ~ s g , where

sufficient case,

such club

so that we may as p* P ~ , ~: i ~ hs,

D, h:

for its c o m m u t a t i v i t y

sufficient,

S ~ hsS.

s:

if D

in CAY/~;

*

well

the

theorems".

In the first of those results is Do-

in so far as these

if ~S is an isomorphism,

and as it is in the when S belongs

"lax"

as

case

to the full

,

subcategory explicitly

~ of D

Since

and directly;

commutativity class

.

in the The

s is a club-map,

sf can be c a l c u l a t e d

so we have an effective

of any d i a g r a m

in the

"pseudo"

test

case,

for the

and of a large

"lax" case. same

is true

in the

second

result

of §i.i,

in wh i c h

it is first

stated (the D - a c t i o n

If f,g:

T ~ S in D

with S in the full r e f l e c t i v e

then f = g if and only if sf = sg.

an a r b i t r a r y

in the form

D'-morphism).

DoD',

subcategory

In the case where

the D - a c t i o n ,

is to be a stron~

D'-morphism,

comes

in CAT/~ when D and D' do.

from a club

indicated

in §1.5 of [9];

Laplaza's

problem

distributivity

d:

of two

it is no longer

in the special symmetric

A~(B¢C)

~

and we can take it to be in ~ by p a s s i n g But

if we ask d to be an isomorphism,

its type neither

in ~, and dd-l:

(A~B)¢(A®C)

in S nor in soP;

the d o c t r i n e

any sense we can at present

This

structures

to

and a

the type of d lies to the opposite

its inverse ~

is for the reasons

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

monoidal

(A~B)¢(A~C),

the case that D

does not

come

give to that notion.

doctrine.

d -I already

(A~B)¢(A~C)

in ~op,

has

has its type from a club

in

288

What we can do is to r e t u r n

to the club D

in the n o n - s t r o n g

,

case,

and consider

a model A

image of d happens which the image

(that

is, a D -algebra)

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

of n happens

(or more

in w h i c h

generally

to be a m o n o m o r p h i s m ) .

the

one in

Then although

,

sf = sg does not their

images

whose

objects

general

f,g:

f = g in D

in {A,A},

the

at the m o d e l - l e v e l

involving With

category"

So for

of [ 91

are

§9,

suitably

such a m o d e l A we have

for c o m m u t a t i v i t y

in D

of

- which m e an s

a

of any d i a g r a m

(in Laplaza's

case)

a

d but not d -I.

this

said,

case the doctrines

and with

D and D'

mo~oidal

aategories,

original

result

which

[23],

includes

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

that

are both the doctrine

is equivalent

so that

it is easy to see that our result

imply the e q u a l i t y

"rich e n d o - f u ~ c t o r

transformations.

T ~ S describable

diagram

, it does

are functors A n ~ A and whose m o r p h i s m s

natural

crlt~rion

imply

his theorem

as a map D

"distortion"

in [19]

P for symmetric

to ~ by Mac Lane's

s can be r eg a r d e d

s is L a p l a z a ' s

in L a p l a z a ' s

§4.

([19]

~ ~o~,

§2),

We give more

and that

details

below. 1.5

We end this

coherence

theorem

introduction

exemplified

by o b s e r v i n g

by the results

that the kind

of

of

§i.i

is pretty

result

in [231,

common. First,

if we take Mac Lane's

just at the a s s o c i a t i v i t y a:

(A~B)~C ~ AS(BSC)

part,

original

we see that the i n v e r t i b i l i t y

is not central

to the m a i n argument.

and look of

What

is

,

really p r o v e d

is that

s:

D

~ D is a reflexion,

where D is the

,

subcategory bracketed that

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

wholly

club R

from the right.

the r e f l e x i o n

result

T ~ S whenever

in that

by L a p l a z a

case.

in [17].

S lies

of the objects

It is only b e c a u s e

is in fact an equivalence.

even when a is not an isomorphism, for f = g:

, consisting

a is i n v e r t i b l e

It follows

sf = sg is n e c e s s a r y in D.

This

A total one r e q u i r e s

and sufficient

is a partial

more

work,

that,

and

coherence is g i v e n

2~

To get back to the partLal one, virtually

predictable,

of the present Identifying functor means h:

however:

I say that

it is

and in a sense automatic.

The m e t h o d

of proof

paper r e q u i r e s

a D -functor

P with the set of natural

by h(1)

giving

= !, h(n)

h:

h2 ~ hl ~ h3, and this is of

course

axiom if it is to be a D -functor: axiom for a.

but note that axioms"

similar

maps l:

situation,

but arises

A final say for m o n o i d a l monoidal

categories,

Mac Lane's

result

D

~ D.

But Mac

they are e~uivalent This

equivalence

pairs h: ([21]

with hardly was needed

variables

is not

an artificial

in the old, unsolved, closed

original

D' is not

gives

towards

But we can get a p a r t i a l

[61

from A, not

in

of the

form hA.

CAT/~ club

result

h.

Lewis

completely

for a ~eneral

the d i a g r a m s

they

above.

and c o n s i d e r

by a C-functor

in the m o n o i d a l

from A';

as e n d o - 2 - f u n c t o r s .

the end of §3 of [13]

connected

that

equivalence

to our D ; a l t h o u g h

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

Observe

categories,

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

a similar

isomorphic

result, for strict

for m o n o i d a l

(P) cases were always rather special:

had c o d o m a i n

coherence

D' ~ D that

let C be a club

by E i l e n b e r g - K e l l y

only

category.

as d o c t r i n e s , and not m e r e l y

any effort.

coherence

If D : ~o- is the d o c t r i n e

has d e t e r m i n e d

when C : P or N.

monoidal

A~I ~ A; this

A ~ A' of C - c a t e g o r i e s

and [221)

observation;

I and non-

a doctrine-map

example,

at once to the

object

on Mac Lane's

is d i s c u s s e d

As a last

an

have a constant

Our first result

Lane's

has to satisfy

what

and if D' ~ No- is that

gives

of e n d o - 2 - f u n c t o r s .

~Fedlct

non-symmetric

categories.

of

"coherence

precisely

comment

components

the

I~A ~ A, r:

for a non-monoidal,

a; the other (h,h)

h as a

we need

and this reduces

considerations

sh = i.

it to a D -functor

is in itself a l i g h t w e i g h t

should be when we also

isomorphic

problem

This

Enriching

But

with

~0, we define

in p a r t i c u l a r

are then g i v e n by an easy induction.

pentagonal

D ~ D

numbers

= ~ ~ h(n-l).

hn 0 hm ~ h(n+m);

h:

whose

commutatlvity

(N) and

they

C

symmetric

contained

lay in A'; and they always

A commutativity

criterion

for

such

290

diagrams only

from A means

algebra the

can be g i v e n

looking

on A = I and A'

free

algebra

it is i m m e d i a t e "unknown". l:

at once, not

at the w h o l e

= I where

on A = I and A' that

Since

its d o m a i n

h

is free

C ~ C is a m o d e l ,

rendering

for a g e n e r a l

there

C.

Having

club,

I is the unit = empty.

Let

is C itself, on the g i v e n

are u n i q u e

which

is the

category, this

while

C'

free

but

be h:

at

C ~ C';

is the

generators

strict

variables

and

C-functors

since

n,s

commutative C

n

• C

1

L

IC

C'

s

and having

n(~)

free

C-category

show

that

= ~.

But

on ~,

gives

s is in fact

embedded byh;

so that

Lewis

proves

incredibly

Since

a doctrine in the w i d e A:

A,B,

if and

this

sense:

sh = I.

of C' onto image

only

since

the

C is the

It is easy full

to

subcategory

C,

of h, the d i a g r a m

if sf = sg.

A lot

but m u c h m o r e

less

than

general,

very

cheap.

A method

section

on an a r b i t r a r y

L ~ K, not

L = I).

so that

for C = P or N, a d m i t t e d l y ;

2.

2,1

n = i;

requirement,

if S is in the

T ~ S in C' c o m m u t e s

and

last

a reflexion

f,g:

useful,

this

of c o n s t r u c t i n ~

is p u r e l y

necessarily

As far as p o s s i b l e

formal,

2-category

an a c t i o n

a:

we take

(D,m,j)

K, and w e u n d e r s t a n d

DA~A

to be

"D-algebra"

of D on a 2 - f u n c t o r

on an o b j e c t we use

D-reflexions

a,b,

A of K(the etc.,

special

for a c t i o n s

case of D on

etc. We a s s u m e

and d o c t r i n e s

given

acquaintance in [13]

with

§3 above;

the

general

we r e c a l l

facts

about

in particular

monads that

the

291

free D - a l g e b r a DA has action mA: a:

D2A ~ DA, and that an action

DA ~ A is a strict D-morphism.

We add a few enrichments

a p p r o p r i a t e to the doctrine case. First, (2.1)

Dy:

if y:

r ~ r':

Dr ~ Dr':

not only are Dr, Dr'

A ~ B is any 2-cell, then in

DA ~ DB

strict D-morphisms but Dy is a D-2-cell.

Next,

for a free D-algebra DA and a D-algebra B, there is an isomorphism not only of sets but of categories (2.2)

[L,K]

¢:

(A,B) ~ D-AZ~,(DA,B).

(Here D-Ag@, has as objects the D - a l g e b r a s of domain L; its m o r p h i s m s are strict D-morphisms,

and its 2-cells are D-2-cells.

We recall

that we replace D-~Z@, by D-A£@ when we a l l o w all D-morphisms as 1-cells.)

In detail, S

r

(2.3)

@ sends

A

/ ~ y ~ B

where (2.4)

to

r v

A ~ y ~ " ~

B

=

A

B

S'

~ DA.//@B~B, S t

r * S

DA . .

~8

S

JA

(2.5)

DA

Dr

#e

,

B

DA~/~

=

~Dy

~'DB

• B. b

S v

Now c o n s i d e r an a r b i t r a r y D - m o r p h i s m G = (g,g):

(i.e. not n e c e s s a r i l y strict)

A ~ B, so that g is a 2-cell a

(2.6)

DA

~ A

g

Dg

DB

~ B b

292

is a D-2-cell

Theorem 2,~

a (2.7)

DA

~A

Dg

DB

bB

. . . . . . . . . . .

,

all the edges now bein$ r e s a r d e d as D-morphisms~

of which Dg,a,b

are strict. Proof

The D - n a t u r a l i t y axiom (3.19) of [13]

a s s o c i a t i v i t y axiom (3.18) of [13] Observe that,

for g reduces to the

for a D - m o r p h i s m

(g,g). D

since b.Dg = Cg by (2.5), we can also write

(2.7) as a D-2-cell

(2.8)

g:

Cg ~ Ga.

Observe further that the other axiom for a D-morphism, axiom (3.17) of [13],

(2.9)

the identity

gives

A ~

DA

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

jA

• B

--

A

II i

B.

~g g

2.__~2

Now,

in the situation of T h e o r e m 2.1, replace A by a free

D - a l g e b r a DA.

Define the strict D - m o r p h i s m WG and the D - 2 - c e l l

@G by: DA

(2.10)

D

A

B

=

D

~ DjA

~ B ¢g

Had we replaced DJA by iDA on the right side of this definition, should by (2.9) have got merely the identity. have

Since, however, we

we

293

JA DA

A

JA

i JDA DA

we can conclude

.... DJA

~ D2A

from (2.9), not that @G = i, but that DA G

(2.11)

DA~

A

B

A

i

jA

oAf

~G

Note in particular (2.12) Theorem

that, by (2.4),

~G = ¢(g.jA). 2.2

D-morphisms

@G: YG ~ G is the coreflexion DA ~ B.

That is to say, if s:

D-morphism and ~:

s ~ G is a D-2-cell,

(2.13)

s

~

=

i ~G

(2.14)

8.

DA ~ B is a strict

then

i~ G

B for a unique D-2-cell

of G into the strict

~a ExPlicitly,

~ = ¢(~.JA),

and B is the unique D-2-cell

satlsfyin@

TG

(2.15)

A

g

~ D A ~ ~ B jA

=

A

~

~ D ~ ~ B . jA

S

Proof

Since CG.JA = i by (2.11),

turn gives

S

(2.13)

implies

(2.14) by (2.4), proving the uniqueness

(2.15), which in of ~ satisfying

(2.13). Define

B therefore

value of B is b. D(~.jA).

by (2.14),

so that by (2.5) the explicit

Using this, the definition

the fact that mA. DJA = i, we get (2.13)

(2.10), and

if we compose with DJA the

294

diagram expressing

the D-naturality

of a, to wit mA

mA D2A

D2A

~ DA

DA

Ds

DB

,~ B

s

g

DB

b

The naturality Proposition

(2.16)

2,3

of ~,~ is summed up in:

Let G be the composite

DA

• C

where H -- (h,h),

~B

w E

s

H

t and s are strict,

t and s = ¢r for r:

Then

A~C.

G (2.17)

Proof

DA ~ ~ G G ~G

~G =

Corollary (2.18)

B

=

DC

DA

• B.

h

t

Dr

g.

by

DJA

=

t.h.Ds.DJA

=

t .~.

Dr

2.10) by (2.16)

by (2.4).

L,e~ G be the composite

2.4 DA

DC

rE

Dq

H

~ B;

t

then (2.19)

D

A

B ~G

Proof

=

DA

~B.

P D Dq

In this case the r of Proposition

~H

2.3 is jC.q.

t

[]

295

2.~

In [i0]

adjunction given

s~

§3 a b o v e

h:

B ~

we c o n s i d e r e d

the

problem

B' to a D - a d J u n c t i o n ,

in the

B' as a D - a l g e b r a .

enrichment

to exist;

D-morphism;

A condition

when

it does,

and we c o n s i d e r e d

must

be

case

satisfied

s necessarily

in p a r t i c u l a r

of e n r i c h i n g sh = i, for the

becomes

the

an

a stron~

special

case w h e r e

s is to be strict. Here starting

point, For

that r:

we are A ~ B;

that

we a p p r o a c h in the

case

same

situation

where

B'

is a free

this

section,

we s u p p o s e

that

given

a strict

D-morphism

s:

that

we are g i v e n

from a d i f f e r e n t D-algebra

B is ~ i v e n DA ~ B,

DA.

as a D - a l g e b r a ;

say

s = Cr for

a D-morphism

H = (h,~):

B ~ DA;

else

in o r d e r to get

and

we have

(2.20)

sH

=

i

as D - m o r p h i s m s . n:

the

1 ~ Hs

data

We a s k what

satisfying

sq = i and n h

for a D - r e f l e x i o n Write

which

D-morphism, between

(mere)

by

and

C:

2-cells

(2.4).

~:

¢(jA) ~:

then

by

On the o t h e r

hand

i:

By T h e o r e m

1 = Hs = G and

JA

so as to c o m p l e t e

Hs;

~ ¢(hr)

(2.21)

= I,

are

~ hr,

as

a D-2-cell the

B.

composite

is ¢(jA).

D-2-cells

D-2-cells

of DA onto

G for the

is ¢(hr)

we n e e d

2.2,

~G

= ¢(h.s.jA),

DA ~ DA is a strict

there

D-2-cells

in t u r n

(2.12)

is a b i j e c t i o n

C:

1 ~ YG;

in b i j e c t i o n

by

such

(2.2)

with

in

A

~ DA.

jA Using to w r i t e

(2.5)

YG and $G'

to w r i t e we

~ = ¢~,

see by T h e o r e m

and u s i n g

(2.17)

2.2 that

n is the

(with

t : i)

composite

296 B

(2.22)

of

DA

course

Dr

the

the b o t t o m

top

~ DB

leg

of

Since s~

this

=

s.mA.D~

=

b. DS.

=

b.D(s~)

=

¢(s~)

s~.

By

D~

Theorem

Summing 2.5

by

and with

D-2-cells

if a n d o n l y

n:

method

in o n l y widely

one

(2.23)

•

sn = i if a n d o n l y

if

up:

i ~ Hs a n d

s:

DA ~ B a n d H:

2-cells

~:

B ~ DA with as in

jA ~ hr;

and

(2.22) s~ = i

G reflexion,

this,

is a n i m p o r t a n t

of the

and

and

we a l s o n e e d n h = i; I see no

and have

it v e r i f y

it by a n ad hoc

case.

in s i m i l a r

~ G = ¢(hr)

(2.5),

strict

sH = i, t h e r q is a b i j e c t i o n

way of expressing

There

= s by

sh = I, so

s is

D-morphisms

i,,f, s~ = I.

in e a c h

b.Dr

two p r o b l e m s problems.

I = ¢(JA),

;

= i.

of c a t e g o r i e s ,

To get o u r d e s i r e d general

since

(2.20),

(2.5)

all thus Given

s = Cr s t r i c t between

hs

since

¢ is an i s o m o r p h i s m

= i.

is

l e g is I s i n c e m A . D j A

Now consider

sn

DA

special

case which,

studied

in this

It is t h a t w h e r e

this

A

is t h e

paper, Y G = I.

case where

~ JA

DA

while

it o c c u r s

does

occur

Since

297

co~nutes.

