Basic Concepts of Enriched Category Theory 0521287022, 9780521287029

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LONDON MATHEMATICA-L SOCIETY' LECTURE NOTE SERIES Managing Editor:

Professor I.M.James,

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Basic concepts of enriched category theory, G.M.

KELLY

London Mathematical Society Lecture Note Series.

64

Basic Concepts of Enriched Category Theory

GREGORY MAXWELL KELLY, F.A.A.

Professor of Pure Mathematics, University of Sydney

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON

NEW YORK

MELBOURNE

SYDNEY

NEW ROCHELLE

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 lRP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia ~Cambridge University Press 1982

First published in 1982 Printed in Great Britain at the University Press, Cambridge

Library of Congress catalogue card number: 81-18015

British Library cataloguing in publication data Kelly, G.M.

Basic concepts of enriched category theory.-(London Mathematical Society Lecture Note Series 64 ISSN 0076-0552.) 1. Categories (Mathematics) I Ti t1e. II Series 512'.55 QA169 ISBN 0 521 28702 2

Q_A/~

1

/ /(Lftf

!9tZ

To my wife and my children

TABLE OF CONTENTS page

Introduction

Chapter 1.

9

The elementary notions

1.1

Monoidal categories

1.2

The

2-category

1.3

The

2-functor

1.4

Symmetric monoidal categories; the t€nsor

21

V-CAT

for a monoidal

( ) : V-CAT --+ CAT

symmetric monoidal

V-CAT

for a

V

28

1.5

Closed and biclosed monoidal categories

1.6

V as a

V;

representable

Extraordinary

1.8

The

1.9

The (weak) Yoneda Lemma for

1.10

Representability of

V-functors

35

V-naturality

39

V-naturality of the canonical maps

object as a

Chapter 2.

33

V-category for symmetric monoidal

1.7

1.11

23 26

0

product and duality on

closed

V

43

V-CAT

45

V-functors; the representing

V-functor of the passive variables

Adjunctions and equivalences in

47 49

V-CAT

Functor categories V

54

2.1

Ends in

2.2

The functor-category

CA,BJ

2.3

The isomorphism

B,C]

[A®

for small ~

[A,[B,CJ]

A

58

61

page

2.4

The (strong) Yoneda Lemma for Yoneda embedding

A--+

2.5

The free

2.6

Universe-enlargement cases;

Chapter 3. 3.1

as a

the

0

[A P,VJ

V-category on a

[A,BJ

V-CAT;

65

Set-category

V c V'

68

in concrete

V'-category for large

A

69

Indexed limits and colimits Indexing types; limits and colimits; Yoneda isomorphisms

71

3.2

Preservation of limits and colimits

74

3.3

Limits in functor categories; double limits and iterated limits

3.4

The connexion with classical conical limits when

3.5

76

V

=

Set

81

Full subcategories and limits; the closure of a full subcategory under a class of colimits

85

3.6

Strongly generating functors

87

3.7

Tensor and cotensor products

91

3.8

Conical limits in a

94

3.9

The inadequacy of conical limits

3.10

Ends and coends in a general

V-category

V-category;

completeness 3.11

99

The existence of a limit-preserving universe-enlargement

3.12

96

V c V'

104

The existence of a limit- and colimit-preserving universe-enlargement

V c V'

107

page

Chapter 4. 4.1

Kan extensions The definition of Kan extensions; their expressibility by limits and colimits

112

4.2

Elementary properties and examples

116

4.3

A universal property of

LanKG;

its

inadequacy as a definition 4.4

Iterated Kan extensions; Kan adjoints;

[A 4.5

120

0

P,V]

as the free cocompletion of a small

V;

initial and final functors when

V = Set

127

Filtered categories when commutativity in

Set

V =Set;

the

of filtered colimits 131

with finite limits 4.7

125

Initial diagrams as the left Kan extensions into

4.6

A

The factorization of a functor, when

V

Set,

into an initial functor and a discrete 136

op-fibration 4.8

The general representability and adjoint-functor theorems; the special case of a

140

complete domain-category 4.9

Representability and adjoint-functor theorems when

4.10

145

V = Set

Existence and characterization of the left Kan extension along a fully-faithful K:

A --+

C in terms of cylinders in

to colimits by the

C(KA,-)

C sent

150

page

Chapter 5. 5.1

Density Definition of density, and equivalent 156

formulations 5.2

Composability and cancellability properties of dense functors

5.3

159

Examples of density; comparison with strong generation

5.4

Presentations of the density of a fully-faithful K

5.5

164

in terms of

K-absolute colirnits

170

The characterization of functor-categories 0

[A P,VJ;

small projectives and Cauchy

completion; Morita equivalence 5.6

174

Existence and characterization of the left Kan extension along a fully-faithful dense functor in terms of a density presentation; adjoint-functor theorems involving density

5.7

The free completion of

A

with respect to

all colimits with indexing-type in 5.8

178

F

182

Cauchy completion as the idempotent-splitting completion when

V

Set

5.9

The finite-colimit completion when

5. 10

The filtered-colimit completion when flat and left-exact functors

186

v

Set V

188

= Set; 190

page 5.11

The image under general dense

[K,l]: [C,B]--+ [A,BJ, K,

for a

of the left-adjoint functors;

examples and counter-examples 5.12

193

Description of the image above in terms of K-comodels; generalization to an equivalence Lanz

5.13

A

--1

dense

[Z,l]

between categories of comodels

K: A--+ C,

with

A small and

197

C

cocomplete, as an essentially-algebraic theory with 5.14

C as the category of algebras

The image under a dense

K,

[K,l]: [C,B]--+ [A,BJ,

for

of the equivalences;

characterization theorems for

Chapter 6.

202

C

203

Essentially-algebraic theories defined by reguli and sketches

6.1

Locally-bounded categories; the local boundedness of

6.2

V

The reflectivity in

in all our examples 0

[A P,VJ

of the category

of algebras for a regulus 6.3

213

The category of algebras for a sketch, and in particular for an

6.4

W8

F-theory; algebraic functors

and their adjoints

218

The

224

F-theory generated by a small sketch

page 6.5

The symmetric monoidal closed structure on the category of

F-cocornplete categories and

F-cocontinuous functors

228

Bib 1 i og ra phy

231

Index

239

Introduction (i)

Although numerous contributions from divers authors, over the

past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it.

As the applications of the

theory continue to expand - some recent examples are given below the lack of such an account is the more acutely felt. The present book is designed to supply the want in part, by giving a fairly complete treatment of the limited area to which the title refers.

The

bcs~c

concepts of category theory certainly include

the notions of functor-category, of limit and colimit, of Kan extension, and of density; with their applications to completions, perhaps including those relative completions given by categories of algebras for limit-defined theories.

If we read "V-category" for "category"

here, this is essentially the list of our chapter-headings below, after the first chapter introducing

V-categories.

In fact our scope is wider than this might suggest; for what we give is also a self-contained account of basic category theory as described above, assuming as prior knowledge only the most elementary categorical concepts, and treating the ordinary and enriched cases together from Chapter 3 on.

(ii)

In order to include the enriched case we begin in Chapter 1 by

introducing symmetric monoidal closed categories elementary properties, and defining the

V,

2-category

examining Lheir

V-CAT

of

Introduction

10

V-categories,

V-functors, and

with the forgetful easy and brief.