The

condition

if we n o w c h o o s e

~ = 1.

on us if we want n h n.jA

~h

= l;

= l; since ~ G . J A

is I~ = 1 by

(2.4).

= l; this

special

case

Theorem

2.6

If the d a t a

It m u s t and

this

3.

We are

applying

Theorem

that

going

(2.23)

gives

gives

~.jA

~ = 1 is not

is f o r c e d

= l; w h i c h

sufficient

ad hoc

verification.)

Summing

are

in T h e o r e m

and

if h . D r . h

it is a u t o m a t i c

What

going

~:

its

= ~. Dr o b t a i n e d satisfies

for

up this

and d e a l

It s a t i s f i e s

case

announced

Theorem

we a s s e r t e d

1.5 of [i0]

in §i.i

2.6.

in §i.i; Once

as it is in b o t h

the

by

So we are the a d j u n c t i o n

given,

however,

results

of §i.i,

above

that

any

So our w o r k i n g

with

D -adJunctions

of our r e s u l t s

this m e t h o d

~ = i

as d o c t r i n e - m a p s

to be a D - a d j u n c t i o n .

by T h e o r e m

settin~

D

special

D -morphism,

if m o r e o v e r sn = 1 and

by

s~ = 1.

our r e s u l t s

than

2.5,

1 ~ Hs w i t h

and D - m o r p h i s m s

to get m o r e

we can r e c o g n i z e

n

= I.

to a D - a d j u n c t i o n . not

as

to o b t a i n

2.5 and

s is a strict

requirement

case,

automatically

going

is a c t u a l l y

enriches

this

one D - 2 - c e l l

D-al~ebras

3.1

s--4h

qhr

(2.11),

be the D - 2 - c e l l

= i if and only

seemingly

= 1 and

then

choice

satisfied

gives:

~h

~h

for

requires

there is at m o s t

i_~n (2.22),

hand t h i s

(Even l u t h i s

still

automatically

On the o t h e r

= i by

hr = jA, = I.

s~ = i is of c o u r s e

but

of p r o o f with

such a d j u n c t i o n

of our m e t h o d

of proof.

does

however,

require,

non-strict

D -morphisms,

is a

is that

when

all we

,

are

told

D-Alg,

originally

about

of D - a l g e b r a s ,

Moreover

! here m e a n

an a c t i o n

of D

D

the u n k n o w n

strict

-Alg,

on an o b j e c t

doctrine

D -morphisms, in the n a r r o w of K,

not ,

However

in s a y i n g

we are also

given

that the

we are

given

forgetful

D

and

D

is the

D -2-cells.

sense:

a D -algebra

on a 2 - f u n c t o r

-Alg,,

fun ctor

from

-Alg,

is

L ~ K.

I do m e a n D

2-category

to

to K;

imply

that

and

it is

298

classical purpose allows

that all this determines

of this

section

us to r e c o g n i z e

~.2

once

"small h o m - c a t e g o r i e s "

from [4] these and

limits

As is well known,

is,

pullbacks

that

that

the

can then

and the cotensor

each K(A,B)

construct

2-comma-objects

extends

2-category

to 2-cells).

as does Cat/A

The

Do-

to doctrines

for a small

size of A.

Kan extension.

these

extensions,

category

property

category

It fails

the right

Kan e x t e n s i o n

extension

is V. However

E×K ~ K, sending

all those

and

E(TS,R)

K, in the

to DA,

D on Cat

here

~ E(T,V)

sense

IA = A ( a l o n g Moreover

of the

to CAT by

the p r o p e r t i e s What

to small

as tensor

category;

of

is

categories.

product

It and

only for certain

for some V, namely when then this right

that there

and s a t i s f y i n g

2-cells).

of the form

of endo-2-functors.

composition

of R a l o n g S exists;

(ED)A = E(DA),

for 1-cells

extend

[K,K]

extend

independently

categories

to be a closed

E acts on (D,A)

small

with

on Cat

new in the club case.

2-category

S,R in E is it the case that

things

~ is the a r r o w

common d o c t r i n e s

we do not pursue

D takes

E for the

monoidal

laws

in K, u s i n g

itself has these properties,

certainly

which add n o t h i n g

We write

identity

A E K.

(the u n i v e r s a l

we can always

However

for us is that

i as identity.

of the

on CAT;

For that m a t t e r

is a strict

for n E Cat,

club D do so - one can form RoA

right

important

K ~ Cat;

for a set A.

It is true that many canonically

Cat

(b) that

K(A,-):

where

0 ~ i; and these are then

We recall

limits;

comma objects

[ ~,AI

Kan

is a 2-functor

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

and

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

.

is a small

all small

In,A]

of D

K has

i.e. C a t - c o m p l e t e .

products

product

knowledge

2-category

by the r e p r e s e n t a b l e s

cotensor

we

the simple m a c h i n e r y

(a) that K admits

are p r e s e r v e d

(c) that K admits

The

for all that

- that

this means

isomorphism.

from the implicit

and that K is 2-complete, that

to within

is to introduce D -morphisms

We now assume

category;

D

2-natura!ly

299

(3.1)

K(DA,B)

where

{A,B}

is

m E(D,{A,B}),

the

right

Kan extension

of

B:

I ~ K along

A:

I ~ K.

We know e x p l i c i t l y what this is, namely for C in K (3.2)

{A,B}C = [K(C,A),B],

the c o t e n s o r product of K(C,A) C Cat with B ~ K.

The counlt of the

a d j u n c t i o n (3.1) is a 2-natural e v a l u a t i o n e:

{A,B}A ~ B, and there

is similarly a 2-natural unit d:

We have formally the

D ~ {A,DA}.

same kinds of things as we have in a closed category, a 2-natural m u l t i p l i c a t i o n H: l:

in p a r t i c u l a r

{B,C}{A,B} ~ {A,C} and a unit

I ~ {A,A}, which make K into an E-category.

(That the u n d e r l y i n g

2-category of this E - c a t e g o r y really is K itself is immediate, K(1,{A,B})

since

m K(A,B) by (3.i).)

It is m o r e o v e r easy to check that the. E-valued e n d o m o r p h i s m object unit

{A,A}, with its m u l t i p l i c a t i o n p:

i:

i ~ {A,A}, is a doctrine;

action a:

DA ~ A corresponds under

{A,A}{A,A} ~ {A,A} and its

and that for any doctrine D, an (3.1) precisely to a d o c t r i n e - m a p

D ~ {A,A}. 3.3

We now write K' for the functor 2-category [2,Kl

the 2-category which is like the [~,K~

of [9]

and K" for

§2.2 above except that

its m o r p h i s m s are not lax n a t u r a l t r a n s f o r m a t i o n s but op-lax ones; is what we called FUN in [9] is d e s c r i b e d there

§I0.8 above in the case K = CAT, and it

(in the context of just such c o n s i d e r a t i o n s as

follow, but there specialized to the club case). So K' and K" have the same objects, namely m o r p h i s m s f:

A ~ B in K; a typical m o r p h i s m in K" is a triple

form u

(3.3)

A

it

......

f

c~ '~ B

~A'

If ' " r~B'

~T

(u,v,a) of the

3OO a typical q:

2-cell

(u,v,a) ~ (~,~,~)

is a pair of 2-cells

v ~ ~ in K satisfying the obvious

~; and the various compositions

conditions

are by pasting.

of (3.3) commute;

u ~ ~,

with respect

to ~ and

The morphisms

are those of K" in which e = i, and are thus just pairs making the outside

p:

of K'

(u,v)

its 2-cells are those in K"

between these morphisms. There are evident actions being the restriction (D,(u,v,~)) everything

of the second;

goes to (Du,Dv,Da), in sight,

and 2-cells of E. right adjoints;

namely

(D,f) goes to Df,

and in general D is applied to

with appropriate

arrangements

we have 2-natural

K'(Df,f')

~ E(D,[f,f'l),

(3.5)

K"(Df,f')

~ E(D,< f,f') ),

isomorphisms

is the pullback ~0

(3.6)

[f,f']

b {A,A'}

{l,f,}

{B,B'}

~{A,B'} {f,l}

and < f,f'>

is the comma object

DO (3.7)

also for the 1-cells

Just as in the case of K, these actions have

(3.4)

where [f,f'J

of E on K' and on K", the first

< f.f,)

P{A.A'}

I{l,f'}

{B,B ~}

~{A,B'}

{f,1} of course there is a canonical map

;

301

(3.8) with 61

s:

[ f,f']

~0 ~ = 60,

are j o i n t l y Thus

same

kind

observe this

~is

= 81,

K'

that [ f , ~ that

D ~ (f,f)

= I; it is a m o n o m o r p h i s m

and K" b e c o m e

of f o r m a l

case

~e

since

60

and

like K, w i t h

the

monomorphic.

6o,

BI,

~0,

are

~I,

taking

doctrines;

it is easy

s are

which

under

when

just

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

for any d o c t r i n e

corresponds

of D on f in K",

E-categories,

properties. and ( f,f}

Moreover k:

~ < f,f')

to

see in

all d o c t r i n e - m a p s .

D in E,

(3.5)

written

and

f' = f, we

a doctrine-map

to an a c t i o n

(a,b,~):

Df ~

f

out as

a

(3.9)

A

DA r_

Df

f

f DB

is e a s i l y

seen

, ,,

to a m o u n t

~B

to a c t i o n s

on A and B p t o g e t h e r

wlth anenrichment

F = (f,~):

The

A ~ B.

action

a:

DA -~ A and b:

a:

~ of

DB -~ B of D

f to a D - m o r p h i s m

DA ~ A i t s e l f

corresponds

to the

doctrine-map (3.10)

and

D

similarly

D-map; there

the

to a c t i o n s of

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

3~4 go one

a doctrine-map

situation

doctrine-map

(3.9)

that

in o r d e r

further.

morphisms

of K";

Now write

K # for the

D ~ If,f]

of D on A and

between

D ~ (f,f>

Finally, step

m {A,A},

for b.

Similarly in K' a m o u n t s

m

< g,g' )

y- < f,g' > (y,l)

Once a g a i n when y' = ¥ we get a d o c t r i n e [y,Y~, and ~0,~i are then doctrine-maps.

It is clear that a d o c t r i n e - m a p D ~ [Y,Y~,

an action of D on Y in K #, is just a D - 2 - c e l l Y: D - m o r p h i s m s F = (f,~):

A ~ B and G = (g,g):

or equally

F ~ G between the

A ~ B, as defined

in

(3.19) of [13]. 3.5

There is an analogue of (3.7) in which I is an isomorphism:

it can be called either the strong comma object or the pseudo pullback. Each of the main results stated in §i.i has both a "lax" and a "pseudo" case.

In order to make our treatment of these cases formally identical,

we agree to use (f,f) in the strong sense, without i n t r o d u c i n g a new name for it, when h a n d l i n g the "pseudo" case. invertible,

etc.

T h e n the ~ of (3.9) is

3O4

4.

4.1

We first define

between

two d o c t r i n e s

The most E

elegant

then to observe

ff':

a lax d o c t r i n e

(D,m,j)

and

way of doing

map H = (h,h,h°):

(D*,m

it would

,J ) on the be to take

and to form E" from it as we formed K"

= [K,K],

product

Lax a l ~ e b r as

that E" was a strict m o n o i d a l

of the objects

DD' ~ EE';

f:

D ~ E and f':

and finally

to define

D ~

D*

2-category

K.

the

2-category

from K in §3.2;

category,

D' ~ E'

the t e n s o r

being

a lax d o c t r i n e

map as an

,

object

h:

D ~

D

tensor product.

of E" with the However

we

structure

say the

of a m o n o i d

same thing

for this

in e l e m e n t a r y

A lax d o c t r i n e map, then, consists of a 2-natural , ation h: D ~ D t o g e t h e r with m o d i f i c a t i o n s h°,h as in

terms.

transform-

m

(4 .i)

i

hh

~ D D

,

h ,

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

~D

m

satisfying

the axioms Dj

(4.2)

D

m ~ D

• - DD ....

D

~D

Dh

Dh ° D

=

h

~--DD

h

h

Dj hD

D

J-D Dj

D

. D m

~D

D i

,

305 jD (4.3)

D

DD

h

~D

=

D

tD

Dh f,

JD D

•- DD

h

h

> hOD *

hD

D

~-DD

,

j D

Dh

,

mD =

~-D

Dh

~- DD

DDh I

*

DD

~D

**

D

~

2-naturality

a 2-cell axioms

maps

which

with respect

by v e r t i c a l

Thus

becomes

and the

*

*

the axioms

a 2-category

p:

same

2-cells,

and

sub-2-category

but whose

are the ordinary

Doat

1-cells

are

(strict) of these,

as

in [13]§3.6. There

and Doa£. map

from

the obvious

are the m o d i f i c a t i o n s

defined

easily

of doctrines

doctrine

2-cells

.

Lax Doct when we define

h ~ k satisfying It has the

~D

*

m

follows

these

and the

*

D

pasting of d i a g r a m s

those with h and h ° i d e n t i t i e s ; maps,

~D

**

we get a c a t e g o r y

to h°,h,k°,k.

same objects

*

D

mD

compose

to be a m o d i f i c a t i o n

with the

*

D DD

satisfies

of h etc.

maps,

~ D

~

,

hD

m

the c o m p o s i t e

lax d o c t r i n e

f~

,

~ DD

•

Dm

Lax doctrine

mD

hhD •

D DD

the

~Dh

DDD

h

hD

hD D

that

=D

>

DD D - Dm

,

m

DDD

,

4.1);

mD

i

~DD

DDD

Dhh 1 ,

D

m

Dm

4.4)

~D

m

are lots

We call

if h and h ° are

of 2-categories

the lax doctrine isomorphisms;

intermediate

between

map H = (h,h,h °) a pseudo

we call

it normal

Lax Doct doctrine

or n o r m a l i z e d

if

306

is an identity.

Our arguments below are expressed in terms of

the r e l a t i o n between Lax Doct and Doct, u n c h a n g e d if we replace 4.__22

but they go over a b s o l u t e l y

"lax" by "pseudo" or if we require normality.

For A E K, we define a lax action of the doctrine D on A to

be a lax doctrine map K: defined in §3.2. (3.1) to a:

If k:

D ~ {A,A} where

{A,A} is the doctrine

D ~ (A,A} corresponds under the i s o m o r p h i s m

DA ~ A, then the kind of argument familiar in the

context of closed c a t e g o r i e s s h o w s that k°,k

correspond r e s p e c t i v e l y

to a , a in

jA (4.5)

A

n~

~ DA

D2A

,

DA

~

> Da

A ,

DA

D,

A

a (In detail,

we go from (4.1) to (4.5) by m u l t i p l y i n g

(4.1) on the

right by A and composing on the tail end with the e v a l u a t i o n e:

{A,A}A ~ A.

We go in the other d i r e c t i o n by a p p l y i n g {A,-} to

(4.5) and c o m p o s i n g on the front with the unit E ~ {A,EA} where E = i or D2.) (4.2)-(4.4)

Equally simple calculations

are equivalent to the axioms DJ A

(4.6)

DA

show that the axioms

(4.6)-(4.8) below:

mA m D2A

DA

m DA

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

~

A

--

identity,

307

JDA

DA

(4.7)

,, ~.D2A

= identity,

~=~~_ DA =

a

Da JA

~DA

a

*~-~A DmA

(4.8)

mDA

mA

D3A

D2A

D2a

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

J'DA Da

D3A

,,- DA

oa

D2A

a ,~, D 2 A ~

a

-~A a Da

~ g

a

a

We more commonly use the term lax action of D on A not for the lax doctrine map K = (k,k,k°): triple

D ~ {A,A} but for the above

(a,a,a) satisfying the axioms

(4.6)-(4.8).

This latter

d e f i n i t i o n can be used evenwhen A is not an object of K but a m o r p h l s m L ~ K, in which case the first d e f i n i t i o n fails because the right Kan e x t e n s i o n {A,A} of A along itself may not exist; and we do use it in this extended sense. called a lax D-al~ebra;

The A with such a lax action of D is

note that this d e f i n i t i o n agrees precisely

with that of Street in [24]

§2 above.

M o r e o v e r a, for instance,

is

an i s o m o r p h i s m or an i d e n ~ t y precisely when k is ; so we carry over the words algebras.

"pseudo",

"normal" from the lax d o c t r i n e maps to the lax

Finally the strict doctrine maps c o r r e s p o n d of course to

the honest D-algebras, where both a and a are identities.

308

4J3

For f:

doctrine map P: §3.3.