2-functor

V-natural transformations, together ( ) : V-CAT---+ CAT;

this much is

0

Next, developing the basic structure of

tensor products of

V-categories,

V-CAT

V-functors of two variables,

V itself as a

extraordinary

V-natural transformations,

representable

V-functors, and the Yoneda lemma - requires verifications

of diagram-commutativity, whose analogues for trivial equations between functions.

V

V-category,

= Set reduce to fairly

This seems to be an inevitable

cost of the extra generality; but we have been at pains so to arrange the account that the reader should find the burden a light one. With this done, the discussion of representability, adjunction, and equivalences in

V-CAT,

which closes Chapter 1, is simple and direct.

The short Chapter 2 takes up the closed structure of given by the

~B,CJ.

V-functor-category

is to be an object of

Since the hom

V-CAT, [B,C](T,S)

V and not merely a set, more work is once

again required than in the

V

B is quite a simple limit in

= Set case. V

0

-

However this hom for small

now supposed complete - of the

special kind called an end, precisely adapted to the extraordinary V-natural transformations; and from this definition flow easily the desired properties. The indexed limits and colimits of Chapter 3, along with their various special cases, are constructs of the greatest importance even when

V = Set;

and the relations between double and iterated indexed

limits express a rich content. by their reducibility when

This impor_tance is scarcely lessened

V • Set to classical limits - any more

than the importance of these classical ones is lessened by their reducibility in good cases to products and equalizers.

Even in the

Introduction

11

V = Ab/ of additive categories, the indexed limits - although

case

here they e~st when all the classical ones do - are no longer

direatZy reducible to the latter; while for a general

V the indexed

limits are essential, and the classical ones no longer suffice.

V into a bigger universe,

Chapter 3 ends by showing how to expand

without disturbing limits and colimits, so as to allow the free use of functor-categories

[B,CJ with large B.

The remaining chapters 4, 5, and 6, dealing respectively with Kan extensions, density, and algebras defined by limits (or more generally by "topologies"), make use of these limit- and colimit-notions to complete the development of our chosen area of basic category theory. Most of the results apply equally to categories and to

V-categories,

without a word's being changed in the statement or the proof; so that scarcely a word would be saved if we restricted ourselves to ordinary categories alone. of a general

Certainly this requires proofs adapted to the case

V·, but these almost always turn out to be the best

proofs in the classical case

V = Set as well.

This reflects the hopes

with which Ellenberg and the author set out when writing Lawvere's observation in logic "applies directly [symmetric monoidal]

[54]

to structures valued in an arbitrary

closed category

certain results peculiar to the case

V "

Set,

V = Set;

there are of course and we accordingly

an occasional section to this case alone.

the commutativity in

and

that the relevant segment of classical

Because of the special properties of

devote

[26],

Some examples are

Set of filtered colimits with finite limits;

the notions of initial functor and of discrete op-fibration; and the

Introduction

12 classical adjoint-functor theorems.

We treat all of these for

completeness - partly to keep the account self-contained, and partly to compare them, where appropriate, with analogues for a general

V.

The little prior knowledge that we do assume is easily available for instance from

[60] - and to have included it here, with examples

to enliven it, would have involved an unjustifiable length.

increase in the

For Chapter 1, it consists of the basic notions of category,

functor, and natural transformation, including functors of two or more variables and contravariant functors; representable functors and the Yoneda lemma; adjunction and equivalence; and what is meant by a faithful or a fully-faithful functor.

From Chapter 2 on, we also

need the classical notions of limit and colimit; with the names of such special limits as product, equalizer, pullback, and terminal object; and the meanings of monomorphism and epimorphism.

In the

rare places where more is assumed but not expounded here, references are given.

(iii)

We now turn to what we have omitted from the present book; the

list includes many important notions, well meriting an extended treatment, whose inclusion would

howe~er

have disturbed the essential

simplicity of an initial account such as this. First (to return to the title), the basic concepts of category theory concern categories- in our case, objects of discussion. but

.V-categories -as the

These form, or live in, a

2-category

V-CAT;

2-categories are not yet the formal objects of discussion, any

Introduction

13

more than categories are when we study group theory. casts light on group theory, as does theory.

Category theory

2-category theory on category

Hence the step to this next level, where weaker notions of

natural transformation naturally arise, is an important one; but it is quite properly deferred pending experience of some particular 2-categories, and we do not take it here. aspects of it in the forthcoming

We make a start on some

[46].

Closely connected to this is our decision not to discuss the "change of base-category" given by a symmetric

V

---+-

V1

and the induced

V

induced by the canonical

V-CAT ---+- V 1 -CAT.

2-functor

we said, consider the forgetful --+

2-functor

Set;

monoidal' functor

( ) : V-CAT 0

We do, as

--+

CAT

but this is entirely elementary,

not involving even the definition of a symmetric monoidal functor. The general change of base, important though it is, is yet logically secondary to the basic

V-category theory it acts on.

properly needs a careful analysis of the

To treat it

2-category of symmetric

monoidal categories, symmetric monoidal functors, and symmetric monoidal natural transformations - including adjunctions therein and the dual concept of op-monoidal functor.

There is evidence in

[43]

that this itself is best studied in the more general context of categories with essentially-algebraic structure, which draws on the matter of this book together with

[45]

and

[46].

Since the natural setting for the important work of Day [14],

[16])

([12],

on the construction of symmetric monoidal closed

categories as functor-categories, or as reflective subcategories of these, involves the

2-category of symmetric monoidal categories,

this too has been omitted.

Introduction

14

One thing we particularly regret omitting is the theory of monads; it could certainly be seen as basic category theory, yet there was no convenient place to put it, and it would have required an extra chapter to itself.

Luckily, besides the account of Mac Lane

in the classical case, we have the articles of Linton [22],

and Kock

[51],

We also have the elegant

[57],

2-categorical treatment by Street

have been more closely studied.

[70], 2-categories

A consequence is our failure to

discuss the completeness and the cocompleteness of the of small

Dubuc

covering various aspects of the enriched case.

which provides some argument for deferring the topic until

V-Cat

[60]

2-category

V-categories - which is most easily referred back to

the completeness and the cocompleteness of the algebras for a monad. Finally, our account covers only what can be said for every well-behaved

V (except for those things special to V =Set).

Results valid only for a special class of

V are best treated in

separate articles, of which the author's forthcoming

[45]

is one

example.

(iv)

Our concern being to provide a development of basic category

theory that covers the enriched case, we have given illustrations of many of the results in isolation, but have not thought this the place to discuss detailed applications to particular areas of mathematics. In fact, applications needing the enriched case are rapidly proliferating; the following is a selection of recent ones that have come to the author's notice.

Introduction

15

The discussion of dualities in topological algebra, taking for

V

a suitable category of topological or quasitopological spaces, and

initiated by Dubuc and Porta

[23],

articles by Dubuc, Day, and Nel.

V

Kan extensions where

has been continued in numerous

Pelletier

is Banach spaces.

[67]

has made use of

Recent Soviet work in

representation theory (see the articles by Kleiner and Roiter, and by Golovashchuk, Ovsienko, and Roiter, in graded modules for

V.

[64])

has differential

The present author, as part of a study of

essentially-algebraic structures borne by categories, has extended in

[45]

the classical results on "cartesian theories" (finite-limit

theories) from the case where

V is sets to those where it is groupoids

or categories. The study of homotopy limits by Gray

[32]

takes him into the

area of change of base-categories, involving the connexions between the closed categories of categories and of simplicial sets.

The work

of May on infinite loop spaces, involving from the beginning the base category him

[62]

[61]

V of compactly-generated spaces, has of late led

to very general considerations on categories

V with two

symmetric monoidal structures and a distributive law between them. Mitchell

V

[63]

has related monoidal structures on

to low-dimensional cohomology groups of the group

[G,VJ G.

for suitable These latter

applications go beyond the basic theory presented here, but of course presuppose the relevant parts of it, Recent work of Walters theory, but with

([75],

[76])

does concern the basic

V generalized to a symmetric monoidal closed

bicategory; in this context he exhibits the sheaves on a site as the

In traduction

16 symmetric Cauchy-complete

V-categories.

It is plain that continued

expansion of the applications is to be expected.

(v)

Writing in text-book style, we have not interrupted the development

to assign credit for individual results.

To do so with precision and

justice would in any case be a daunting task; much has been written, and many insights were arrived at independently by several authors. Such references as do occur in the text are rather intended to show where more detail may be found.

What we can do, though, is to list

here some of the works to which we are particularly indebted. Perhaps the first to advocate in print the study of enriched categories was Mac Lane

[59];

although published in 1965, this

represents basically his Colloquium Lectures at Boulder in 1963. There were some early attempts by Linton make a start;

Benabou

([4],

[5])

[56]

and Kelly

[40]

went further; while Eilenberg

and Kelly wrote a voluminous article

[26]

covering only the

"elementary" notions, but including change of base-category. present Chapter 1 draws mainly on

to

[26]

Our

and on the author's article

[ 42].

The principal source for Chapter 2 is Day and Kelly see also Bunge

[ 10].

[20];

but

The ideas in Chapter 3 of co tensor products,

of ends, and of limits in a

V-category, go back to

[42]

and

[20];

and the concept of their pointwise existence in a functor-category to Dubuc

[22].

However the general notion of indexed limit, which

makes Chapter 3 possible in its present form, was discovered independently by Street

[72],

Auderset

[1],

and Borceux and

In traduction Kelly

[9].

articles

17 The last two sections of Chapter 3 call on Day's

[ 121

and

[ 14].

Chapter 4, on Kan extensions, is certainly indebted to the three articles

[20],

Dubuc in

[ 22]

[22],

[9]

already mentioned; in particular it was

who pointed out the importance of the "pointwise"

existence of Kan extensions, which we make part of our definition. The chapter also contains many more-or-less classical results for

V =Set;

all but the best known are credited in the text to their

authors. It is especially in Chapter 5, on density, that we have been heavily influenced by writers who themselves dealt only with namely Gabriel and Ulmer

[31]

and Diers

presentation" we have taken from Day meaning somewhat.

[21].

[16],

V = Set;

The term "density

although modifying its

Again, writers on particular aspects are given

credit in the text. Chapter 6 shows its debt Gabriel and Ulmer.

to Ehresmann

[24],

as well as to

The form of the transfinite construction used

here to prove the reflectivity of the algebras, although it is taken from the author's article

[44],

ultimately depends on an idea from

an unpublished manuscript of Barr (reference

[3]

of

[44]),

which

makes cowellpoweredness an unnecessary hypothesis, and so enables us to include the important example of quasitopological spaces. The author's original contributions to the book are perhaps most visible in the arrangement of topics and in the construction of proofs that apply equally to the classical and enriched cases.

For instance,

Introduction

18

many readers will find the way of introducing Kan extensions quite novel.

Beyond

thi~

the work in 5.11- 5.13 on Kan extensions along

a non-fully-faithful dense functor seems to be quite new even when

V = Set,

as is its application in 6.4; while the whole of Chapter 6

is new in the enriched setting.

(vi)

In the early chapters the formal definition-theorem-proof style

seemed out of place, and apt to lengthen an essentially simple account. Deciding to avoid it, we have accordingly made reference to a result either by quoting the section in which it occurs, or (more commonly) by quoting the number of the displayed formula most closely associated with it.

In the later chapters, now more concerned with applying the

logic than developing it, it seemed best to return to formally-numbered propositions and theorems. The end-result is as follows. refers to section 6 of Chapter 5.

A symbol 5.6 without parentheses All other references occur in a

single series, consisting of displayed formulae and proposition- or theorem-numbers, with the first digit denoting the chapter.

Thus

in Chapter 5, formula (5.51) is followed by Theorem 5.52, itself followed by formula (5.53), which in fact occurs in the statement of the theorem.

To assist cross-referencing, each left-hand page bears at its head the number of the last section to which that page belongs, and each right-hand page bears the numbers of the formulae, propositions, and theorems that occur on the two-page spread (in parentheses, even if they are theorem-numbers).

Introduction (vii)

19

It remains to thank those who have made the book possible.

Over the last twelve months I have had the happy opportunity to present early drafts of the material to acute and stimulating audiences at the Universities of Sydney, Trieste, and Hagen; their encouragement has been of inestimable value.

Equally important has been the support

of my colleagues in the Sydney Category Theory Seminar - including the visiting member

Andr~

Joyal, whose presence was made possible by

the Australian Research Grants Committee.

I am indebted to Cambridge

University Press, for suggesting this Lecture Notes Series as suited to such a book, and for their careful help with the technical details of publication.

It was they who observed that the original typed

manuscript was already suitable as camera-ready copy; I regard this as a compliment to the intelligence and taste of my secretary Helen Rubin, whose first mathematical job this was, and I thank her very sincerely.

Finally I thank my wife and children, who have

borne several months of shameful neglect.

Max Kelly, Sydney, January 1981.

(1.1)-(1.2)

1

The

1.1

21

elementary notions V = (V0 ,0,I,a,~,r)

A monoidaZ category

a functor

0•

isomorphisms

V0 x V0 aXYZ:

--+

(X

V0 Y)

0

an object

,

0

Z--+ X

0

consists of a category

,

0

I (Y

V

V

of

0

Z),

0

,

and natural

tX: I

0

X--+ X,

subject to two coherence axioms expressing the

rX: X ® I --+ X,

commutativity of the following diagrams: ((W®X)®Y)®Z ---~a'---- (W®X)®(Y®Z) _ _ _.:::a_ _--r W®(X®(Y®Z))

r

a®l

ll®a

(W®(X®Y))®Z - - - - - - - - - a - - - - - - - - - + W®((X®Y)®Z), (1.1)

(X®l)®Y

a

X®(I®Y)

~~ X®Y

I t then follows (see

[58]

and

.

( 1. 2)

[ 39 J) that every diagram of natural

transformations commutes, each arrow of which is obtained by repeatedly applying the functor

® to instances of

here an "instance" of

of the functors

0

and

their inverses, and

is a natural transformation, such as

in the diagrams above, formed from

or

see

a

a,~,r,

I

to its variables.

a

1•

w®X,Y,Z

by repeated application

For a precise formulation,

[58]. A special kind of example, called a cartesian monoidal category, is

given by taking for

V0

'

~-

any category with finite products, taking for

1.1

22 0

and

a,~,r

the product

I

and the terminal object

X

the canonical isomorphisms.

Set,

are the categories

Cat,

and taking for

1,

Important particular cases of this

Gpd,

Ord,

Top,

CGTop,

HCGTop,

QTop,

Shv s, of (small) sets, categories, groupoids, ordered sets, topological spaces, compactly generated topological spaces, hausdorff such, quasi-topological spaces

[69],

and sheaves for a site

S;

here "small"

is in reference to some chosen universe, which we suppose given once for all.