A ~ B in K, and a doctrine D on K, D ~ (f,f)

where (f,f)

consider a lax

is the doctrine defined in

Because the passage from (4.1)-(4.4)

to (4.5)-(4.8)

formal, d e p e n d i n g only on the action of E on K s a t i s f y i n g

is purely (3.1), we

can repeat it all at the level of the action of E on K" satisfying (3.5).

We conclude that P corresponds to a lax action of D on f

in K", given by precisely the data (4.5) and the axioms

(4.6)-(4.8),

but with A replaced by f and with the 1-cell a and the 2-cells a,a of K r e p l a c e d by 1-cells and 2-cells of K" Now consider what this means. r e p l a c i n g a:

(4.9)

f in K"

DA ~ A in K be (a,b,~) as in

DA

Df

Let the m o r p h i s m Df ~

~ A

~

I

'

DE

.~ B .

^

_

Let the 2-cells in K" replacing the 2-cells a,a of K be the r e s p e c t i v e

pairs

(a,~),

(a,~).

By t h e

definition

o f what a 2 - c e l l

is

in K",

these

have to satisfy

jA

(4 .io)

A

jA ~ DA~

A

B

• DA

jB

~ DB

f

1

3O9

mA (4.1i)

D2A

D2A

~ DA

~a

D2f

A

f

~

~

)"

DB

f

~B

-

B.

b

When it comes to the axioms

(4.6)-(4.8), we observe that these are

2-cells in K", w h i c h are only pairs of 2-cells

in K - s a t i s f y i n g conditions once given

Df

D2B

f

? b

essentially about

~ DA

a

D2f

D2

mA

indeed, but these latter automatic

(4.10) and (4.11).

Thus

(4.6)-(4.8)

for the lax action

of D on f reduce to the original (4.6)-(4.8) for a,a and for b,b. Putting all this together, we see that a lax action of D on f in K" is the same thing as a lax action

(a,a,a) of D on A (corresponding of

course to the lax doctrine map

D .....(f,f) P a c o r r e s p o n d i n g lax action satisfying

(4.10) and

- {A,A}), ~0

(b,b,b) of D on B, and an f as in (4.9)

(4.11).

We call the pair F = (f,f) a m o r p h i s m of lax D-algebras F:

A ~ B; it is just what Street calls a "lax h o m o m o r p h i s m "

in

i

f241

§2 above.

We call it a strong m o r p h i s m if f is an isomorphism,

and a strict one if f is an identity.

Of course an argument p r e c i s e l y

similar to that above identifies a strict m o r p h i s m f:

A ~ B of lax

D-algebras as a lax a c t i o n of D on f in K', and hence as a lax doctrine map Q:

D ~ [f,f].

This shows that,

for lax doctrine maps

just as for strict ones, there is no difference between one into [f,f|, and one into (f,f) that happens to factorize through ~:

E f,f| ~ ; which could a l t e r n a t i v e l y have been seen by g o i n g

310

back to the definitions of [f,f] comma object.

and (f,f} as a p u l l b a c k and a

This point needs to be kept in mind for our argument

below. (Note that we get the stron~morphlsmsby using the different (f,f) of §3.5.) Finally,

just as we agreed to extend the d e f i n i t i o n of lax

algebra given by (4.5)-(4.8)

to the case where A is not an object of

K but a 2-functor L ~ K, so we extend the d e f i n i t i o n of m o r p h i s m of lax algebras to this case, by (4.9)-(4.11), where f is now 2-natural and ~ is a modification. 4.4

Lastly in this hierarchy,

map R: ~y,y~

D ~ ~y,y~, where y:

we have to consider a lax doctrine

f ~ g:

A ~ B in K as in (3.11), and

is the doctrine defined in §3.4.

A r g u i n g as above, we see that

this is the same thing as a lax action of D on y in K #. analogue of the a:

DA ~ A of §4.2 be

(a,b,~,g):

Let the

Dy ~ y, so that as

in (3.12) we have a (4.12)

DA

..... ~

A

=

DA

A v~

Df

Dg

~,

DB

g

...... r

Df

B

f

DB

Y

~-- B b

b

Let the analogues of the 2-cells a,~ of §4.2 bethe 2-cells (a,b) of K #. The axioms

Then ~ and a are as in (4.5), and

(3.14) and

(a,b) and

(b,b) similarly.

(3.15) that these must satisfy to be 2-cells

in K # reduce to (4.10) and

(4.11), with their analogues for g,g.

Finally the lax-action axioms to (4.6)-(4.8)

g

(4.6)-(4.8)

for a,~ and the analogues

for (a,b) and (a,b) reduce for b,b.

Summing up, we

have lax D - a l g e b r a structures on A and B, m o r p h l s m s F,G:

A ~ B of

these where F : (f,~) and G = (g,g), and finally a 2-cell y: satisfying

(4.12)

f ~ g

(which is identical with the (3.19) of [13], d e f i n i n g

D-2-cells in the case of honest D-algebras).

311

We therefore even in this where L ~ K,

call

lax case;

such a y s a t i s f y i n g

and we extend

the lax D - a l g e b r a s

Now that we have we make

the various

definition

A and B are not objects

in which case ~ is of course

D-2-cells,

this

(4.12)

a D-2-cell to the case

of K but

2-functors

a modification.

lax D-algebras,

morphisms

of these,

and

Lax-D-Alg of these elements, d e f i n i n g

a 2-category

c o m p o s i t i o n s by the evident

pasting

operations;

restrict

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

categ o r y

Lax-D-Alg,. There is a similar r e s t r i c t i o n to the strongmorphisms.

4.5

It is easy to see,

inclusion

and c o t e n s o r object

although

Doat in Lax Doct

of

products.

of lax D-algebras,

if we

tedious

to write out,

is 2-continuous~

It follows

that

we get the

limits

D, r e g a r d e d

Lax Doct, admits a 2 - r e f l e x i o n H = (h,h,h°):

of

that the

it p r e s e r v e s

a doctrine

sub-2-

as an

D ~ D

into

Doct, p r o v i d e d that the a p p r o p r i a t e solution-set condition is satisfied

for the given

a lax d o c t r i n e

D.

map H:

That

D ~ D

is to say,

such that

there

is a d o c t r i n e

any lax doctrine

map

D

K:

and D ~ E

,

is of the form K : tH for a unique We get the

same

2-continuity

we require

normality;

I do not which,

for a given

doctrine

if we replace

and exactly

intend

strict

to study

the

"lax" by

map t:

"pseudo",

same c o n s i d e r a t i o n s

in this

D, the s o l u t i o n - s e t

paper

condition

D

~ E. or if

apply.

the conditions

under

is satisfied

and the

,

reflexion

D

exists.

Certainly

Cat/~ it is immediate that D the m e t h o d s

of Barr Ill

say K is locally question

too because

exists,

or of Dubuc

presentable

aside b e c a u s e

when D is Do-

for a small

as we see b e l o w in §4.10.

[5]

will

show D

and D has a rank.

I haven't

thought

I suspect that it doesn't

club D in

to exist

I suppose

if

In part I leave

out the details;

really matter:

but

it seems

the

in part to me

,

likely

that D

Cat, or that D universe

always exists

exists

as a d o c t r i n e

on a suitable

when D is g i v e n on K.

of e x t e n s i o n s

of D a d u m b r a t e d

on CAT when D is one on

completion

The q u e s ti o n in the

second

of K in a b i g g e r

is c o n n e c t e d paragraph

with that

of §3.2,

and

312

really Our

deserves

concern

a fuller

in this

treatment

paper

is w i t h

in its o w n r i g h t the

consequences

at a n o t h e r of the

time.

existence

,

of D , a n d

the

club

case

is

sufficient

to s h o w that

our c o n s i d e r a t i o n s

a r e not v a c u o u s . A final the p r o o f

point

of T h e o r e m

exist.

If it d o e s

affects

its u t i l i t y

not,

there

by

merely

be d e n i e d

should

have

which

would

purely

ly of the

"size"

4.6

"normalized the u s e

theorem

called

universe,

that

theorem.

And

remains

true, and

of [i0]

above,

which

no-one clearly

to p s e u d o - D - t e r m s ,

would

doubt

we

should

stand,

and

case, that

such

such when

be v a l i d

the m e t a t h e o r e m

does

replacing

§i as t h e y

remain

D')

if it d o e s

so on;

would

for

in no w a y

in the p s e u d o - a l g e b r a

But

though

( there

D-category"

of the r e s u l t s

nuisance.

D

in [I0]

independent-

involved

is felt

stated. b a c k to the m a i n we h a v e

is g i v e n

in b i J e c t i o n

Thus

of that

pseudo

of D - e v e n

A in K - s o m e t h i n g

are

that

bigger

them over again

D'-terms

Coming

an algebra

that

dlagram-arguments,

from

than

in some

that

I assumed

paper,

for the p r o o f

be a n a w f u l

translated

rather

so o n l y

to p r o v e

formal

vein:

3.1 of that

is no d o u b t

" D~category"

in t h i s

the D - a l g e b r a s

as b e f o r e

to t h e h o n e s t

action

a = a

*

Otherwise

put,

every

D -action

a

.hA,

a

just

to the :

*

~

.hA,

lax D - a c t i o n

identify

map

last t:

the

the D - a l g e b r a s

paragraph.

D

lax d o c t r i n e

~

maps

{A,A}; K:

lax D - a l g e b r a s .

lax a c t i o n

D A ~ A,

a = a

we

in the

doctrine

K = tH w i t h

in K are

K corresponds

(4.13)

anticipated

by a strict

under

point,

(a,a,a),

the r e l a t i o n

a = a

*

D ~

Such and

{A,A}.

Moreover

while

these

if

t corresponds

K = tH t r a n s l a t e s

into

0

.h A.

is of the

form

(4.13)

for a u n i q u e

.

Similarly

a morphism

F*

(f,f):

A ~ B of D -algebras

is

,

given map

by a s t r i c t

D ~ (f,f),

lax D - a l g e b r a s .

doctrine

which The

map

D

corresponds connexion

~ (f,f)

and hence

to a m o r p h i s m

between

F and F

b y a lax d o c t r i n e

F : (f,~): is g i v e n

A ~ B of by

313

a (4.14)

DA

=

A

I

Df

DB

hA

DA

DB

B

factorizes

~ = i

~

If,f]

if and only

if and only

is, F is strict Finally

maps

If,f]

through

(4.15)

that

e:

D ~ [y,y~,

(4.16)

~

trivial

is that

produces we have

also

the

...... b

F

of v e r t i c a l diagram the

if F

map,

does

.

~

so; w h e n c e

correspond

to lax d o c t r i n e

that

~ G

are just the D - 2 - c e l l s

(4.14)

the only

pasting

case

and

that

of diagrams

(4.15)

D * -Alg [resp.

Lax-D-Alg

2-functor

[resp.

y:

F ~ G.

respect

the

is not a b s o l u t e l y (4.14),

which

of the same kind by the n a t u r a l l t y

2-category

D

F

is so.

~ ~y,y~

2-category

know the forgetful

for a unique

doctrine

if D ~ < f , f >

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

identified

essentially

y:

of composition;

another

D B

A

~* = I,

doctrine mapsD

Moreover kinds

if

and we conclude

(4.14)

is a strict

if and only

the D -2-cells

various

a

hB

we can say that any F is of the form

Since m o r e o v e r

* D A

D f

Df

b Again,

~

clearly

of h.

Thus

D * -A l g,], namely

Lax-D-Alg,l.

to K, we do indeed

have

as

Since we an implicit

,

d e t e r m i n a t i o n of D . (This whole section remains v a l i d for the stronger of §3.5.) 4.7 For the purposes of our proof, we need the results of §4.6 not only

for a l g e b r a s

2-functors

A:

A that

L ~ K (L being

for such an A the right not exist, argue

we cannot

object-wise,

are o b j e c t s K itself

Kan e x t e n s i o n

argue d i r e c t l y

using

the r e s u l t s

of K but

for a l g e b r a s

in our a p p l i c a t i o n s ) . {A,A}

of A a l o n g

as in §4.6. of §4.6.

itself

Instead

that are Since need

we have

to

314

Let A:

L ~ K then

(a,a,$).

For e a c h

and h e n c e

as in

X E L we get a lax D - a c t i o n

(4.13)

a unique

*

aX = a X.hAX, is that

a

To Because

aX = a X.hAX,

the

unique show

D -action

with

lax

action

(aX,~X,aX)

a X on AX such

on AX, that

~

is 2 - n a t u r a l ,

of c o u r s e

be a lax D - a l g e b r a ,

one

and

aX = a * X . h ° A X .

so that

a

satisfying

the n a t u r a l i t y

a is n a t u r a l ,

is i n d e e d

we have

What

has

to p r o v e d

a D -action

on A, and

(4.13).

of a , c o n s i d e r

commutativity

@:

X ~ Y in L.

in

aX (4.17)

DAX

-

~-AX

, ,

A¢ DA$ I

DAY

~-- AY

;

aY and A S is in fact

a strict

(4.10)

being

Hence

and by

(4.11)

(4.14)

and

morphism

satisfied

(4.15)

of lax D - a l g e b r a s ,

because

we have

a and

the

axioms

a are m o d i f i c a t i o n s .

cormmutativity

in

a X (4.18)

D AX

D

AX

A¢

A~

DAY

, aY

~

AY

,

,

showing

the n a t u r a l i t y

X ~ Y in L. whence

Then

of a

As:

A S ~ A@

it is a D - 2 - c e l l Next,

of these.

D-algebras

in K, w h i c h

morphism

by

let A,B:

a morphism

.

its

2-naturality,

is a D - 2 - c e l l

(4.16),

giving

by the

the

FX = (fX,fX):

therefore

¢ ~ ~:

2-naturality of a

and F = (f,~):

AX ~ BX is a m o r p h i s m

corresponds

of D * - a l g e b r a s

let ~:

2-naturallty

L ~ K be lax D - a l g e b r a s

Then

F * X = (fX,~ *X)

For

as

in

in K, w h e r e

(4.14)

of a, . A ~ B of lax

to a u n i q u e

~X = ~ * X . h A X .

If

315

F is strict,

so is F

, and t h e r e

general

we m u s t

s h o w that

case

is no m o r e

f

to prove;

is a m o d i f i c a t i o n .

but

in the

This m e a n s

the

equality

,

a X .................

D AX

D fX

a X AX

~

t " 1 f X

D BX

~

, b

we h a v e

because

~

this

because

the

correspondence

correpondence

, b Y

(4.14)

that

~--- BY

respects

f X corresponds

~ is a m o d i f i c a t i o n

fY

f Y

D BY

BY

At the next level, nothing

D fY

Y

because

under

(4.17),and

....

~- AY

DAY

Be

D BY

which

a Y

BX

D Be

AX

D A¢

fX

,

~

D AX

;

composition,

to ~X and

(4.18)

to

by h y p o t h e s i s .

of D - 2 - c e l l s ,

there

is by

(4.16)

to prove. We

conclude

that

the

results

of

§4.6,

expressed

in the

form

^

that

every

D -action

lax D - a c t i o n a

, etc.,

present, w i d e r , 4.8

We

(a,~,a)

remain

true

is of the when

form

we take

(4.13)

for a u n i q u e

the a l g e b r a s

in the

sense.

can n o w p r o c e e d

rapidly

to the p r o o f

of our

first

main

result. Theorem

4.1

Suppose

that~

Let

considered

H = (h,h,h°): doctrine (4.19)

map sH

D be a d o c t r i n e

D ~ D*

=

1,

as an o b j e c t

into

satisfyin~

on the

Doat.

Let

of s:

2-complete

2-cate$ory

Lax Doct, D a d m i t s

K.

a reflexion

D * ~ D be the u n i q u e

strict

316

which

exists

a lax one. and qh "lax"

because Then

= i.

there

The

same

D ~

The

comments

at the

and

the d e t a i l s

from

last

is t r u e

maps,

(4.20)

sh

(4.19)

=

i,

in

to t h i s u n d e r the

The (4 • 13)

The

(4.20)

be w r i t t e n

sh

=

m: m

lax a c t i o n

DD

,~-

that

the a x i o m

morphism

to H

(4.4).

s~ = i

proof,

first

in the

from

the

paragraph light

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

of

of §4.5,

§3.5.

of lax

as

map,

(4.20),

:

D

D

i.

m.sD:

the

D D ~

D is a D - a c t i o n

on D w h i c h

lax action

~ D

of D

(m* . h D * , m* . ~ ~D*, of

m

=- D

DD

corresponds

(m,l,l);

that

is to

(4.1)

may

m

on i t s e l f

corresponds

* . hOD * ) of

be w r i t t e n

D

on

D

by

*.

as

h

~-- D

m

H = (h,h)

(4.10)

that

or if we r e p l a c e

is i m m e d i a t e

lax D - a c t i o n

diagram

hD

reduces

The

such

normality~

in the

sh ° =

h

DD

I ~ hs

and afortiori

D 2 ~ D of D o n i t s e l f .

Dh

I assert

i,

is, b y

action

second

those

may

map

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

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

D -action

to the

from

following

of [13] .

(4.13)

strict

case ~

s is a d o c t r i n e

(3.8)

q:

of the t h e o r e m

of the

that,

Because on D, as

sentence

doctrine

if we r e q u i r e

last

end of §4.1,

Observe doctrine

D is a s t r i c t

is a m o d i f i c a t i o n

by "ps___~euJg"; i n t h e

Proof.

say,

i:

is a m o r p h i s m

in t h i s

case reduces

It f o l l o w s

= (h,h):

D ~ D

as

in

D ~ D to

of lax D - a l g e b r a s .

(4.3),

(4.14)

that

of D - a l g e b r a s

and

the a x i o m

there

For (4.11)

is a u n i q u e

satisfying

317

(4.21) m

hD

DD

D

, ~-D D

DD

=

sD

m ~D

~DD

2Y Dh

h

h

Dh

D*h

h

@@ DD

,

~DD

other

and hence

a strict

and

the

(4.22)

sH

replaced

The

.

take

The

so that

respects

morphism Since

of D - a l g e b r a s ,

sH = i by

composition,

(4.1).

to

,

2.5;

(4.20)

we h a v e

D since

Since

by

the q:

i ~

of c o u r s e

by D, a n d H

(2.4),

the

s is a d o c t r i n e

sh ° = i by we h a v e

with

B replaced

to be a m o d i f i c a t i o n

2.5 a g a i n

(2.22),

Theorem

2.5 n o w b e c o m e s ,

is j: i ~

2.5 is t h e r e f o r e

by Theorem

to a p p l y

by D i = D

r of T h e o r e m

D, w h i c h

According is the

of lax D - a l g e b r a s . (4.14)

~D

,

m

-~ D is a s t r i c t

in a p o s i t i o n

it to be the h ° of

st = i,

s: D

DA r e p l a c e d

: i ~

~ of T h e o r e m

DD

1.

by D ,

sj

hand

correspondence

--

by H

composite

,~

hD

morphism

W e are n o w D replaced

DD

m

On the

since

D

,

hD

sq

j

(4.20),

map.

~ hj;

a n d we

we h a v e

= i.

H s produced

by T h e o r e m

2.5

composite

(4.23)

DD D *j

,

D

Y

,

~D

h ,

*

D

D

Dh ~

It r e m a i n s

of

only

h we h a v e

D

to g i v e

D*h°.h

a proof

= hD*.Dh °

D

that

so that

q h = i.

(4.23)

But

by the

2-naturality

composed with

h becomes

318

D

(4.24)

Dj

hD

DD

* ~ D D

Dh Dj*~

sD

~

D h

, DD

,

~

the

composite

(4.21),

whence

, , DD

h ,

,

~' D

m

o f the

(4.24)

~-~ D

_, h

hD

however

m

DD

is the

two rectangles identity

by

o n the r i g h t

(4.2).

This

is h as in

completes

the

proof. 4.9 of

In m y p a p e r §3.1, a n d u s e d

[I0]

in the

above

proof

in t h i s

volume,

of T h e o r e m

3.1,

I stated

only

at the

one v e r y

end

small

part

,

o f the

above

theorem:

namely

that

s:

D

~ D had

a right

inverse

h,

,

as

in

(4.20).

pseudo

case, In

The

D

which

in q u e s t i o n

is the c a s e

~3.3 of II0]

here

that

I called

is t h a t

arises

for the

normalized

in [i0] .

the d o c t r i n e

D flexible

if s:

D

~ D

,

had a right notion

inverse

was u s e d

doctrine

D

is a l w a y s

equally

D ~ D

in T h e o r e m

The p r o o f applies

t:

that

a

3.3 of [i0] .

flexible:

we n o w

is e l e m e n t a r y ,

in t h e

was

lax case

as

not

(strict) It was

stated

justify

map;

there

this

that

the

this.

depending

in the

doctrine

on T h e o r e m

pseudo

case,

4.1,

and

w i t h or w i t h o u t

normality. Proposition

4.2

as an o b j e c t reflexion q:

D

of

Let H:

Lax D o c t ,

D ~ D into

of D , c o n s i d e r e d

~ D

be the u n i q u e

there

is a s t r i c t

Proof

Since

Doct;

and

as a n o b j e c t strict

doctrine

be the r e f l e x i o n

map

p:

let K: of

doctrine D

~ D

D

o f D,

considered

~ D

be t h e

Lax D o c t , map

such

with

into Noct.

that

qK = i.

Let Then

qp = i.

** KH:

D ~ D

is a lax d o c t r i n e

map,

and

since

,

H:

D ~ D

doctrine since

is t h e r e f l e x i o n , map

qK = i.

p:

D

~ D

Since

f o r the r e f l e x i o n

Then qKH

qp a n d

H gives

we h a v e K H = p H for a u n i q u e = qpH;

I are b o t h

qp = i.

D

but

strict,

again

qKH

strict = H = iH

the u n i q u e n e s s

property

319 In the pseudo case,

since q is an e q u i v a l e n c e of endo-2-

functors by T h e o r e m 4.1, it follows that pq a i. have not proved,

I suspect, but

that the i s o m o r p h i s m pq m 1 can be chosen to be a

doctrine-modification,

so that D

This would be very convenient

and D

are equivalent doctrines.

if it were true, and the q u e s t i o n should

be looked into. 4.10

We c o n s i d e r finally the special but important case w h e n

D = Do-

for some club P in Cat/~.

and §i0 of [9]

We again refer the reader to §i

above for general facts about clubs of this kind.

To give a lax D - a l g e b r a structure, w h i c h we shall also call a lax P - a l g e b r a structure, functor a: {A,A}

on a c a t e g o r y A, we have first to give a

PoA ~ A, or equally a m o r p h i s m P ~ {A,A} in Ca£/~; here

is now the "rich e n d o - f u n c t o r category" of [9]

right Kan e x t e n s i o n of (3.2) above. ITI:

So for T C P we have to give

A n ~ A where FT = n; and for f:

g e n e r a l i z e d natural t r a n s f o r m a t i o n

§9, and not the

T ~ S in P we have to give the

IfJ:

ITJ =

ISI of type Ff.

Since

this is to be a functor, we must require

(4.25)

Ifgl = Ifr

Igl and

Iii = i.

Next we have to give a as in (4.5), a natural t r a n s f o r m a t i o n with components

ILl(A), that is, a natural t r a n s f o r m a t i o n

In the normal case we require ~ = I, which involves d e m a n d i n g that It1 = i.

In the pseudo case we have to provide a with an inverse,

we must also give a natural t r a n s f o r m a t i o n

$':

ltJ

and demand that

i,

so

320

a Similarly

'

=

i

and

a'a

=

for the a of (4.5);

aT[SI...Sn|[AI...Am]

i.

it is to have components

:

ITI(ISII(AI''') ..... ISnl(...Am))

IT(SI...Sn)I (AI...Am),

with inverses a' provided

in the pseudo

case.

All these data are

then to satisfy the further axioms c o r r e s p o n d i n g It is clear from [9]

of D

such a lax D - a l g e b r a

is itself

in Cat~l, which we have in effect just

an algebra for a club O described

§I0 that

to (4.6)-(4.8).

by its generators

and relations.

Explicitly,

the objects

are generated by objects T in bijection with the objects T of

D, and with FT = FT. the normal

These are subjected

to no relations

case, where we impose the relation

for the morphisms

of D

consist

of an f:

[ = i.

except

in

The generators

T ~ S with r~ = rf for each

^

f:

T ~ S in D; of an a:

~ ~ ;, with ra = i, which is to be omitted

in the normal case; and of an

aT[SI...S n]

:

T(SI ..... ~n )

~

T(SI ..... Sn)'

with Fa = i, for each object T[SI,...,S n] of PoD. these are to be augmented by further generators in the reversed these generators

senses.

are fg = fg and ^

relations

The relations

In the pseudo case,

a' and a'T[SI...Sn]

between expanded

i T = I~

corresponding

instances

to (4.25);

the

_ _

aa' = i, a'a = i, aa' = I, a'a = 1 in the pseudo

and finally the relations

of

corresponding

of these for instance asserts

to (4.6)-(4.8).

the commutativity

of

case only;

The first

321

l-

aT[ 1 . . . II

T D'DD

D'D

identity,

~

~ DD v

m'DD (5.10)

D'D'DD

D'DD

~

~

D'D'DD

~ D'DDD' |

l

D'~

DD v

~

1

~D' Dm '

........

~ D ' D D ' ~

DD'D'

DD'

pDD (5.11)

D'DDD D'Dm ~

...........

~ DD'DD

pD

DD'm~ Dp

D

'DD

...........

,L

~

.... ~ DDD'

DD'D

DVD

~

DD v

P DDp DDD'D .....

D 'DDD ~rD

DDDD '

I mDD '

mD 'D Dp

D 'DD

DDD '

DD'D

> D'D ....................

,ol,

327

We now Justify the above assertion. in (5.1) is by [13] DU',.

By [13]

§3.3 to give an action q:

Proposition

give a 2-natural

p:

p:

D'D ~ DD'

and for a D'-algebra

extension D, and write DA

In doing so we use A both for an object of K (A,a')

consisting

D'A ~ A.

of the object A together with

We then have

DA = (DA, D a ' . p A ) ,

the D'-algebra

with object DA and with D'-action

D'DA

~ DD'A pA

~ DA. Da'

For a morphism F' = (f,~'):

A ~ B of D'-algebras,

D'-morphism

(5.14)

DF'

=

(nf, n['.pA)

as in pA (5.15)

D'DA

Da ' ....... ~ DA

~ DD'A

)D'Df I D'DB

DD'f 1

pB

for

form of D, in terms of p; but we may as

well give at the same time its canonical rather than D,A, etc.

The conditions

of

at once into (5.3) and (5.4).

We give the explicit

(5.13)

q as above is to

D'D ~ D'U',F', where F', is the left adjoint

q to be an action translate

(5.12)

D'DU', ~ DU', of D' on

2.1 to give a 2-natural

U',; that is, to give a 2-natural

the D'-action a':

First, to give D, as

~ DD'B

Df

D~'

Db '

DB

we have a

328

Finally a D'-2-cell y:

F' = G' gives a D'-2-cell Dy:

DF' =~ DG'

directly by

(5.16)

~

~.

=

The 2-functor D, is what we get by r e s t r i c t i n g D to strict D'-morphisms;observe

that if f:

A ~ B is a strict D ' - m o r p h i s m then

D,f = Df is just the strict D ' - m o r p h i s m Df:

DA ~ DB.

Since what we

have said about D, is all justified by our general remarks above, all that needs separate v e r i f i c a t i o n is that when F' is not strict, that are not strict,

(5.15) really is a D ' - m o r p h i s m

(5.16) really is a D'-2-cell w h e n F',G'

and that D really is a 2-functor:

The next thing we w a n t e d was 7,: The last requirement jA:

all this is easy.

I ~ D, with U',J, = jU',.

forces the component j,A:

A ~ D,A to be

A ~ DA; the desired J, exists p r e c i s e l y when JA:

indeed a strict D'-morphism, Once we have this,

and this c o n d i t i o n reduces to (5.5).

it is immediate that JA:

A ~ DA is 2-natural not

only for strict D ' - m o r p h i s m s A ~ B but for all; 2-natural j:

(5.17)

~A

A ~ DA is

so we in fact get a

i ~ D with the same components:

=

],A

= jA;

of course we have U'j = jU' Then we wanted m: requirement to be mA.

(5.18)

~2 ~ $ w i t h U'm = mU'.

forces the m o r p h i s m - p a r t

The last

of the component mA:

D2A ~ DA

Thus

mA

=

(mA, m'A)

for some invertible

2-cell m'A

(since mA is to be strong).

of m'A is d e t e r m i n e d by the r e q u i r e m e n t

that m be 2-natural.

The form Indeed,

329

mere f:

naturality A ~ B,

in A of mA,

suffices

to be a':

and

to fix the

D'A ~ A, we e a s i l y

that m e r e l y f o r m of m'A.

see that

pDA

(5.19)

D'DDA

,,

for

m'A

strict

First,

taking

is of the

form

DpA

,

~ D D ' D A

DDD'A

~

~A

mA

~ DD'A

~ DA

pA

we e a s i l y more

~A;

see that

is n e e d e d The

w must

to m a k e

further

for a D ' - m o r p h i s m (D,m,j) and

satisfy

(5.11)

pseudo

in e l e m e n t a r y

the a x i o m s

look

terms

with

= (a,a'):

D'a

.......

are that mA

(5.7)

and

classical

case

satisfy

reduce

w = i, but

w.

2-category

involving

D,D',p,w

A is a D ' - a l g e b r a a:

DA ~ A. in v i e w

of

D-Alg and A

elements

(5.9),

analysis

its

of a

elements

of K. a':

D'A ~ A)

is to be a D ' - m o r p h i s m

(5.13)

the

has

~ DA

a'

a

" 1 ~ A

a t

(5.8),

not here).

D'-action

Da '

l

the a x i o m s

The l a t t e r

~DD'A

D'A

to

and d e s c r i b e

(with

no

- and that

in terms

of p and

Then

is simple.

(5.10)

these

D ' A ~ D'B,

(5.6).

of the a b o v e

at the

a D-action

in

our j u s t i f i c a t i o n

DA ~ A, w h i c h

D'DA

to

as

the v e r i f i c a t i o n

for a d o c t r i n e ;

pA (5.20)

A ~ B to be D'f:

we do n e e d

in the

law

A D-algebra together

m 2-natural:

completes

We n o w

f:

be a m o d i f i c a t i o n ,

these reduce

distributive

5.~

Da'

taking

things

(automatic

This

next,

A ~ B

~ DDA

mD'A

D'DA

for an i n v e r t i b l e

f:

DDa'

......

D'mA

D'-morphisms

form

330

There

are

really

four axioms

to be s a t i s f i e d :

is a D ' - m o r p h i s m ,

and

the

second

the

first

two

two

say t h a t

say t h a t

a

it is a D - a c t l o n :

j 'DA (5.21)

DA

~ D'DA

~ DD'A

DA

D'A

=

identity,

"~A

m'DA (5.22)

D'D'DA

,

~-

D'DA

~ DD'A

D'A

D'D'DA

~ D'DD'A

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

~ D'DA

~

~ DD'A

D'~' D'D'A

-

D'A

~ DA

~ A ,

T

D'A

=

D'jA Da '

D'DA

A

~'

~

a

(5.23)

~DA

DD'A

g') ]

~- D A

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

D'a

a

D'A

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

a'

~A

identity,

331

(5.24)

pDA ....... DD'DA

D'DDA

DDa '

DpA DDD 'A

~- DDA

> D'mA

~A

D'DA

......~ DD'A

~ DA Da '

pA

a

D'a

D'A

~A a'

pDA

DpA

D'DDA

DDa'

DD'DA

DDA

DDD'A

DD'a[

D'Dla D'DA

D~'

DD'A

~

pA

f

DA

Da'

> a

D'a[

a'

D'A

Of these,

(5.21) and

a.jA = i using using

(5.20),

D'-action.

(5.22) are straightforward;

(5.20) and (5.17 (5.19),

the equalities

and (5.15).

(5.21),

expresses

a.mA = a.Da,

since a' was given as a

level already gives new information; that a:

1-cell equality

in (5.23) and

see by composing

expresses

Note that at the level of 1-cells

(5.22) are automatic,

the information

equalities,

; (5.24)

(5.23)

The same is not true of (5.23) and (5.24),

at the 1-cell

convenient

A

DA ~ A is a D-action. (5.24);

(5.23) with J'A and

where equality

in fact precisely

(Clearly this implies

and it is implied by them, as we (5.24) with j'D2A.)

It is very

to separate the 1-cell from the 2-cell information for example

in the kind of reasonings

in such

used in §4.3 above.

332

Thus

in fine to give a D - a l g e b r a

A of K, w i t h a D - a c t i o n a' as in (5.20)

A and a D ' - a c t i o n

satisfying

write the D - a l g e b r a

A as

the four axioms

(A,a,a',~').

we d e s c r i b e d such an algebra, both a D - a c t i o n structure

of a D'-morphism. Now we consider

It is a pair F : (F',~) D'-2-cell

and to give a 2-cell

(5.21)-(5.24).

terms,

the D - a c t i o n

being given

with

the

this precise.)

it is to give a D - m o r p h i s m

where F'

We may

as one p r o v i d e d

We have now made

what

is a D ' - m o r p h i s m

F:

A ~ B.