Other cartesian examples are obtained by t&king for

ordered set with finite intersections, such as the ordinal

V

0

an

2 = {0,1}.

All of these examples are symmetric in the sense of 1.4 below, and all the named ones except

Top

are closed in the sense of 1.5 below.

A collection of non-cartesian (symmetric, closed) examples is given

Ab,

by the categories groups,

G-R-~lod,

DG- R-Mod

R-modules for a commutative ring

differential graded ®j

R-Mod,

the category

R,

of (small) abelian

graded

R-modules, each with its usual tensor product as

Ban of Banach spaces and linear maps of

with the projective tensor product;

the category

CGTop*

compactly-generated spaces with the smash product for ordered set

R+

R-modules, and

0'

'

norm~

1,

of pointed and the

of extended non-negative reals, with the reverse of

the usual order, and with

+ as

®.

A non-symmetric example is the category of bimodules over a non-commutative ring

R,

with

®,

Another is the category of

endofunctors of a small category, with composition as a,~,r

0;

here

are identities, so that the monoidal category is called strict.

(1. 3)

23

the tensor produat, and

~

In general we call

I

We suppose henceforth given a particular monoidal ~s

locally small (has small hom-sets).

V0 (I,-): V0

functor

Set,

Ord,

Top,

--+

Ab,

CGTop*'

X;

(Set,

Ab,

reflecting).

Gpd,

Ban,

VX

is the

Ban) it is even conservative (• isomorphism-

Yet

is not faithful in general; in the cases

V

Cat and

VX is the set of objects of X, and in the case DG-R-Mod,

is the set of

convenient to call an element V0

an element

)

1.2 A V-category A(A,B)

E

V0

VX

0-cycles.

In spite of the failure of

in

In such cases as

V is faithful, while in some of them

in these cases

R-Mod,

V0

it is (to within isomorphism)

the ordinary "underlying-set" functor; in the case of unit ball of

V such that

We then have the representable

Set, which we denote by V.

R-Mod,

the unit object.

f

V

f

of

to be faithful in general, it is VX

A consists of a set

identity element

®

f: I --+ X

of X.

for each pair of objects of

M = MABC: A(B,C)

(that is, a map

A(A,B)

--+

A(A,C)

jA: I--+ A(A,A)

obA

of objects, a hom-object

A, a composition law

for each triple of objects, and an for each object; subject to the

associativity and unit axioms expressed by the commutativity of (A(C,D)~(B,C))®A(A,B)

----=a=-----+ A( C,D) 0(A(B ,C) ~A(A,B))

j

M01

A (B,D)o.I(A,B)

~

!OM

/o.I(A,C)


avariant functor from

B·• while a functor T: A 0 B --+ C may be thought of as a

to

funato1' of two variables. T(A,-): B

E

Such a

B--+ C for each A

B. Here T(A,-)

E

T

A and

gives rise to partial. funato:t>s T(-,B): A--+ C for each

is the composite

B- I

08---:~A0B----+

A®1

T

fromwhich can be read off the value of

c,

( 1. 20)

T(A,-)BB''

Suppose, conversely, that we are given a family of functors T(A,-):

B--+ C indexed by obA and a family of functors

T(-,B): A--+ C indexed by T(A,-)B T: A 0

= T(-,B)A, =

T(A,B)

obB,

such that on objects we have

say,

Then there is a functor

B --+ C of which these are the partial functors if and only if

we have commutativity in

1.4

32

A(A,A')®B(B,B')

T(-,B')®T(A,-)

C(T(A,B'),T(A',B'))®C(T(A,B),T(A,B'))

1M C(T(A,B) ,T(A' ,B'))

c

lM 8(B,B )®A(A,A') 1

I

, -)

®T (- 'B)

T is then unique, and

Moreover (1.21). ( [ 26]

T(A

C(T(A',B) ,T(A' ,B'))®C(T(A,B) ,T(A' ,B)).

T(A,B),(A',B')

is the diagonal of ca~

The verification is easy, and the details Ch. II I,

§ 4)

T,S: A® 6--+ C,

constitutes a

a family

B,

each fixed

a

A, a

T --+ S

V-natural

in

C

0

if and only if it

T(A,-)

--+

T(-,B)--+ S(-,B).

V-natural

V-naturaZity may be ver>ified

C) to verify that,

aAB: T(A,B)--+ S(A,B)

V-natural transformation

constitutes, for each fixed for

be found in

.

It is further easy (using (1.21), and (1.3) for if

( 1. 21)

S(A,-),

and,

In other words,

variab~e-by-var>iab~e.

In relation to the underlying ordinary categories, we have of

(A 0 p)

course not

A0

A

o

X

0

B • o'

x B0 --+

(A

= (A ) 0 p

and

0

(T0 p)

=

0

(T ) 0 P. 0

However

(A

8

B)

is

0

rather there is an evident canonical functor 8

B) 0 ,

and a similar one

1 --+

I0 .

For

T: A

8

B

--+ C,

the partial functors of the composite ordinary functor

A

o-

are precisely

®

T(A,-)

0

6

(A

0

and

T(-,B)

0

8

6)

0

•

We could discuss, in the present setting, the extraordinary

V-naturaZ families of 1.7 below.

However, the techniques that make

( 1. 22)

(1.21)-(1.26)

33

this simple arise more naturally, and at a higher level, when closed; which is the case of real interest. [17]

V is

Moreover it is shown in

that any symmetric monoidal category can, by passing to a higher

universe, be embedded in a closed one; see also 2.6 and 3.11 below.

1.5

The monoidal category

V (symmetric or not)

is said to be closed

(cartesian closed, when V is cartesian monoidal) if each functor -®Y: V

0

--+

V

0

has a right adjoint

[Y,-J;

so that we have an

adjunction (1.23) with unit and counit (the latter called evaluation) say d: X--+ [Y,X ® Y], Putting

X= I

in (1.23), using the isomorphism

recalling from 1.1 that

V = V (I,-): V --+Set, 0

(1.24)

e: [Y,Z] ® Y--+ Z •

0

~:I®

Y ~ Y,

and

we get a natural

isomorphism (1.25)

since

[Y,Z]

V (Y,Z), 0

is thus exhibited as a lifting through

it is called the internal hom of

Putting

Y • I

y

and

V of the hom-set

z.

in (1.23) and using the isomorphism

r: X® I~ X,

we deduce a natural isomorphism (1.26)

i: Z - [I,Z]

it is easy to verify that of (1.25).

Replacing

Vi: VZ- V[I,Z]

X by

W ®X

is just the case

in (1.23), and using the

Y

I

1.5

34

a: (W ®X)® Y

isomorphism

~

W 0 (X® Y), we deduce a natural

isomorphism (1.27)

p: [X® Y,ZJ- lX,:Y,ZIJ

and it is easy to verify that

Vp

modulo the isomorphisms (1.25). and convenient to replace

agrees with the

n

of (1.23)

In many concrete cases it is possible

V by an isomorph in such a way that (1.25)

becomes an equality; this should be clear from the examples of closed categories given in 1.1, and we do not labour the point. The monoidal category every

-®

Y have a right adjoint

right adjoint with

V is said to be bialosed if, ·'not only does

rrx,-].

[X,-] = [X,-].

When

v

but also every

X®-

bialosed biaategory of left-R-,

has a

is symmetric, it is biclosed if closed,

The non-symmetric monoidal category of

mentioned in 1.1, is biclosed.