A ~ B and f is a

as in

~

(5.25)

a

DA

~A

DF'

f

~B If F'

a',

(In §i.I of the I n t r o d u c t i o n

in rough

and a D'-action,

A is to give the object

is itself

the pair

(f,~'

F' ~B

:

A ~ B then as a 2-cell

~ is of the

form

a

(5.26)

DA

~A

Df

f DB

f ~

B.

b

The r e q u i r e m e n t

that

f actually

be a D'-2-cell

is

(cf. [13] (3.19))

333

Da '

pA D'DA

(5.27)

.....

DA

~ DD'A

_> Df' D'DB

DB

~ DD'B

D,b~~B

f

A

Db' b'

D'B

,.b B

b'

pA =

D'DA

D'DB

Da ' . . . . . .

~ DD'A

D'f

~ DA

D'A

~ A

D'f

f,

D'B

~-B b'

Finally they

the a x i o m s

assert

axioms

only

F:

= (f,f,f')

s u c h that

we

y:

the

condition

y:

~

in fact

level

(B,b,b',b')

of

what

for

(g,g,g')

is,

2-cells,

Thus

reduce

since to the

in f i n e to g i v e

is to g i v e

and

(5.27)

a D-2-cell

since

y:

it is a p u r e

a 2-cell

F ~ G and a D'-2-cell

y:

F' ~ G';

it to be a D - 2 - c e l l is just

a D-morphism,

a

a triple

A ~ B is a D - m o r p h i s m ,

of all to be a D ' - 2 - c e l l

(f,f,f')

D-2-cell

~

F = (f,~):

consider

for it to b e a D - 2 - c e l l just

at t h e

A ~ B is a D ' - m o r p h i s m ,

Lastly first

to be

to be a D - m o r p h i s m .

(A,a,a',a')

= (f,f'):

has

an e q u a l i t y

for F = (f,~)

D-morphism

F'

for F = (F',f)

y:

y:

y:

is s a t i s f i e d . F ~ G: but

A ~ B is.

then the

equality

It

condition

of 2 - c e l l s ,

F ~ G. So a D - 2 - c e l l f ~ g that

F' ~ G'

is at o n c e a

334

~.4

We intend

doctrine

on K.

so that vertex a:

to exhibit

The

it becomes the {A,A}

first

D ~ {A,A}

of §3.2;

we get

If we compare

§§4.2-4.4,

while

will

with

the present

now suppose K to have This

together

small homs

k:

D ~ D

doctrine

{A,A}

maps

corresponding

similarly.

§5.3 above

However

to a'

we abstract

by an a r b i t r a r y is analogons

to §4.1.

to

Of course we

and to be 2-complete.

leads us to the f o l l o w i n g

of d o c t r i n e - m a p s

not the actions

something

replacing

§4, then

(5.20)

vertex D'D and with t e r m i n a l

§5.4 is analogons

, j ) on K, we define

modification

with

for a

is to t r a n s f o r m

corresponding

transform

in this way,

doctrine.

this

with leading

D'A ~ A but the

(5.21)-(5.24)

the t r a n s f o r m

in doing

as the algebras

it will then involve

and D' ~ {A,A},

The axioms

(D , m

step

a diagram

DA ~ A and a':

D-algebras

notion.

a map K from and k':

For any d o c t r i n e

(D,D',p,w)

t_2o D

D' ~ D , t o g e t h e r

with

to consist a

k as in

(5.28)

D'D k'k

P ~

D D

~ DD' kk'

D~D

D, satisfying 1-cell

the following

level):

four axioms

(which are all automatic

at the

U

U

S . z °-

U

~J

ct l-J.

j,J. C~

t~ t~

~3

t~

tZJ

jJ°-

U

W~ t~

~3

~t

C'F

C~ (1)

U U

i

U

U

t',J v

'J1 L~J 0 v

f,~ 01

336

pD'

D'p

(5.31)

D' ~'D

DD'D'

D'DD'

I

m'D

Dm '

P

-~ DD v

D D

I

kk'

k'k ^

k

D D

D D

D'p

pD' ....~... D'DD'

D'D'D

D'k'k I

£ D'k

kk

D'D D

D'D D

kD '

^ ~ kD'

D D D'

'm* D'D

~ DD'D'

D*k D D D

DD

m\ /" D

D D D' ' D D'

DD

kk' D '

D'

337

Dp

pD D'DD

(5.32)

DDD '

'~ DD'D ~>

D'm

imD'

:~' DD'

~'D >

k'k

kk'

^

k

DD

D D

Dp ..

pD : DD'D

D 'DD

k

'kD1

....;D ... >

DDD

/ k D, ~ D D D

~ DDD'

> ^ Dk

'k

DD D

DD D D k

,

mk D D,

D ,D ,D ,

DD

~,~DD ~

DD

D

Dkk'

338

If K = (k,k',k) and H : (h,h',h) D

,

we

define

a

2-cell

H ~

K to

consist

of

are maps from (D,D',p,~)

to

doctrine-modifications

0:

h ~ k and p': h' ~ k', satisfying the evident axioms with respect ^ ~ ~ to h and k. Thus we get a category Map((D,D',p,~),D ),or M(D ) for short.

A doctrine map t: M(t):

M(D*) ~ M(E),

D

~ E induces a functor

sending (k,k',k)

to (tk,tk',tk).

A doctrine

m o d i f i c a t i o n ~: t ~ t I induces a natural t r a n s f o r m a t i o n M(t) ~ M(t I) ^ whose (k,k',k)-component is (Tk,Tk'). Thus M constitutes a 2-functor Doct

~

CAT.

Now it is easy to see that a map K: corresponds

exactly to a D-algebra

(5.20)-(5.24);

(D,D',p,~) ~ {A,A}

structure on A as described in

we have only to replace D

by {A,A} in (5.28),

m u l t i p l y on the right by A, and compose with the e v a l u t a t i o n e:

{A,A}A ~ A, to get the s i t u a t i o n of (5.20).

(5.30),

(5.31),

respectively;

(5.32) easily reduce to (5.21),

The axioms (5.23),

(5.29),

(5.22),

(5.24)

of course they were set up to do just this.

But now a map from (D,D',p,~)

to the doctrine { f,f} of §3.3

gives the analogue of (5.20) with f r e p l a c i n g A a n d the diagram now living in K".

So a,a' get r e p l a c e d by actions of D,D'

that is to say, by enrichments of f to a D - m o r p h i s m D'-morphism (f,~'), and a' gets r e p l a c e d by a 2-cell

on f in K",

(f,~) and to a (a',b')

in K";

the c o n d i t i o n for this to be a 2-cell in K" is precisely

(5.27).

axioms

separately.

(5.21)-(5.24)

Hence to give a map

reduce of course to those for a',~'

(D,D',p,~) ~ (f,f) is to give D - a l g e b r a structures

to A and to B and to enrich f to a D - m o r p h i s m Finally,

The

for T:

f ~ g:

(f,f,f'):

A ~ B.

A ~ B, a map from ( D , D ' , ~

to the

doctrine ~y,T~ of §3.4 is equally easily seen to be just what makes y a D-2-cell.

339

Note that a map from §3.3,

or equally

[f,f],

a map to ( f , f }

corresponds

identities. clearly

(D,D',p,w)

This

that happens

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

in which both

We now pass on to the a n a l o g u e s

that the

2-functor M:

is likely

Doct ~ CAT

particular

equally

for symmetric

of §§4.5-4.7.

which

It is clear

it is t h e r e f o r e condition

the conditions

here.

in §6, where D comes

club

D-morphism,

solution-set

in §4.5 c o n c e r n i n g

to be so apply

applications

through

f and ~' are

is 2-continuous;

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

The remarks

of

f' arbitrary.

5.5

fied.

[ f,f]

to factorize

is not the same as a strict

has ~ = i but

2-representable

to the d o c t r i n e

under w h i c h this

It is certainly in C a t / ~

from a club

monoidal

categories.

is satis-

so in our and D' from the

We h e n c e f o r t h ,

suppose

it to be so, and we h e n c e f o r t h

the r e p r e s e n t i n g unique

doctrine

map

We want case we have because

map.

Thus

t:

D

we have

be an isomorphism, isomorphism. difference

two cases

be stron$,

D'-A/g**

in the former;

This

corresponds

and hence

simultaneously.

the lax case.

There

with

D restricts

to the to

to be an

is no formal

M is 2-continuous

them t o g e t h e r

However,

a' in (5.20)

k in (5.28)

case.

The

where we retain only

to r e q u i r i n g

to r e q u i r i n g

the two cases;

we treat

to be

~ E is tK for a

the d o c t r i n e

of D'-A/g

We call this the Pseudo

between

~ D

~ E.

supposedmAto

D'-morphisms.

(D,D',p,~)

(D,D',p,w)

so far may be called

one on the s u b - 2 - c a t e g o r y strong

any map H:

in fact to consider

considered

take K:

in the latter as

identical

notation,

,

although

of course

it is a d i f f e r e n t

D

in the latter

case.

However,

,

in our prime club,

application,

as it does

will not in the pseudo

in the lax case:

direct

proof

course

in the pseudo

and all our

D

of its existence

2-cells

(cf.

and

case come from a

I have therefore

not the

§1.4 of the Introduction).

case we ~ive (f,f) are isomorphisms.

its

stronger m e a n i n g

same Of

of

§3.5,

340

In v i e w

D-Alg

and

between the

D

-A19.

D-Agg,

Not,

D -algebra

where

(5.33)

=

D -morphism

= (f,~,~')

(5.34)

~

with

strict

,

~* = (f,

):

a

last

exactly

which

the

D A ~

of

ones

they

some are

of

(those

implicitly.

isomorphism

~'

§5.4,

are o n l y

A corresponds

.k'A,

between

paragraph

we do k n o w D

:

=

isomorphism

D -morphisms

we g i v e

a

an

the

we k n o w

§4.6,

a'

F

in

and h e n c e

.kA,

a

,

The

we h a v e

we s a w

A with action

(A,a,a',a')

a

as

but

~ = i and f' = i), In a n a l o g y

The

then,

and D * - A g g , ; the

D-morphisms;

strict

with

of §5.4,

=

a

A -~ B c o r r e s p o n d s

explicitly.

to

the D - a l g e b r a

.kA.

to the

D-morphism

where

~* . k A ,

=

>'

-* f .k'A;

=

and we have

(5.35)

>*

=

!

y:

f ~ g

if and only

if

>

=

i

and

>'

=

I.

Finally,

(5.36)

is a D - 2 - c e l l

We can m a k e every

D-algebra

D -action extend objects

this

statements

(A,a,a',~')

, and

so on.

to the m o r e

A of K but

our proof; done.

a

these

we l e a v e

is of the

The

same

general

2-functors the

if and o n l y

easy

A:

in the a l t e r n a t i v e form

(5.33)

techniques

case

where

as

form

that

for a u n i q u e in §4.7

the a l g e b r a s

L ~ K, w h i c h

details

if it is a D - 2 - c e l l .

we n e e d

to the r e a d e r ,

and

allow are

us

to

not

to c o m p l e t e suppose

this

341

5,6

In this

prove

our

second

analogous

main

to that

proceeding avoid

section

endo-2-functors

Theorem

5.1

2-complete doctrines DD'

2-categ0ry

Define

(5.38)

K,

sj

h:

=

and ~h

l.atter b e i n ~

is not could

but

state

and

in d e t a i l

make

it so by

as we give

it we

direct.

"underlying

objects"

of all

our a l g e b r a s

be a p s e u d o suppose

distributive

that m a p s

s:

D

~

DD'

from

law on the

(D,D',p,w)

(D,D',p,w)

b__e the u n i q u e

~

D*.

into Then

strict

jj '

DD'

= 1.

one

to

that

DD' -* D

as t h e

~ D D

sh = i, and

manner,

by K = (k,k',k):

Let

composite

,- D

kk '

Then

and

are r e p r e s e n t e d

such

less

(D,D',p,w)

is a D - a l g e b r a .

(5.37)

the

itself

doubtless

direct

we p r o c e e d

of K.

Let

D -morphism

4.1;

by b e i n g

section

to §4.8)

The proof

is a m o r e

diagrams

In this are

theorem.

of T h e o r e m

as t h e r e

large

(analogous

.

m

there

is a m o d i f i c a t i o n

This

appl.i.e.s to b o t h

the

that

where...the t h i r d

element

n:

I ~

hs

such

lax and..the p s e u d o qf a m a p

from

that

sO = 1

cases

- the

(D,D',p,w)

t__oo

^

a doctrine

is required........t..obe an isomorphi.sm;

.isomorphism ~ and

so is q.

in the

latter

case

k is an

342

Proof case

As in the p r o o f take

below

care

(h,h)

of

thus

§3.5

the d e t a i l s

below

D

as

.kD

D'

is a D ' - a l g e b r a

,

a'

D-algebra

on it, n a m e l y

By

the u n d e r l y i n g

b

(DD',

because

in the p s e u d o

whence

~'

below

we use

the a' are

the

stronger

n is i n v e r t i b l e

b, b',

DD'

were

action

from

m

, and

(5.33)

we h a v e

=

m

,

.k'D

with

action

with

D-action of DD'

hence

actions

on D

and on

A and B r e s p e c t i v e l y .

by

object

b');

the v a r i o u s

names

with

where

=

D-algebra

whence hand

so f i n a l l y

of n a m i n g

a

(5.12)

case;

if t h e i r

(D*,a,a',a')

m

below,

is i n v e r t i b l e

is a D - a l g e b r a

D-algebra

the d i f f e r e n c e s

2.6.

the p u r p o s e these

4.1,

k is an i s o m o r p h i s m ,

(5.39)

in the p s e u d o

we treat

(5.39)

h

by

of T h e o r e m

For DD',

of t h e m s e l v e s :

is i n v e r t i b l e

invertible;

of T h e o r e m

a'

m',

is t h e r e f o r e

=

m

.kD

so we can

mD'(also

is DD'.

it is also

.

form

called

Thus

a

DD'

the free

b = (b,b')). is a

a D*-algebra

with

action

. Since

statement

is the

free

of the t h e o r e m

satisfying

a unique

shall

D'

need

below

r D

D'-morphism.

Moreover

is a strict

way

D -morphism

s:

of the

D

~ DD'

composite

from

with

s is a f o r t i o r i

the d o c t r i n e a'

on D

j':

whence

map

given

the d o c t r i n e

D'-morphism;

composite

the v a l u e

D -morphism,

the D ' - a c t i o n

in the u s u a l

by its

is as in the

s

As a strict

that

strict

on i, t h e r e

~ DD'.

k'

note

D -algebra

(5.37).

We

(5.40)

D

I ~ D'

by

map

as in

k'

is a s t r i c t (5.39)

k'. (2.4)

But

a strict

k'j'

Thus and = j

D'-morphism;

is just

that

arising

the c o m p o s i t e (2.5)

(5.40)

it is d e t e r m i n e d

since

k'

is a

~3

doctrine may

map,

so that

be w r i t t e n

using

(5.37)

we h a v e

~ D'

too

composite

(5.41)

(5.40)

and

(5.42)

(2.5)

H.jD'

and g i v e n

(5.43)

D'-morphism

is g i v e n

=

Since (2.4)

D

is a D - a l g e b r a a unique

=

k':

strict

D' -*

we w r i t e

and Dk'

Substituting that

h has

We

D-morphism.

is

• D

is a free H:

one,

DD' ~ D*

there

is by

satisfying

.

h

by

(5.14),

[u'

=

(5.38)

being

a = (a,~') strict

as

~.'.D'Dk'.

composite

in the

of a f r o m statement

(5.39),

strict

D-morphism

~ ~ DD'.

~ D s

we

see

of the theorem.

s is a f o r t i o r i

~ D'

k' a l r e a d y

it

that

D -morphism,

the

Because

As a D ' - m o r p h i s m ,

for h the v a l u e

the v a l u e

a strict

f o r m H = (h,h,~').

(5.25).

= 1,

value

Consider

extended

(Dk',l)

conclude

in this indeed

DD'

D-morphism

h = I by

a.Dk',

Being

the

D*

H in the

as in §5.3;

=

that

a

D-morphism,

h

and

~ DD

is a strict

(5.44)

we c o n c l u d e

as the c o m p o s i t e

DD'

a D'-morphism.

§5.2,

jD'.

explicitly

§5.3,

by

by

Dk ' As in

which

JD'

is a strict

sk'

= jj',

~ DD'.

j'

jD'

= sj

as

I

Since

sk'j'

a strict

344

As in By

(2.4)

(5.42)

another

and

and

name

(5.45)

(5.41),

si

:

Now

let H

(5.46)

--

now

D

~ DD'

prove

DD' ~ D

composition,

and

H.

since

Since

(5.17)

is

this

s corresponds

(5.37)

the

of §2.3,

r:

with D

i ~ DD'

,i r e p l a c i n g

corresponding

to

D,A

to

So

=

h .jj '

=

m

.kk' .jj'

--

m

.j j

=

m

.D j

=

j

by

since

(5.38) k,k'

m

.D j

in the s p e c i a l

case

(5.47)

h

.D j j ' . h

result

The

of the p r o o f

=

maps

.j

since

reflexion

are d o c t r i n e

now

=

of

follows

I

§2.3

(doctrine

in w h i c h

from Theorem

that

rest

by

be the D - m o r p h i s m

to the D - m o r p h i s m

situation

is jj'

we are

Our d e s i r e d

which

jD'.

i.

By

hr

Therefore

is jD';

with

we h a v e

in the

respectively. s:

(5.34)

preserves

(5.35),

So we are

its c o m p o s i t e

Hence

= (h,):

as in

sH

composite

by

I.

correspondence by

it is d e t e r m i n e d this

for JD'

corresponding

itself

(2.5),

i.

consists

in v e r i f y i n g

this.

axiom).

(2.23) 2.6,

commutes.

once

we

345

W e d o this,

(5.48)

by hindsight,

a' .D'JD

=

in s e v e r a l

small

steps.

First,

by

(5.23),

i.

Compose

this

w i t h D'k':

replace

D'jD

.D'k'

D'D' ~ D ' D

and use

by D ' D k ' . D ' j D ' .

Using

the n a t u r a l i t y

the t h i r d

of j to

equation

of

(5.44),

we n o w h a v e

(5.49)

h'.D'JD'

U s e the

second

(mentally so t h a t

--

equation

replacing (5.49)

(5.50)

h

i.

of

f:

=

h'

DD' ~ D ) ;

in t e r m s

we h a v e

mDD'

satisfied

,

,

~ DD'

,

Db

,

= D D DD'

-* h h

D D

D *D *h

and u s e

D

--*

(5.52)

(That

h

is,

on the

(5.50)

of

~

to

front

simplify

*

D D D

~ D

we h a v e u s e d

end w i t h

the

--~

.m DD' .D k'jD'

and t h e n r e v e r t e d

.k'DD',

=

h

(5.50)

to l i n e a r

*

right

W

.D b

side.

D D'D' We

is

b ~ DD'

-* h

D D

, ~ D m

~ D D DD',

end u p w i t h

*

.D k'jD'.

to k i l l t h e D h

rather

D* h

, , ~ Dm

D k'jD':

H

, -~ D DD'

Dh

m

sides

= h

by the D - m o r p h i s m

b

~ D DD'

D *h

both

h'

1.

D D DD'

Compose

of h

to

N o w o n e o f the a x i o m s

(5.51)

to e x p r e s s

A ~ B by h:

simplifies

.k'jD'

(5.34)

on t h e r i g h t

than diagrammatic

of

notation).

(5.51),

h

~8

Now be w r i t t e n

compose

(5.52)

with

kD'j':

~ D D'

, D D'j'

kD'

we t h e n

have

on the r i g h t

h

or h

.D y.kD'

this

is e q u a l l y

this

is I.

right

side.

which

say, h

.D b

where

.D

is

.k'jD'.D'j'

But

composing

Simplifying

D D'D'.

k'jD'.D D'j'.kD',

y = b

.kDD'.Dy.

So a f t e r

side

~

by

(5.34),

(5.52)

a little

what

with

By the n a t u r a l i t y h*.kDD' kD'j',

we get

= ~,

we get

on the

left

and by

h

.m DD' .kk'jj'

Write

kk'jj'

--

as the

DD'

side,

i.

composite

~ D D

, , D D jj'

m D D DD'

to r e p l a c e

m DD'.D

kk'

and the use D Jj'.m

(5.54)

which

.

Then

h

by

the n a t u r a l i t y (5.53)

.D J J ' . m

(5.38)

becomes

.kk'

is the

of m

=

(5.47)

i,

that

we

seek.

D jj'

of k, (5.44)

! on the

have

(5.53)

can also

as

DD'

What

DD' -~ D D'D"

by

we n o w

347

6.

6.1

P s e u d o - c o m m u t a t i v e doctrines and clubs

An endofunctor D of C a t

is m a d e into an e n d o - 2 - f u n c t o r by

giving a natural t r a n s f o r m a t i o n [A,B] subject to two axioms we derive a natural

(6.1)

t:

AxDB

~

~

[DA,DB]

(the "strength" of D)

(multiplicative and unitary).

From this strength

(and in fact 2-natural) t r a n s f o r m a t i o n

D(AxB)

as the image under a d J u n c t i o n of the composite

(6.2)

A ~

[B,AxB]

~

[DB,D(AxB)]

.

It is well known that giving the strength of D is equivalent to giving t; in fact t and the strength are mates, [13]

§2.2, under the adjunctions

-xA -~ [A,-]

in the sense of

and -xDA -~ [DA,-] ; so

that t is often called the "monoidal strength" of D:

of course t must

satisfy two axioms c o r r e s p o n d i n g to those for the strength. If now (D,m,j) is a doctrine on Cat, it is easy to express the m o n o i d a l strength of D 2 in terms of t, while that of I is the identity.

The 2-naturality, as distinct from the naturality, of j and

of m can be expressed in terms of the m o n o i d a l

strengths,

in the form

of a commutative d i a g r a m involving j and t and another involving m and t.

So all told, to make a mere monad

(D,m,j) on C a t

into a doctrine is

to give a natural t as in (6.1) satisfying four axioms. We use the symbol # in a general way to denote conjugation under the symmetry c:

(6.3)

t#:

DAxB

AxB ~

~ D(AxB),

namely the composite

BxA

of C a t .

So alongside t we also have

348

(6.4)

DAxB

D(BxA)

-~ Bx]Zz~ ~

c

t

-* D(AxB). Dc

We can then form the 2-natural

(6.5)

~:

DA×DB ~ D(A×B)

as the composite

(6.6)

DAxDB

t-~ D(AxDB)

Dr-* D2(AxB) "*m D(A×B),

as well as its conjugate

(6.7)

d#:

DAxDB ~

D(AxB),

namely the composite

DAxDB ~t D(DAxB) D~# D2(AxB) ~m D(AxB).

(6.8)

Moreover,

using I for the unit category,

(6.9)

d ° = jI:

I ~

DI.

Kock [14]

shows that d and d ° enrich D to a monoidal

2-functor

(D,d,d°):

monoidal~

the extra condition

(6.1o)

Cat ~ Cat.

This is not in general

needed

symmetric

for this is precisely

~ = ~#.

The unit j: natural

we can set

I ~ D is always monoidally

transformations

2-natural"

there

and "symmetric

2-natural;

is no difference

monoidally

2-natural".

recall

between

that for

"monoidally

The multiplication

349 m:

D 2 -* D is not in general m o n o i d a l l y

2-natural, but is so if and

only if (6.10) is satisfied. Kock called a doctrine - or more g e n e r a l l y a V-monad on a symmetric m o n o i d a l closed V - commutative Because of the last remark, call t h e m m o n o i d a l

V-monads;

they are the m o n a d s on V in the

2-category of symmetric m o n o i d a l

algebras V D is symmetric closed,

U:

V D -~ V, F:

and n e c e s s a r i l y lie in the

V-categories.

completeness assumptions

if it admits coequalizers;

For commutative D,

on V, the V-category of

and indeed symmetric m o n o i d a l closed

and the forgetful and free V-functors

V -~ V D are symmetric monoidal.

if vD,u,F are monoidal,

(6.10).

it has now become rather more common to

2-category of m o n o i d a l V-categories,

nnder the m i l d e s t

if it satisfied

so is D = UF.

Conversely,

of course,

Thus, m o d u l o the a l w a y s - t r o u b l e -

some m a t t e r of the existence of coequalizers in V D, c o m m u t a t i v i t y of D is the n e c e s s a r y and sufficient c o n d i t i o n for vD,u,F to be symmetric m o n o i d a l closed:

it generalizes Linton's c r i t e r i o n

(cf. [61 p.549)

when V = Sets and D comes from a finitary theory. A final remark at this level. monad on Sets.

Suppose D is a commutative

Then, being symmetric monoidal,

D takes a commutative

m o n o i d A to a commutative m o n o i d DA, and in fact lifts to a m o n a d D on the category of commutative monoids.

So if D' is the monad on

Sets whose algebras are commutative monoids, we get an honest d i s t r i b u t i v e law p: 6.2

D'D -~ DD'

In the doctrine case,

c o m m u t a t i v i t y as expressed by (6.10)

seems to be rare in natural examples.

We call a d o c t r i n e pseudo-

commutative if there is instead an i s o m o r p h i s m (invertible m o d i f i c a t ion)

satisfying suitable axioms y#y:

(one of which is that 7 be involutary;

d ~ d# ~ d is the identity).

Then we get a p s e u d o - d i s t r i b u t i v e

350

law

(p,w):

D'D ~ DD'

the last paragraph) the d o c t r i n e

as in §§5.1,

the m o n a d

for symmetric

for in all m y everything

examples

becomes

I succeed

easier

a more

sketch

this

because

compact

structure

but

in the above

Cat/~,

(as in

rather

would

treatment

generality,

and then

all the diagrams

come down

be pleasant,

but unless

of it, my a p p l i c a t i o n s

the extra complication. that

and let D be p s e u d o - c o m m u t a t i v e

satisfying

monoids,

for a club D in

Let then A be a D'-category, category,

now D' is not

categories.

The extra g e n e r a l i t y

in f i n d i n g

do not justify

than

D is Do-

much

by one dimension.

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

monoidal

I shall do no more

5.2, where

axioms to DA.

is, a symmetric

with y as in (6.11)

to be determined.

We give

Its tensor

and

product

monoidal

a symmetric

identity

object

monoidal are given

by

(6.12)

DAxDA ~ D(AxA) d

(6.13)

I

~ DI ~ DA; d° DI

its a s s o c i a t i v i t y those

-~ DA, D@

and right

for A (called

identity

isomorphisms

a and r) by composing

DAxDAxDA

~~ dxl

are obtained

from

Da with

D(AxA)xDA ~ D(AxAxA) d

and Dr with

DAxI

~

ixd °

Its c o m m u t a t i v i t y as the composite

DAxDI ~ D(Axl). d

i s o m o r p h i s m is obtained

from that

(called

c) for A

351

(6.14)

DAxDA

DAxDA

...........

a

T'

d

Dc

D(AxA) ...........

" D(A×A) Dc DA

where y' denotes because

Dc.y.

The m o n o i d a l - c a t e g o r y

(D,d,d °) is monoidal;

(c 2 = i and the hexagonal involutory

YAxB,C

the symmetric-monoidal-category

axiom)

require respectively

and that it satisfy a kind of hexagonal

to ~A,C and

axioms

that y be

axiom

( relating

YB,C ).

The above passage monoldal

axioms are immediate

from A to DA respects

functors and the corresponding

strict

symmetric

natural transformations,

and therefore gives us a lifting of D as in (5.1), whence also a lifting as in (5.2).

The lifting of j at the

automatic,

because

symmetry:

for this it suffices

of y with jxl:

(D,d,d °) is monoidal,

(5.2) level,

(but not strict)

that is, to enrich

symmetric monoidal

functor

here I use m in place of the more usual m, which

unfortunately triple

the

AxDB ~ DAxDB is the identity.

D2A ~ DA to a strong

(m,m,m°);

except as regards

to impose the axiom that the composite

It remains to lift m at the m:

(5.1) level is

has another

(m,m,m°).

sense in §5, being in fact the name of the

We can take m ° to be the identity.

To get an m,

it turns out that we need a m o d i f i c a t i o n ~:

d(mxm) ~ m.Dd.d:

D2AxD2B ~ D(AxB).

t,m,j allow us to write this as ~: we set p = m.Dt.m.D¥.t #. a symmetric monoidal

The four axioms satisfied by

m.Dt.m.Dd.t # ~ m.Dt.m. Dd#.t #, and

Of the three axioms saying that

functor,

(m,m,m °) is

the one involving m ° is satisfied

in virtue

352

of our a x i o m ~(J×l) stating

that

satisfied three

(D,m,j)

in v i r t u e

more

axioms

distributive I

that

or at m o s t

reluctant

to d e l a y

I have

on to the

not

this

club

the n o t a t i o n

F:

D ~

letters

succeeded longer

one.

So

pseudo

of T ( S I ( R I I

laws

T[S I

The

similarly

functor by

t:A

x

~ for rS, we

shall

image

similarly the

club

and

case

we

and we of D

so on;

quite

OoD

... S T ) and under

j:

2-natural similarly

law for m e x p r e s s e s ... RT~T))

T(~

ordered

are w r i t t e n of

for

J ~ D of the

for m o r p h i s m s ;

The

use

or

~ D

the

is

in fact

generally

denoted

equalities

DoB ~ D o ( A x B )

So I

set of a x i o m s

if we use m:

... S ~ ( R T I

of A×B

to do so.

as for p e r m u t a t i o n s

S T ] to T(S I

for m o r p h i s m s .

and the m o r p h i s m s

a n d am

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

associative

... RI~ I)

that

them,

In the

to date m has

the

to

initio.

as w e l l

in [ 9 ] § 1 0 ,

... RT~T) , and

trying

case

for a club D in C a t / ~ ,

that

...

club

a definitive

no c o n f u s i o n

for m and J e x p r e s s

= T, and

while

T for FT,

although

the

for ~ r e d u c e

in so r e d u c i n g

law ab

numbers,

cause

as

with

In p a r t i c u l a r

of J is ~ E D.

T(S I ... S T ) ( R I I

to be g i v e n

j are

our d e s i r e d

doctrine".

of s y m b o l s

sends

Again

object

The

axioms

= i, l e a v i n g

we n o w have

giving

D = Do-

write

of D,

then m

the e q u a l i t y

that

for n a t u r a l

for m o r p h i s m s .

objects

we get

three

involving

y(j×l)

by a n a l o g y

commutative

so short

It should

DoDo- ~ Do-;

the

yet

of [ 9]§10.

the m u l t i p l i c a t i o n

~(T)

two

six a x i o m s

case w i t h o u t

5; we t y p i c a l l y

we are r u n n i n g

unitary

that

volume

suppose

use

functions.

the

axiom

the p s e u d o d i s t r i b u t i v e

We n o w

unique

the

three,

for the y of a " p s e u d o

Greek

Of the final

law.

but

6.3

same

for ~ e n s u r e

follows:

establish

two.

is a d o c t r i n e , of the

suspect

one or two,

pass

= I, l e a v i n g

(6.1)

with the

two

... 2) = T a n d pairs

(A,B)

that

are

and ( f , g ) .

is e a s i l y

seen

353

t( A, S[B 1 ... B ~

t

= h[

for t #.

× DoB ~ ~o(A×B);

they

...

A,B>|,



l;

H e n c e we c a l c u l a t e send the object

{ T [ A I ... AT] , S[B I ... B ]) r e s p e c t i v e l y

to

(6.15)

T(S

... S ) [ { A I , B I)

... ( A I , B a)

... ( A T , B I}

... (A

(6.16)

S(T

... T)[{ AI,B I)

... { AT,B I)

... ( AI,B ~)

... { A ,B ) ] ,

with similar

effects

and T occurs

~

(6.15)

on m o r p h i s m s .

times

in (6.16);

the l e x i c o g r a p h i c a l

lexicographical

order

(6.17)

zTS:

T(S

component (6.16).

of

... S) ~

(6.15)

(6.11)

S(T

is in the

is to give for each

... T)

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

and

(6.16);

of T~ d e m a n d e d

by a

then the ( T [ A I ... A T ] , S [ B I ... B ])-

is the m o r p h i s m

By the n a t u r a l i t y in

,B B)

isomorphism

of y = YA,B

commutativity

in (6.15)

of the (B,~} •

w h o s e type FzTS = ~T,a comparison

of the ( A

(Y

o r d e r of the (~,B) , and in (6.16)

To give an i s o m o r p h i s m ¥ as in T,S E D a n a t u r a l

,B )], a

T

Here S o c c u r s T times

the order

T

zTS[I,I,

of zTS we m e a n

...,I]

of c o u r s e

from

(6.15)

that we have

to

354

zTS,

T(S...S)

(6.18)

S(T...T) l g(f...f)

f(g...g)

S'(T'...T').

T'(S'...S') ZT, S ,

(This diagram

is more complicated

and Ff = ~ is a function morphlsm

than it looks:

from T to ~'; the ~-th g in f(g...g)

from the ~-th S in T(S...S)

in accordance good sense,

in general T' 6 T,

with the conventions

to the @~-th S' in T'(S'...S'),

of [9]

§i0; however

in §6.2;

it remains

only to put such axioms

those we require

for y.

case,

¥ and referring

forgetting

detail what we barely 6.4

However

outlined

we deal henceforth

with an isomorphism

~T,~ described

above,

one axiom:

namely

on zTS as ensure only with the club out in

in §6.2.