S

[Y,-],

R-bimodules,

It is part of a larger structure, the right-S-, bimodules,

where

R

and

vary over all rings; and it seems to be typical that non-symmetric

biclosed monoidal categories occur in nature as a small part of such a bicategory. see of

[6 ],

(The notion of bicategory, for an exposition of which

generalizes that of

2-category, the strict associativity

1-cells being replaced by associativity to within coherent

isomorphisms; a one-object bicategory is a monoidal category, just as a one-object

2-cat~gory

is a strict monoidal category.)

of R.F.C. Walters ([75], [76]) "V-categories

A" where

Recent work

suggests the importance of studying

V is a biclosed- bicategory with an appropriate

kind of symmetry; but that goes beyond the scope of the present account.

(1.27)

35

There also do occur in nature important closed monoidal categories that are not biclosed.

A typical example is given by the strict

monoidal category of small endofunctol'o (in some suitable sense of "small") of a well-behaved large category such as For such a monoidal closed

V,

Set;

see

[45].

we can develop part of the theory of

V-categories; we have the results of 1.2 and 1.3, but not those of 1.4; and we have the Yoneda Lemma of 1.9 below, but not its extra-variable form.

However, we do not pursue this at all, for in practice it seems

that the interest in such a

V lies in V itself, not in V-categories -

which become interesting chiefly when

V is symmetric monoidal

closed. We end this section with two further comments on examples. the symmetric (cartesian) monoidal

Top

is not closed:

-xY

First,

cannot

have a right adjoint since it does not preserve regular epimorphisms; see

[19].

Next, to the examples of symmetric monoidal closed category

in 1.1, we add one more class.

An ordered set

V with finite inter-

sections and finite unions is called a Heyting algebra if it is cartesian closed; a boolean algebra is a special case.

1.6 From now on we suppose that our given V is symmetric monoidal closed, with

V

0

locally small.

The structure of

V-CAT then becomes

rich enought to permit of Yoneda-lemma arguments formally identical with those in

CAT.

Before giving the Yoneda Lemma and its extensions

in 1.9, we collect in this section and the next two some necessary preliminaries: less so here.

results many of which are almost trivial in

CAT,

but

1.6

36

The proof of each assertion of these sections is the verification of the commutativity of a more-or-less large diagram.

This verification

can be done wholesale, in that each diagram involved is trivially checked to be of the type proved commutative in the coherence theorems of

[47]

and

[48].

Yet direct verifications, although somewhat

tedious, are nevertheless fairly straightforward if the order below is followed.

V "makes V itself

The first point is that the internal-hom of into a call is

V-category". V,

[X,Y].

More precisely, there is a

whose objects are those of Its composition-law

V0 ,

M: [Y,Z]

V-category, which we

and whose hom-object [X,Y]--+ [X,Z]

0

V(X,Y)

corresponds

under the adjunction (1.23) to the composite ([Y,Z] ® [X,Y]) ®X~ [Y,Z] ® ([X,Y] ®X)~ [Y,Z] and its identity element ~:

I

0

X--+ X.

jX: I--+ [X,X]

0

Y ~ Z,

corresponds under (1.23) to

Verification of the axioms (1.3) and (1.4) is easy

when we recall that, because of the relation of the definition (1.28) of

M is equivalent to

e e(M

w of (1.23),

to the 0

1)

=

e(1

0

e)a.

It is further easily verified that (1.25) gives an isomorphism between

V0

and the underlying ordinary category of the

V-category

V; we

henceforth identify these two ordinary categories by this isomorphism, thus rendering the notation

V

0

consistent with the notation

A

0

of 1.3.

Next we observe that, for each A E A, B

E

we have the ~ep~esentabZe

A to

A(A,B)

(1.28)

E

V,

and with

V-category V-functor

A and each object

A(A,-): A--+ V, sending

( 1. 2 8) - ( l. 32 )

37

A(A,-)BC: A(B ,C) -

[A(A,B) ,A(A,C) I

corresponding under the adjunction (1.23) to Axioms (1.5) and (1.6) for a

A(A,C).

( 1. 29)

M: A(B,C)

®

A(A,B)

--+

V-functor reduce easily to (1.3)

and half of (1.4). Replacing

Aop

A by

funotor A(-,B): Aop

--+

gives the contravariant representable

v.

The families 0

partial functors of a functor A(A,B);

A(A,-)· and

HomA: A p

®

A --+

A(-,B)

are the

V sending (A, B)

to

for the condition ( 1. 21) again reduces easily to (1.3) for

homA: Ao op

We write

x

Ao

--+

v0

for the ordinary functor

v

A op x A 0

it sends

0

(A,B)

to

( 1. 30)

0

A(A,B),

A(f,g): A(A,B)--+ A(A',B'),

A.

and for its value on maps we write where

By 1.4, the partial functors of

f: A'--+ A and

homA

g: B--+ B'.

are the functors underlying

A(A,-)

and A(-,B), so that our writing their values on morphisms as

A(A,g)

and

A(f,B) is consistent with (1.12).

the definition (1.29) of A(A,C)

we see that

A(A,g): A(A,B)--+

is the composite

A(A,B) ~I while

A(A,-)BC'

Combining (1.11) with

®

A(A,B) ~ A(B,C)

A(f,B): A(D,B)--+ A(A,B)

A(D,B)

-;:y+

A(D,B)

® I ~

®

A(A,B) ~ A(A,C)

( 1. 31)

®

A(A,D) ~ A(A,B)

( 1. 32)

is A(D,B)

From these it follows easily that we have commutativity in

1.6

38

A op

x

When

A = V,

[ , l : V op

A

~.'""'

0

we see at once that x

0

V

homV

v 'set

( 1. 33)

V •

--+

0

0

to

(X,Y)

0

is just the functor

Now we observe that there is also a sending

v

Ten(X,Y)

m

and with

X® Y,

[X,X'] ® [Y,Y']--+ [X® Y,X' ® Y']

V-functor

Ten: V

@

V --+ V,

Ten(X,Y) ,(X' ,Y'):

corresponding under the adjunction

(1.23) to the composite ([X,X']®[Y,Y'])®(X®Y)

([X,X']®X)®([Y,Y']®Y) ----+ X'®Y'

m

when we observe that (1.34) is equivalent to

e(Ten ® 1)

(e ® e)m,

V-functor axioms (1.5) and (1.6) is easy.

verification of the

(1. 34)

The

ordinary functor

V0

X

V0

is at once seen to be

(V

V)

--,T=-e-n--+ Vo

0

®: V o

X

underlying ordinary functor of For any

®

V-categories

A

V --+ V ' o

o'

Ten(X,-) and

( 1. 35)

0

so that, by 1.4, the is

X®-,

B we have a commutative diagram

v Ten

( 1. 36)

(1. 33)- ( 1. 39)

39

in terms of the partial functors, this asserts the commutativity of A®B _ _ _(;,;_A:. : .®: . :B)~(~(A:.:.!..:,B:.!_)..!. .,-_!.):____-+

v

A(A,-)®:S(B,( 1. 37)

which is easily checked.

At the level of the underlying functors

hom,

this gives (A® B)((f,g),(f',g'))

1.7

A(f,f') ® B(g,g').

The point to be made in the next section is that all the families

V,

of maps canonically associated to a

( 1. 38)

V-functor

T,

or to a

a,

such as

or

M: A(B,C)

or

e: [Y,Z] ® Y--+ Z,

or

--+

B(TA,TB),

V-natural in every variable. case of

M,

V-category

V-natural

X® (Y ® Z), T: A(A,B)

or to a

or

a:

I--+

B(TA,SA),

A, or to

a: (X® Y) ®

®

Z--+

A(A,B) --+ A(A,C),

are themselves

To deal with a variable like

B

in the

we must now introduce the notion of extraordinary

V-naturatity; which later plays an essential role in the definition of V-functor-category. First we observe that the formulae (1.31) and (1.32) allow us to write the "ordinary"

V-naturality condition (1.7) in the more compact

form A(A,B)

----=1'---->- B(TA, TB)

s B(SA,SB) ----=~--:-:---+

B(aA,l)

B( TA, SB) •

( 1. 39)

1.7

40

By (extraordinary)

V-naturaZity for an

B,

in

where

K

E

B

obA - indexed and

family of maps

0

T: A p ® A

--+

B,

we

0

mean the commutativity of each diagram T(A -)

A(A,B)

E(T(A,A),T(A,B))

T(-,B)j

B(I3A,1)

E(T(B,B),T(A,B))

E(13B, 1)

---+

Duality gives the notion, for the same family

yA: T(A,A)

--+

K;

T

( 1. 40)

E(K,T(A,B))

and

K,

of a

V-natural

namely the commutativity of

A(A,B) _ _ _....::.T~(B::..L--!-)- 4 E(T(B,A) ,T(B,B))

T(-,A)j + ----.---~-..,----4

E(T(B,A),T(A,A))

Such extraordinary

V

= Set;

of the

B(l ,y A)

B(T(B ,A) ,K)

(1.41)

V-naturality is of course fully alive in the case

it is exhibited there, for instance, by the "naturality in d

and the

e

of (1.24).

V • Set,

In this case

Y"

(1.40) and

(1.41) have more elementary forms obtained by evaluating them at an arbitrary

f

extraordinary

E

A(A,B).

Clearly extraordinary

Set-naturality:

V-naturality implies

we have only to apply

V

to (1.40)

and ( l. 41) . Just as for ordinary SA: K--+ T(A,A)

Q: B--+ C, a

is

V-natural as above, and if

then the maps

V-natural family

V-naturality, it is clear that if

QSP.

P:

V--+

QI3PD(=Q 0 SPD): QK---+ QT(PD,PD)

A and constitute

(1.40)-(1.42) If is

41 0

T: (A ® V) P ® (A ® V) (A,D)

V-natural in

each

fixed

D and in

D for each fixed and

BAD: K ~ T(A,D,A,D)

a family

if and only if it is

T(-,D,-,D)

partial functors

-+ B,

A,

T(A,-,A,-)

V-natural in

A for

with respect to the respectively.

The most

direct proof involves translating (1.40) via (1.32) into a form analogous to (1.7); then,as in the proof for ordinary

V-natural

transformations, it is a matter of combining (1.21) with (1.3) for Thus here, too,

V-naturality can be verified variable-by-variable.

That being so, we can combine ordinary and extraordinary and speak of a

V-natural family

T: A. ® Bop ® B --+ C

C.

v

and

V-naturality,

o.ABC: T(A,B,B) --+ S(A,C,C),

0 S: A® C p ®

c --+ v.

If each of

where

A,B, and

stands for a finite number of variables, this is then the most

general form. The analogue of "vertical composition" for such families is the calculus of

[25].

There are three basic cases of composability, in

addition to that for ordinary these three are dual.

V-natural transformations; and two of

Each of the three, however, is in fact a sequence

of sub-cases indexed by the natural numbers.

The reader will see the

pattern if we just give the first two sub-cases of the only two essentially-different sequences; the proofs are easy using (1.39) - (1.41). For

A and if

T,S: A0 p ®A.--+ B,

if

BAB: T(A,B) --+ S(A,B)

o.A: K--+ T(A,A)

is

is

A and

V-natural in

V-natural in

B,

the

composite

K - - - - > - T(A,A)

S(A,A)

( 1. 42)

1.7

42 is

V-natural in

s: A0 P ®A--+ B, B and if

and

For

A. if

is

aAB: K--+ T(A,A,B,B)

SABC: T(A,B,B,C) --+ S(A,C)

V-natural in

V-natural in

is

A

and

A,B, and

C,

then the composite

K ---,~ T(A,A,A,A) -

:-AAA--+

0

a~

is

V-natural in

A.

( 1. 43)

S(A,A)

" The next one in this series has

aABC: K--+ T(A,A,B,B,C,C)

and

SABCD: T(A,B,B,C,C,D) --+ S(A,D);

and there is of course a dual series.

and

R: A--+ B;

are

V-natural in

if

A and

TA

is

V-natural in

aAB: TA B,

---+

A.

-->-

S(A,B,B)

S: A ® A p ® A --+ B, BAB: S(A,A,B) __,. RB

and

the composite ( l. 44)

S (A, A, A)

For the same

S: A ® A0 P ® A ® A0 P ® A - - + B, BABe: S(A,A,B,B,C) --+ RC

0

T: A --+ B,

The self-dual series begins with

are

if

T

and

R,

but for

aABC: TA--+ S(A,B,B,C,C)

V-natural in

A,B, and

C,

and

then the

composite

TA

-------+

aAAA

is

V-natural in

S(A,A,A,A,A)

( 1.45)

- - - - + RA

8.\AA

A.

In each case a handsome geometrical picture is produced if, to the diagrams (1.42) - (1.45), there are added curves above the diagram linking variables which must be set equal in

a,

and curves below the

diagram linking variables which must be set equal in

s;

the

(1.43)-(1.45)

43

geometrical union of these curves, thought of as meeting at the variables, then links the variables which must be set equal in the composite.

Of

course the associativity of this kind of composition, whenever it makes sense, is a triviality.

1.8

We turn now to the

the beginning of 1.7.

V-naturality of the canonical maps mentioned at Of course such an object as

here thought of as the value appropriate (a) is

Ten(HomA(B,C),HomA(A,B))

First, it follows at once from (1.29) that

a: T --+ S

Next, for a

verify

of the

is

is a

in

T: A--+

B,

the map

V-natural in both variables B;

A and

TAB: A(A,B) --+ B.

It suffices to

and here the appropriate diagram of the

This last gives, for a A

and

this is also

A'

of

V-functor

T: A

@

B --+ C,

T. the

T(-,B): A(A,A')--+ C(T(A,B),T(A' ,B)).

V-natural in

V-natural

in the sense of (1.7).