Cat/S together

natural

situation

only to zTS , and carrying

We define then a ~seudo-commutative

in addition

it does make

both legs having the same type.)

Thus does the club case fit into the general sketched

is a

c!ub to be a club D in

zTS as in (6.17),

in the sense of (6.18), the c o ~ u t a t i v i t y

of the type and satisfying

of

zT,S(RI,. (6.19)

T(S(RI...R~)...S(R1...R))

"R~--S(R1.)..R)(T...T)

I T(S...S)(R I...R~...R I . . . R ) zTS(I,I , . . . ,i)

S(T...T)(RI...R1...~...R )

S(RI(T...T)...R (T...T))

I

S(ZRI T,--.,z R T )

S(T(R I...RI)...T(R

...R a)).

355

We derive some immediate consequences of (6.19). put S = ~ (so that c = I) and R I = ~.

First,

The top and the left edges are

then both the i s o m o r p h i s m ZT~ while the right edge is Z~T , whence

(6.20)

z~T = 1.

Now in (6.19) put T = i, S = i, R I = R.

(6.21)

zR&

Using

i.

Finally just put S = ~, R I = R, T arbitrary.

(6.22)

(6.20) we get

Using

(6.21) we get

ZRTZTR = I.

Hence z is involutary and satisfies

(6.20).

Before giving examples we make the following remark.

Any

club D becomes a strict m o n o i d a l category, with ~ as identity, set T@S ~ T(S...S).

M o r e o v e r F:

if we

~ ~ ~ is a strict m o r p h i s m of strict

m o n o i d a l categories when ~ is given the c a r t e s i a n m o n o i d a l structure T®~ = T×~ ~ ~ .

For a p s e u d o - c o m m u t a t i v e club, zTS:

symmetric monoidal;

T@S ~ S@T makes

the hexagon axiom is got by setting

R I = ... = R~ ~ R in (6.19), and the other axiom is (6.22). F:

Moreover

V ~ ~ is then a strict symmetric m o n o i d a l functor, as ~T,c is

the classical symmetry on ~.

This o b s e r v a t i o n is useful in limiting

our search for examples. Example 6.1

The club ~ itself.

~i + "'" + cT while ~ is i. the p e r m u t a t i o n ~T,~: satisfy the axiom make sense,

We recall that T(al,...,~ ~) is

We take zT

T~ ~ ~T of §6.3.

:

T(c...c) ~ C(T.-.T)

to be

This is easily seen to

(6.19); in fact if it did not the axiom would not

its two legs being of different types.

Note that an

~-algebra is just a category with s t r i c t l y - a s s o c i a t i v e finite coproducts.

356

Example

6.2

coproducts.

The club S w h o s e algebras are c a t e g o r i e s - w i t h - f i n i t e The augmentation

define zTS (as we must)

r:

S ~ ~ is an equivalence,

and we

to be the unique m o r p h i s m such that

FzTS = ~T,~" Example 6.~

The sub-club ~ of ~ with the natural numbers as objects

but with permutations as its only morphisms;

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

is of course the inclusion.

"

Again z

T~

= ~T

,c

The P-algebras are -

the strict symmetric m o n o i d a l categories.

Example 6.4 categories.

The club P whose algebras are the symmetric m o n o i d a l By Mac Lane's original coherence result [23],

a u g m e n t a t i o n F is an equivalence of P with F(P) the unique m o r p h i s m with FzTS ~ ~ ,

Example 6.~

= ~ C ~.

the Again zTS is

.

The full sub-club of ~ determined by the objects 0 and

i; it is the arrow category ~.

Its algebras are c a t e g o r i e s - w i t h - a n -

initial-object.

Example 6~6

The full sub-club of ~ d e t e r m i n e d by the objects

it is the discrete category 2.

0 and i;

Its algebras are c a t e g o r i e s - w l t h - a -

distinguished-object.

Example

6~7

Our r e m a i n i n g examples are all of the kind where the

a u g m e n t a t i o n F:

D ~ ~ is the constant functor at i E ~.

A club of this

kind is nothing but a strict m o n o i d a l category D, with T(S) = T~S and with ~ as the identity for @.

It is p s e u d o - c o m m u t a t i v e

it is symmetric monoidal,

in which case we set zTS:

equal to the symmetry c:

T~S ~ S~T.

D-algebra

Since DoA

T(S) ~ S(T)

is just D×A,

is a category A with a strictly associative

action DxA ~ A

if and only if

and unitary

(as for example E acts on K in §3.2 above).

the club whose algebras are c a t e g o r i e s - b e a r i n g - a - m o n a d , is not an example,

for the m o n o i d a l

category ~ is not symmetric.

a

Note that

namely 6'

structure on the simplicial

We pass to some p a r t i c u l a r cases of this

~7

example.

Example 6.7,1

Let C be a symmetric m o n o i d a l category and consider

the club D whose algebras are categories A together with a coherently (but not strictly) instance,

associative and unitary functor @:

any tensored C - c a t e g o r y A is such an algebra

C×A ~ A.

For

(where by

C - c a t e g o r y I m e a n "category enriched over C " - a remark n e c e s s i t a t e d by the fact that I sometimes, D-category).

for a club D, call a D - a l g e b r a a

Since all the operations on A are unary, the a u g m e n t a t -

ion of D is constant at i.

We have to show that D is symmetric, and

is thus a p s e u d o - c o m m u t a t i v e club. the canonical strict m o n o i d a l

But we know just what D is:

it is

category equivalent to C; its objects

are n-ads ( C I , . . . , C n) of objects of C, its tensor product is given on objects by concatenation,

and its m o r p h i s m s ( C I . . . , C n } ~ ~ BI,...,B m}

are the m o r p h i s m s CI®(C2@...(Cn_I@Cn) ) ~ BI@(B2@...(Bm_I@Bm) ) in C. It is now clear that D is symmetric when C is.

ExamDle 6.T.2

Let D be the discrete category ~ of natural numbers,

which is symmetric m o n o i d a l with + as its tensor product. is a category A with an e n d o f u n c t o r E:

Example 6.7. 3 monoidal

A D-algebra

A ~ A.

Let D be the discrete category 2 with the symmetric

structure having the usual m u l t i p l i c a t i o n of its objects 0

and i as tensor product.

This is not the same as Example 6.6, for

the a u g m e n t a t i o n is now constant at i, whereas there it was r0 = 0, FI = i. E 2=

An algebra is a category A with an endofunctor E such that

E.

Examole 6.7.4 monoidal

Let P be the arrow category 2 with the symmetric

structure given on objects as in the last example.

This is

again different f r o m Example 6.5, as the a u g m e n t a t i o n is again constant at i.

An a l g e b r a is a category A together w i t h an

indempotent comonad.

R e p l a c i n g ~ by ~op gives another example,

an algebra is a category bearing an indempotent monad.

where

358

6.5

Let D be a p s e u d o - c o m m u t a t i v e club as in §6.4, giving the

doctrine D : go_., let D' be the d o c t r i n e Po- whose algebras are symmetric m o n o i d a l

(s.m.) categories.

We exhibit a pseudo d i s t r i b u t i v e

law of D' over D, as described in §§5.1,

5.2.

(We could equally

produce a pseudo distributive law if we took D' to be the doctrine ~ofor strict s.m. categories;

this requires only the easy o b s e r v a t i o n

that if the s.m. structure on A is strict,

so is that we construct

below on DoA; we shall not refer further to this case.) The first step is to produce the lifting extension

(5.2).

Let

(A,@,l,a,r,c)

DA = DoA a s.m. structure to get

(6.23)

D

(5.1) of D, and its

be a s.m. category; we give

= (DoA,~,~,~,~,~).

We set

T[ A I ... A T ] @S[ B 1 ... Bo]

= T(S...S)[ A I @ B I , . . . , A I @ B o , . . . , A T @ B I , . . . , A T @ B o] ,

and define @ similarly on morphisms.

We define the rest of the

structure by

(6.24)

(6.25)

[ = l[I] ,

(T[A I...]@S[B I...] )@R[C I...] ) -~ T[A I...]@(S[B I...]@R[C I...] )

T(S...S)(R...R)[

(6.26)

TIAI...I~

IF

(AI~B?)~C 1 . . . . I - ~ ( S . . . S ) ( R . . . R ) 1[ a , a , . . . , a l

Y

T[ k I ... ]

II T [ A I . . . I ® i[I]

Jl T[ AI@I .... ]

T[ A I . . . 1 ,

i[ r,r ..... rl

II

[AlO(BI®C l) . . . . 1,

3,59

(6.27)

m- S[B I...]@T[A I...]

T[ A I. . .] @S[ B I. ..]

fl

H

S(T...T)[B10A1,...]

T(S ....S)[ AI@B 1 .... l

zTS[ c , c , . . .

.

,c]

The axioms for a,r,c follow from those for a,r,c together with

(6.22)

(needed for ~2 = l) and the special case of (6.19) where R 1 = ... = R~ = R (needed for the hexagonal (Note that when A is the unit category on Dol, %:

= P itself,

described

A ~ B is a strict

TIA1...ATI mations.

^

functor

structure If

RoA ~ R o B

sending

for s.m. natural

transfor-

(5.1) of D.

(5.2) sends the non-strict

o

(~,¢,~),

to (¢,$,¢°)

extension

a and ~).

6.1 above.)

so too is DoS:

and similarly

Thus we have the lifting Its automatic

I this is the s.m.

just before Example

s.m. functor

to T[¢A1...$AT],

axiom connecting

s.m.

o

where ¢:

~A ® CB * ~(A@B)

where ¢ = Roe, where S h a s

and where ¢°= ~I~°].

and ~ : I B ~ ¢IA,

components

T(S...S)/$ ..... 2],

(Recall that we are using $ for the more

familiar ¢ to avoid confusion

with the m etc. of §5.)

sends the s.m. natural transformation with T[A1...AT]-component The functor joA:

~:

Simiarly

(~,$,~°) ~ (@,~,~°)

to that

T|aA1,...,~AT]. A ~ DoA is the funetor

and is clearly a strict s.m.

functor because

z~

sending A to SIAl, = 1 by (6.20).

This

gives us the j, and the j of §5.1. The next thing is to enrich moA: functor mA = (moA,~A,m°A), m,m,m

(moA

)~ =

Since ~ = ~[~] l[II

~ poA

to a strong s.m.

as in (5.18); we abbreviate mA, ~A, m°A to

where no confusion is likely.

on D o D o A .

RoDoA

= l[![Ill

= ~; so we take m

o

Use ~ etc. for the s.m. structure = ~[~][II , we have

= 1.

Now let

360

X = T[ PI [ All...AI~I]

, . . . , PI:[ ATI" • .ATw ] ] T

Y = S[RI[BII...BIo I] , ...

be objects (moA)X

of poNoA.

We need m:

= T(PI...PT)[AII...]

(6.28)

(moA)X ~

(moA)Y ~

(moA)X @ (moA)Y

T(MI...MT)[AII®BII

)...S(RI...Ro))[AII@BII .... ] ,

M I = PI(S...S)(RI...R

On the other

...RI...Ra).

hand,

xSY = T(S...S)[PI[AII...]~RI[BII...] =

T(S...S)[PI(RZ...RI)IAII~Bll

.... ] .... ! .... ],

so that

(moA)(X@Y)

=

T(S...S)(PI(RI...RI)...)[AII~Sll

= T(NI...NT)[AII®BII

where

(6.31)

.... |

for example

(6.29)

(6.3O)

(moA)(XSY).

and

T(PI...PT)(S(RI...R

where

, Rq[ B(~I " " "B°Po] ]

.... ]

for example

N I = S(PI...PI)(RI...RI...Ro...Ra).

.... ]

Now

361

We now take the X,Y-component

(6.32)

T ( Z P i s ( l , l .... ,i),

of m to be

..., Z p T s ( l , l , . . . , l ) ) [ l , l , . . . , i ] ,

where for example ZPis(l,l,...,l): Using the properties

M I ~ N I.

(6.18)-(6.22)

of z, it is trivial if

somewhat tedious to verify that m = ( m o A , m , m ) for a s.m. functor.

satisfies the axioms

It is clearly 2-natural in A by its construction.

Finally, using the same properties of z and a large sheet of paper, we easily verify that

(D,m,j)

satisfies the axioms for a doctrine.

Thus we have our d i s t r i b u t i v e law p,~ as in §5.2. We shall not need the explicit values of p and ~. however,

should be noted.

One point,

As a 2-natural t r a n s f o r m a t i o n t

poDo- ~ D~Po-, p comes from a map PoD ~ DoP in CatJ~, that we may as well still call p; see for example T h e o r e m 8.1 of [9]. a l t h o u g h P and D are both clubs in Cat/~, not lie in Cat/~.

For instance,

the map p:

it sends ~[T,S]

However,

PoD ~ DoP does

to T(S...S)[®,®,...,@] ;

the type of the former is T + ~, while that of the latter is 2T~; so p does not commute with the a u g m e n t a t i o n £.

This is in contrast with

say §4.10 above, where we did not have to move out of Cat/~;

and it is

perhaps the first example to show that, even if we are only concerned with clubs in C a t / ~

we need the fuller theory of [9].

(This was

adumbrated in 510.7 of [9].) We sum up the result of this section in:

Theorem 6.8

I f D is a p s e u d o - c o m m u t a t i v e

club, the c o n s t r u c t i o n

above defines a pseudo d i s t r i b u t i v e l a w of D' over D = Do-, where D' = Po- is the doctrine for syrmmetric m o n o i d a l c a t e g o r i e s .

Similarly

if D' is the doctrine ~o- for strict symmetric m o n o i d a l categories.

382

The examples in §6.4 of p s e u d o - c o m m u t a t i v e clubs now give examples of pseudo d i s t r i b u t i v e

6.6

laws, as promised in §5.1.

For the special pseudo d i s t r i b u t i v e laws obtained in this way, ,

the D-algebras, and hence the D -algebras, than that given in §5.3.

admit a simpler d e s c r i p t i o n

We work out from first principles what they

are. To give such an algebra B is to give a s.m. category B together with an action b: = (b,~,b°):

DoB

-~ B that

DB -* B.

So b is to be a s.m. functor

satisfies the axioms for a D-action.

(we are using B , b rather than A, a to avoid c o n f u s i o n with the a s s o c i a t i v i t y a.)

The s.m.

structure on D o B

described in §6.4, which we employ without At the level of ordinary functors, that b:

DoB

-~ B is to be a D - a c t i o n

further comment. the action axioms tell us

on B in the ordinary sense.

write the image under b of TEBI...BT] E D o B T(BI...BT)

T(SI...ST)(AII...AT1...)

We

in the usual way as

E B, and similarly for morphisms.

express the equality of T(SI(AII...)

is of course that

The action axioms now

... ST(AT1...))

with

and of I(A) with A, together w i t h the

similar equalities for morphisms. Since joB: B -~ D o B

is the strict s.m. functor sending B to

i| B|, the u n i t a r y axiom for an action immediately forces b ° to be the identity:

(6.33)

b ° = i:

I ~ ~(I) = I.

Since m ° = I too, the a s s o c i a t i v i t y axiom for an action is satisfied at the level of °-components. The only extra piece of structure,

then, that a D -algebra B

has, over and above its symmetric m o n o i d a l structure and its D-algebra

structure,

is the natural t r a n s f o r m a t i o n

363

(6.34)

T(AI'''AT)®S(BI'''B

c

)

IDTIAI...AT! ,S[BI...B] T T(S...S)(AI®BI,...,AI@Bo,...,AT@BI,.-.,AT@Bo), ^

which

we a b b r e v i a t e

^

to bTS or just

to b.

^

The axioms following

(6.35)

for

(b,b,l)

to be a s.m.

functorreduce

to the

three:

T(A1. • .AT)@I_(I)

T(AI...A

)®I T

T(1.

~)(AI~I ..A OZ)

--

T

H

r

T(AI~-r...A ~I)

~-

T

T(A

. .A ).

l(r,r .... ,r)

(6.36) a

(T(A I. .. )@S(B I. .. ) )®R(C I. .. )

T(A I.. • )@( S ( B l ...)@R(C I. . . ) )

T(s...s)b (i I~B1.... )°mCa"" )

T(A I . . . ) @ S ( R . . . R ) ( B I @ c

T(S...S)(R...R)((AI®BI)®C

T(S...S)(R...R)((AI@(BI®C

1 .... ) ~

I .... )

l) ....

l(a,a ..... a)

(6.37)

T(AI...)®S(BI...)

T(S...S) (AI®B 1 .... )

c

~-- S ( B I . . . ) @ T ( A I . . .

_~

zTs(c,c ..... c)

)

S ( T . . . T ) ( B I @ A 1 .... ).