V-functor

V-naturality in

a

form (1.39) reduces, in the light of (1.29), to (1.5) for (c)

is

aA: I --+ B(TA,SA)

V-natural in the sense of (1.40) precisely when

B(TA,TB)

A(A,B)

@

V-functor.

transformation (b)

A(B,C)

B·,

V-naturality However

the appropriate instance of (1.40)

reduces via (1.29) to (1.21). (d)

In particular, for a

[A(A,B),A(A,C)]

V-category

of (1.29) is

(e)

Composing the

the

V-naturality in

A(A,-)BC A of

A,

the

A(A,-)BC: A(B,C) --+

V-natural in all variables. of (d) with

g: I--+ A(B,C)

A(A,g): A(A,B)--+ A(A,C).

gives, by (a),

1.8

44 (f)

The

V-naturality in both variables of

e: [Y,Z]

~

Y--+ Z

is an

easy consequence of (1.29) and (1.34). (g)

Since, by (1.29),

M: A(B,C)

0

A(A,B)

->-

A(A,C)

for a

V-category

A is the composite A(B,C) it is (h)

0

A(A,-)®l ~(A,B),A(A,C)]

A(A,B)

i.: I

A(A,C)

V-natural in all variables by (d), (e), (f), (1.42) and (1.44). The

V-naturality of

follows from (a), since (i)

--e-+

A(A,B)

0

The ~

jA: I--+ A(A,A) 1: 1 --+ 1: A --+ A

V-naturality in every variable of

X ---+ X,

for a

r: X ® I ---+X,

is

V-category

A

V-na tural.

a: (X ® Y) ® Z ---+X® (Y ® Z),

c: X ® Y ---+ Y ® X,

follows easily

from (l. 34) . (j)

From the definition (1.27) of the isomorphism

definition (1.34) of

Ten(-,Y),

p- 1 : [X,[Y,Z]]---+ [X® Y,Z]

p

and the

we easily see that

is the composite

[X' [ y' Z]] -=Te_n_(r--,~Y")-+ [X® Y,[Y,Z] ® Y] -,-[-.-l-,-=-e-r]-' [X® Y,Z] . It follows from (c), (e), (f), and (1.44) that

p -1

all variables; whence the same is true of its inverse (k)

is

V-natural in p.

The composite X ----;;--:y+ I®X

~

IC

]""

1

[X®Y,X®Y]®X - - + [X,[Y,X®Y]]®X--+ [Y,X®Y] p® 1 e

is easily verified to be the is therefore

d: X--+ [Y,X ® Y]

of (1.24); the latter

V-natural in every variable by (i), (h), (j), (f), (e),

(1.42), and (1.44).

(1.46)

(2)

45

The composite X----.. d

is the isomorphism

[I,X

0

I]

i: X--+ [I,X],

[I ,X]

l1,r]

which is therefore

V-natural.

This completes the list of canonical maps, and we end this section with a general principle,

Since a map

f: X--+ [Y,Z]

~

=

and

f

e(f

for

V-functors

0

under the 1),

f: X

0

Y --+ Z and its image

of (1.23) are related by f

it follows from (f), (k), (e)

T,S,R

= [1,f]d

and (1.42) - (1.44) that,

of suitable variances with codomain

V,

we

have: (m)

A family

f: T(D,D,A,B)

0

S(E,E,A,C) --+ R(F,F,B,C)

is

V-natural

in any of its variables if and only if the corresponding f: T(D,D,A,B)--+ [S(E,E,A,C),R(F,F,B,C)]

1.9

is so.

The form of the Yoneda Lemma we give here is a weak one, in that

it asserts a bijection of sets rather than an isomorphism of objects of

V;

a stronger form will be given in 2.4 below. Consider a

each

V-functor

F:

V-natural transformation

element

n:

I--+ FK

of

I

FK

~

A --+ V

and an object

K of

A.

To

a: A(K,-) --+ F we can assign the given by the composite

A(K,K)

K

The Yoneda Lemma asserts that this gives a bijection between the set

V-nat(A(K,-),F)

of V-naturat transformations and the set

( 1. 46)

1.9

46 V (I,FK) (=

VFK)

of elements of

FK;

the component

0

a

A

being given

in terms of n as the composite [ FK , FA]

A(K,A)

----::-[n-,-:1--::]-+ [I , FA]

-i--~1c-+

For the proof, we first note that, for any V-natural in by (1.47),

F,

the

its

A, aKjK

by 1.8 (b), (e), and(£).

n;

is in fact

Set-naturality of

ji

aA' when

A(K,-)

A(K,A)

a

is defined

i,

and the easy verification

It remains to show that (1.47) is in fact

is given by (1.46).

n

Next, if

(a consequence, as we saw in (1.7), of

j

ii: I--+ [l,I].

=

n, (1.47) is indeed

this follows easily from (1.6) for

V-naturality), the naturality of

that

(1.47)

FA •

In the diagram [j '1]

[A(K,K) ,A(K,A)]

[I,A(K,A)]

I[1,aAJ

F

+

[FK,FA] ---,[~__,,...,];-+ [A ( K, K) , FA] ---r[J-.,--:,.]-->-

aK' 1

1

the left square commutes by the

[I,FA]

V-naturality of

--~~-+

i-

a

FA

in the form (1.39),

the middle square trivially, and the right square by the naturality of i.

The proof is complete when we observe that the top edge of the

diagram is the identity; which follows from the definition (1.29) of the axiom (1.4) for

A(K,-), terms of

A, and the definition (1.26) of

i

in

r.

There is an extra-variable form of the result, involving functors F:

0

BP

®

which is

A--+

V and K: B--+ A,

V-na tural in

A

and a family

for each fixed

aBA: A(KB,A)

--+

F(B,A)

B, so that it corresponds

(1.47)-(1.48)

47

as above to a family

nB: I--+ F(B,KB).

It then follows from (1.46)

and (1.42) that

n is

V-natural in

B if

a

is, and from (1.47)

and (1.44) that

a

is

V-natural in

B if

n

is.

It is worth noting an alternative way of writing (1.47) when has the form

B(B,T-)

n: I--+ B(B,TK) image of

1K

for a

V-functor

is then a map

under

T: A--+

n: B --+ TK

in

VaK: A (K,K)--+ 8 (B,TK)). 0

B,

so that

B0

(in fact, the

Then

0

F

is the

composite A(K,A)

B(TK,TA)

B(B,TA) ,

B(n,l)

which is easily seen by (1.32) to coincide with (1.47).

every V-natural

a: A(K,-) --+ A(L,-)

and clearly

k: L --+ K;

1.10 We have called the

for a unique

V-functor K

E

is.

A(K,-): A--+ V representable.

V-functors

representable if there is some Then the pair

A(k,-)

In particular,

is an isomorphism if and only if a

k

More generally, however, a

is

(1.48)

F: A--+

V is called

A and an isomorphism a: A(K,-) --+F.

(K,a) is a representation of

F·

it is essentially ' unique if it exists, in the sense that any other representation

a': A(K' ,-)--+ F k: K--+ K'.

has the form

The corresponding

the representation. contravariant

(We call 0

F: A P--+

For a general n: I --+ FK

a'= a.A(k,-)

V,

n: I--+ FK n

for a unique isomorphism is called the unit of

the counit when we are representing a

V in the form A(-,K).)

there is no simple criterion in terms of an

for the corresponding

a: A(K,-) --+ F

It is of course otherwise in the classical case

V=

to be an isomorphism.

Set;

there we have

l. 10

48

the comma-category are pairs

(A,x)

and only if

1/F

with

=

e"L F

A

~

of "elements of

A and

e"L F

FA;

E

e"L F,

(K,n) is initial in

i f and only i f

x

F",

whose objects

and

a

is invertible if

so that

F

is representable

has an initial object.

However, any such kind

of "universal property" criterion ultimately expresses a bijection of sets, and cannot suffice in general to characterize an isomorphism in

V • 0

0

Now let

F: B p ®A--+

admits a representation

V be such that each F(B,-): A--+ V

aB: A(KB,-)--+ F(B,-).

one way of defining

~c: B(B,C) --+ A(KB,KC)

V-funator for which

aBA: A(KB,A) --+ F(B,A)

as weU as

that nv.kes is

K a

V-natural in

B

A.

For by 1. 9, the that of

Then there is exactly

V-naturality in

B of

is equivalent to

nB' which by (1.40) means the commutativity of

B(B,C)

~

A(KB,KC) F(B,-), [F(B,KB),F(B,KC)]

F(-,KC)

+ [F(C,KC) ,F(B ,KC)] --...,.----=----+ [I,F(B,KC)] .