364 ^

The

-component

of the u n i t a r y

axiom

for an a c t i o n

reduces

to

^

(6.38)

b~IA],~[B] :

Finally

the

reduces,

~(A)@~(B)

-component

~ !(!)(A@B)

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

in v i e w of (6.32),

is I:

axiom

A®B ~ A@B.

for an a c t i o n

to

(6.39) T(P r •.P~ ) (An...)~(~. • .% ) (Bli. • •)

^

bTS

T(PI" '"PT ) (S(RI" " .R )...S(RI...Ro )) (AII@BII .... )

T(S..S)(PI(AII...)@RI(BII...) .... )

T(Pi(S...S)(~I...~..ml... ~) .... )(AliOBll .... )

T(S...S)(bPi~-.)

T(S...S)(PI(RI...RI)(AII@BII .... ) .... )

T(ZPis(l ..... i) .... )(i .... ,i)

T(S(PI...PI)(RI..mr..~... ~) .... )(Am~li,...) where

in the topmost

a r r o w T(P)

stands

for T ( P I . . . P T ) ,

etc.

Now w r i t e

^

(6.40)

eB

T[AI...A I

abbreviating

(6.41)

for

e:

it where

bT[Ai...A I,~[B],

desirable

T(AI...AT)@B

to e TB

or e T or e, so that

~ T ( A I ® B ~ . . . , A T @ B ).

365

We show that b can be given in terms of e alone, the axioms on reducing to simple axioms on e. Write

(6.42)

e#:

B@T(AI...A T) ~ T(B@AI, .... B®A T

for the conjugate (6.43)

e # = T(c,c,..c).e.c

of e under c.

Then (6.37) gives

^

(6.44)

b__iT _- e # .

Now in (6.39) set each P

equal to I and set S = ~.

Since z ~

=

1

^

by ( 6 . 2 0 ) ,

(6.45)

we f i n d t h a t

bTR i s t h e c o m p o s i t e

T(AI...A T) @ R(BI...B p) le

V T(RI@R(B l..[Bp),i T(e#'

• .-,AT@R(BI...B p) . , e#)

T(R(AI®BI,..,AI®Bp),...,R(AT®BI,...,A~B

0 ))

T(R...R)(AI@BI,...,AI@B p, .... AT®BI,...,AT@Bp).

Consider the following five axioms on e:

(6.46)

T(SI...ST)(AII...ATa )@B

=

T(SI(AII...)...S

(A I...))@B

SeT

eT(SI...ST)I

T

T(St(All...)~B . . . . S (Arl...)~e)

ST(esI, ...,esT) • (Sl...S~)(AII~B

.... )

=

T(SI(AII@B, •.. )...ST (A~I@B .... ) ).

366 (6.47)

el:

~(A)®B ~ ~(A@B)

is l: A@B ~ A@B.

(6.48)

T(AI'eTI ' ' A T ) @ I ~

T(AI@I ,...,A @I)

m T(A I...A T ). T(r,r ..... r) a

(6.49)

(T(A I...AT)®B)@C eT@l

T(AI...AT)@(B®C)

1

T(AI@B , . • •)@C

eT

T( (AIeB)OC .... )

T(a,a ..... a) (6.50)

T(AI@(B@C) .... ).

T(A I. ..AT)@S(B I. . .Bq)

T(AI@S(BI...Bo),...~"~ ' )

S(T(A I...AT)@BI,...)

T(es#.. .e~)I

I S(eT.--e T)

T(S(AI@BI, •.. )...S(AT@B I .... ))

S(T(AI®BI,...)...T(AI®B~,...))

T(S...S) (AI®BI,...)

S(T..]T)(AI@B I .... ). zTs(l,l,...,l)

367

Proposition

6.9

Let B be a symmetric monqidal

g-algebra for the pseudo-cOmmutative between natural transformations and natural transformations

e as in (6.41)

of §5.5

(lax case).

by supposing b• or equivalently Proof

We have already

by (6.40) then

satisfyin~

(6.35)-(6.39),

(6.46)-(6.50)

We ~et e from b by (6.40) and b from

A B with all this structure

for the doctrine D

There is a bijec%ion

b as in (6.34) satisfying

where e # is defined by (6.43). e by (6.45).

club 9.

cate~or~ and also a

is precisely

an algebra

We pass to the pseudo case

e, to be an isomorphism.

shown that if we start with b and define e

(6.45) gives us back b.

If on the other hand we start

with e and define b by (6.45)• then

(6.40) gives us back e; just put

S = ~ in (6.50), use

of)

(the conjugate

(6.47), and use

The only other point to be verified b-axioms with the e-axioms. implies its conjugate (6.48);

(6.x) #.

(6.37) identical

Next we get

(6.46)

and using

(6.21);

and using

(6.38).

from

with

is the equivalence

Note that each of the e-axloms First,

then•

(6.50);

(6.35)

and (6.38)

(6.50).

(6.49)

from

given the e-axioms,

identical with

Finally

to get

(6.36).

(6.49) and

(6.49) # .

(6.49) we next get the special case

from this•

from (6.46) # • and from

The reader will have no trouble

(6.36) and

D -algebras.

(6.39).

led

The

(6.46) and (6.46) # , (6.36)'

From this together with (6.36)" where R = ~.

(6.49) # , we get the desired in supplying the details.

The first four of the e-axioms admit conceptual ions, and we are

(6.47).

(6.36) by setting S = R =

As for (6.36) we easily get the special case

where T = R = ~ by using (6.46) and

(6.x)

is identical with

latter is easy using the definition (6.45) of b, using and using

of the

(6.39) by setting S = R I = ~ in the latter

and we get

It remains,

(6.21).

D

interpretat-

to the following definitive description

of

368

Theorem

6.10

doctrine Theorem

Let D be a pseudo-commutative

club and let D

arising as in §5.5 from t h e pseudo distributive 6.8.

category~

together with for each B E B a__nn

enrichment

of the functor

(-@B,eB):

B ~ B, s atisfYin~ axioms to be given below.

abbreviate

the ~ Al...ATJ-component

B eT[ AI...AT] :

-@B:

B ~ B to an op-D-functor

T(AI...AT)@B

symmetry c of B produces

6.52)

(B@-,e#B).

For f:

We agree to

~ T(AI@B,...,AT@B )

o_~f e B t go e T where convenient; and we observe

functor

law of

Then a D -al~ebra is a D - a l g e b r a B that is also a

symmetric monoidal

(6.51)

be the

that conjugation by the

from e B an enrichment

of B@- to an

op-D-

The axioms now are:

B ~ C, the natural

transformation

-@f:

-@B ~ -@C

is op-D-natural.

6.53)

a:

(-@B)@C ~ -®(B®C)

6.54)

r:

-@I ~ i is op-D-natural.

6.55)

e and e # are related b[ the c o ~ u t a t i v i t y it suffices @enerators

to impose ' for T,S belonging of the discrete

The above is for the lax c ase~ op-D-functor

is oR-D-natural.

of (6.50), which to a set Of

c!u b IDI.

for the pseudo case we require the

-@B to be stron$, tha ~ i s ~

e to be an isomorphism.

Proof

By [9]

§10.8 to say that

(-®B,e B) is an op-D-functor

say that the e of (6.51) is natural satisfies [9]).

(6.46) and

The axiom

T and the A .

(6.47)

(corresponding

(6.53) and

to (6.49) and (6.48) respectively. Proposition

in T and the A , and that it to (10.25) and

(6.52) above just makes e natural

The axioms

consequence

in B as well as in of [9],

The result now follows from

of (6.50)

(6.46) and is a

for R,S and for P ,S.

(Qf. [9]§10)

description

using

(6.50) for T,S, where T = R(PI...Pp),

When ~ is presented

by a small number of generators

and

the above theorem gives a very compact

of D -algebras.

6.11

categories,

of

6.9, except for the last clause in (6.55).

(6.19) we see that

Example

(10.26)

(6.54) reduce r by (10.27)

The proof of that last clause is easy:

relations

is to

Let D be itself the club P for symmetric monoidal which is pseudo-commutative

by Example

6.4.

Then a

,

D -algebra B has two symmetric monoidal denoted as above by (B,®,l,a,r,c), (B,$,N,a',r',c'). op-monoidal

be natural

(6.54) for r. three cases

(AI~A2)@B ~ (AI@B)@(A2@B),

eN:

N®B ~ N,

The axiom

for the objects

of 9, subject to the

(6.52) merely asks these components The condition

reduces as usual to two diagrams, Finally

(6.55) requires

(T,S) = (~,~),

the conjugate

e B of -®B to a symmetric

e~:

in B as well as the A s.

op-s.m.-natural

denoted by

we need as usual only to give its components

to the generators

usual three axioms.

the ~'-structure

and the g-structure

To give the enrichment

functor~

corresponding

structures:

(~,N), and

of that for (@,N)).

(6.53) that a be as does similarly

us to impose (N,N)

to

(6.50) in the

(that for (N,@) being

All told we have ten simple axioms

370

involving the

e~ and eN; and

independent

axioms

it is easy

among

the

24 given

Laplaza

[18].

6.7

If we take the d e s c r i p t i o n

6.10,

and m o d i f y

at the

same time

a D-functor

it by c h a n g i n g changing

rather

the a l g e b r a s

for

In the

D'-Alg

but

the sense

its name

to

some new doctrine,

situation

on the

are p r e c i s e l y

situation

of the a r r o w

e in

-@B to

a description

w h i c h we shall

call

got

with D ' - a l g e b r a s

of

D.

D rather

law we had c o n s t r u c t e d

2-category

(6.51),

e enriches

we obtain

have

by

given by T h e o r e m

e, so that

of §5 we should

distributive

for this

of D -algebras

than an op-D-functor,

if from the pseudo not on

to see that these

than D

a doctrine

as objects

and

,

op-D'-morphisms

as 1-cells. Since ~ is invertible this D case , like the D -case; there is n o t h i n g to change except the

exactly

is sense

,

of the

2-cells.

co__reflexion

s:

In p a r t i c u l a r D ~ DD',

Theorem

with h:

5.1 in the

DD' ~

D-case

D, sh : i, ~:

gives hs ~

a

i,

sn = i, n h : i. The point

of our i n t r o d u c i n g

it is of the form general

Do-

for a club

p r i n c i p l e s of[ 9]§i0,

this

D is that,

Cat/~.

D in

is clear

in the lax case, In the light

from T h e o r e m

6.10;

of the for

,

the type

of the natural The

pseudo

D for the pseudo

cases,

type of e in since this F:

D ~

transformation

sometimes (6.51)

arise

case,

in ~.

and the D

from clubs

is in ~, w h i c h

in

for the lax and the

Cat/~,

only happens

is to be so for all T E D,

~ is constant

e lies

namely

when the

when T = i in (6.51);

it only happens

when

at i.

We can bring Example 6.11 above,

the one

studied

by Laplaza,

,

under effect

the

D-setting,

by passing

just by r e p l a c i n g

to the opposite

because B °p is symmetric

monoidal

the m o d e l B by B °p (i.e.

doctrine). when B is.

This

works

in this

in case

371

Example

6.12

Another

D-setting. categories also

being maps

Let D be the club with finite

symmetric

object

important

f~(1)

is a coproduct f :

naturally

6.2,

C o n s id e r

For T E S with

for suitable

occurs

S of Example

coproducts.

monoidal.

T(AI...AT)

example

whose

in the

algebras

are

any S - a l g e b r a

B that~is

rT : T, and for A

E B, the

in B of the A

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

~ ~ T in S whose

types

are the various

i ~ T in S. It follows

S-algebra.

that

there

is a unique

For the n a t u r a l i t y

in T of

e turning e demands

B into a

commutativity

in

eT T(AI@B'''''AT@B)

~ T(AI'''AT)@B

T

F eI

and

e~ is to be I by

~-th component this does

(6.47).

is f (I)@I.

satisfy

So the only possible

On the other

hand

e is that whose

it is easily

seen that

the axioms. ,

We are coproduct, Po-; in

led to the c o n c l u s i o n t h a t

in the c a t e g o r y

or equally

Cat/~,

6.8

that,

of doctrines,

in the category

S is the coproduct

We can apply T h e o r e m D

~

Dopo-,

a reflexion

s:

equivalence

in the pseudo

the d o c t r i n e

of S and

of the doctrines

of clubs

case:

Cat/~

in

is the So- and

and club-maps

P.

5.1 to the present or sA:

So-

D A ~ DoPoA,

or equally

situation, which

to obtain

is an

a coreflexion

,

s : s:

DA ~

*D

DopoA

which,

in the

lax case,

t a k i n g A = I, can be written

~ DoP. I shall

however,

observe

independently

just that

indicate this

briefly

special

of §5, taking T h e o r e m

how s is calculated.

case of T h e o r e m

First,

5.1 can be proved

6.10 as the d e f i n i t i o n

of

372

D -algebra,

and a p p l y i n g

of the present of L a p l a z a ' s

§6 alone.

result;

"distributivity distributive

of Beck.

in (6.51).

The axioms

calculation

these

(6.50)

is in fact

is not

i when e = i; but easily

given by

satisfied

the pseudo

case,

*D-category.

Since

also

and only

s(A)

if s(A) does

expressed

in terms

* e; the functor particular

satisfied;

because

e # given

q with

we have The

and

by

(6.43)

of c in 9opoA

we make

it strict

o

is a D -category;

(resp.

and equally D)-morphism,

A, in the

even for

a we can write

free D * - c a t e g o r y

the usual more objects

T E 9; the m o r p h i s m s

D *A ,

case A commutes

of T h e o r e m

Then

flexible

if

results

of "9 are g e n e r a t e d by a,r,c,

all this

In the other

sq = i, qh = i.

independent

9oP)

and in the pseudo

to T(VI...VT).

a proof

(whose

are trivially

set e = i

so.

e to I.

sends T [ V I . . . V T]

D

"9 ~ 9oP respects

s: ,

sends

data;

of clubs.

and the objects

(h,h,h)

same

and of

We can therefore

for the lax case;

of any diagram

In the lax- D-case

^

At the

is

For T E 9 and for

in view of the value

s is a strict

out of the elementary

suitable

in §6.5.

of T ( A I . . . A T ) @ B

(not trivially

(and in p a r t i c u l a r

hence

at once the image

@,I,

whence DopoA

(6.27)). Thus 9opoA

made

given

to be equal.

(6.46)-(6.49)

"(pseudo)

a free one,

structure

a short proof

between

and

in fact a free one.

an easy

shows

indeed

(6.51)

the techniques

quite

the c o n n e x i o n

sense

category,

using

for instance

sense of

is a 9 - c a t e g o r y ,

T(AI®B,...,AT®B)

gives

in the

category, with the

AI,...,AT, B E 9opoA

2.6 directly,

it conceals

in the

PoA is a s.m.

time 9opoA

This

but

of -~B"

law"

also a s.m.

Theorem

those

structure,

direction

h:

sh = i and q:

by

of 9, and and

in

9op ~ "9 hs ~ i for a

To find q directly,

and to provide , 5.1, we enrich h to an op- 9-functor:

as a 9-functor,

and as an o p - m o n o i d a l - f u n c t o r

o

it has h sense

= i and h formally

is now reversed).

the

same as the b of

The q is then formed

(6.34)

as in T h e o r e m 2.6

373

from the ~ of the resulting

op-*D-functor

(h,~).

In the ease D = P studied by Laplaza, the composite

P ~ PoP with the equivalence

of s:

gives a funetor

it is easy to see that PoP ~ ~o~

rot:

P ~ ~o~ that is precisely his distortion

So as indicated

in §1.4 above,

theorem on page 231 of [19] •

our coreflexion

result

(Note that Laplaza

(|191§2).

implies his

is considering

,

diagrams

writable

in *D - that is, involving

e but not its inverse

-

,

but studying their commutativity be an isomorphism.

in a model

He actually says "monomorphism",

have dualized would be "epimorphism", stands,

and is too rare in examples

Proposition

we observe

that,

i0) follows

all the m [A i...] A I,...,A m

since we

given in his earlier paper

from the same coreflexion theorem, f,g:

are different

and,

if

X ~ Y in

if Y = n[ml[All...],...,mn[Anl...|]

co~ute

are different

then determined

which

to be worth pursuing.)

for a discrete A, two morphisms

po~oA necessarily

e happens to

but this is not enough as it

The other main result of Laplaza, (|18]

in which

where

for each ~, all the

(for the permutations

expressing

f,g are

and equal).

REFERENCES

[z]

M. Barr,

Coequallzers

and free triples,

Math.

Zeit.

116

(1970),

307-322.

[2]

J. Beck,

Distributive

laws, Lecture Notes

in Math.

80 (1969),

119-140. [3]

J. Beck, On coherence fur Math.~

[4]

B.J.

isomorphisms,

E.T.H.

Z~rich,

Day and G.M. Kelly, Enriched Notes in Math.

[5]

E. Dubuc,

[6]

S. Eilenberg

106 (1969),

Free monoids,

For schun~sinst.

(1971). functor categories,

Lecture

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