Cnc,1J

Now the composite

[nB,l]F(B,-)

( 1. 49)

in this diagram is, by (1.47), the

composite isomorphism A(KB,KC) - - - - + F(B ,KC) aB,KC so that (1.49) forces us to define

[I,F(B,KC)]

i ~C

as

( l. 50)

(1.49)-(1.52)

49

This composite (1.50) is

V-natural in

B, by 1.7 and 1.8; and from

this the proof of 1.8(b), read backwards, gives the (1.5) for

K;

while the remaining

V-functor axiom

V-functor axiom (1.6) is a trivial

consequence of (1.49).

1.11 As in any

2-category (see

n,e:: S-l T: A-+ B

adjoint) and

S:

(the unit) and

equations

Te:.nT

adjunctions in

in

V-CAT

an adjunction

[49])

T: A-+ B

between

(the right

B-+ A (the left adjoint) consists of n: 1-+ TS (the aounit) satisfying the triangular

e:: ST--+ 1

=1 V-CAT

and

e:S.Sn

= 1.

By the Yoneda Lemma of 1.9, such V-natural isomorphisms

are in bijection with n: A(SB,A) = B(B,TA)

for by (1.48) a n = B(n,l)T A(SB,A) nn • 1

V-natural map

for a unique

has the form and

nn = 1

n,

n: A(SB,A) while a

n = A(1,e:)S

(1.51) B(B,TA)

--+

has the form

V-natural map· n: B(B,TA) -+

for a unique

e:,

and the equations

reduce by Yoneda precisely to the triangular

equations. The

2-functor

) : V-CAT -+ CAT 0

into an ordinary adjunction

carries such an adjuction

n,e:: s0 - t T :0A -0- + B0

in

CAT.

It

is immediate from (1.33) that the corresponding isomorphism of hom-sets is Vn:

A0 (SB,A)

~

B0 (B,TA)

•

( 1. 52)

Note that the unit and counit of this ordinary adjunction are the same n

and

e:,

now seen as natural rather than

V-natural,

1.11

50

By 1.10, a

V-functor

T: A--+ B has a left adjoint (which is

then unique to within isomorphism, as in any each

8(B,T-)

is representable.

2-category) exactly when

More generally, if those

8(B,T-) is representable constitute the full subcategory we get a

V-functor

v

When

is conservative - faithfulness is not enough

5

0

s0

for

---1 T0 ),

(1.52) is.

A by

n

= 8(n,l)T

and deduce that Thus, taking

n

B,

of

-

0

define a map

n

the

that is

as above (using the unit

n

is an isomorphism since the

V = Ab

T.

implies that of a left

T

s for T: we set SB = S0 B'

V-natural in

8'

for which

called a partial left adjoint to

S: 8' --+A

existence of a left adjoint adjoint

B

for Vn

of

for instance, an additive functor

has a left adjoint exactly when its underlying functor has one. Again by 1.10, in the extra-variable case of a T: C p ~A--+ 8, 0

S

if each

T(C,-)

cutomatically becomes a

has a left adjoint

V-functor

n: A(S(B,C),A) ~ 8(B,T(C,A))

is

V-functor

8

~

S(-,C),

then

C --+ A in such a way that

V-natural in all three variables.

An evident example, with

A = 8 = C = V,

p: [X~ Y,Z] ~ lX,[Y,Z]]

of (1.27).

is the adjunction

For a typical adjunction (1.51) as above, we have a commutative diagram

(1.53)

(1.53)

51

this follows from Yoneda on setting

j,

with

(Vn)(EA) ~ 1.

since

conclude that

A'

Because

=A

and composing both legs

n

is an isomorphism, we

is fully faithful if and only if

T

In consequence, a right adjoint

T

is an isomor>phism.

E

is fully faithful exactly when

is so.

T

0

As in any

2-category, a

equivalence, and we write

A ~ B,

T:

n: 1 ~ TS

and isomorphisms any

V-functor

and

A --+

B.

B is called an S: B --+A

if there is some

p: ST ~ 1.

2-category) by the isomorphism

E = p.Sn- 1T.p- 1ST,

T: A--+

E: ST

Replacing

~

1

p

(still in

given by

we actually have an adjunction

n,E: S

---1 T:

The word "equivalence" is sometimes used to mean an adjunction

in which, as here, In the

n

and

2-category

are isomorphisms; we write

E

S

--1

T:

A ~ B.

T: A--+ B is an equivalence if and

V-CAT,

only if T is fuZZy faithful and essentially surjective on objects; the latter means that every to

TA

for some

For, if

T

A

E

A;

B

E

B is isomorphic (in B

we often omit the words "on objects".

is an equivalence,

T

is fully faithful since

an isomorphism, and is essentially surjective since is an isomorphism. SB

E

A

-in-A

As for the converse, we

and an isomorphism

isomorphism because automatically a

n : B --+ TSB. 5

TE,

T.

S

Since

by the triangular equation

is

E

B an

V-natural-

is then an

becomes n

is an

TE.nT

= 1;

is an isomorphism by (1.12), since a fully-faithful conservative.

B

The corresponding

is fully faithful, so that

V-functor left adjoint to

isomorphism, so is

choose for each

B

T

E

nB: B--+ TSB

n = B(n ,1)T: A(SB,A) --+ B(B,TA)

transformation

of course)

,

0

T

0

whence

is certainly

l. l l

52 In any

2-category (see

[49])

adjunctions can be composed, forming

a category, of which the equivalences are a subcategory. from Yoneda that the composition of adjunctions in to the composition of the

It is further true in any

n,£: S

and

1-cells

--l

2-category (see

and

2-cells

A and

~

~:

S'Q--+ PS.

In

--l

n' ,£': S'

B--+ B',

Q:

V-CAT

( l. 54)

[49 ])

that, given

B'

T': A'--+

there is a bijection

(with appropriate naturality properties) bebNeen and

corresponds

B(B,T'TA) .

=

T: A--+ B and

P: A--+ A'

V-CAT

V-natural isomorphisms in

A(SS'B,A) - B(S'B,TA)

adjunctions

It follows

A: QT--+ T'P

2-cells

it follows from Yoneda that

determine one another through the commutativity of

A(SB ,A) ----.;n~---. B(B, TA) -

Q

A' (PSB,PA)

B' (QB ,QTA)

A'(p,l)j

B' (l,;>..)

A'(S'QB,PA) such a pair

)..,~

S ---1 T

and

A• = A with

s' ---1

have a bijection bebNeen the adjunctions become a to

S' --1 T'

(1.55)

may be called mates,

In particular, when so that

B I ( QB •T 'p A)

n'

T'

p

z

1

and

are both adjunctions

A: T --+ T'

and the

2-category when we define a

to be such a pair

B' = B

(A,~).

~=

A

s'

with

Q : 1'

--+

B,

we

--+

s.

The

2-cell from

s --l

T

(1.54)-(1.55)

Any

53

A--+ 8

T:

full image

A'

of

the replete image some to

TA.

A

if

T,

A"

T

determined by those of

A'

Clearly

B;

determines two full subcategories of

T,

and

B of the form

determined by those

A"

the

TA,

and

B isomorphic to

are equivalent; and each is equivalent

is fully faithful.

A full subcategory which contains all

the isomorphs of its objects is said to be replete; any full subcategory has a repletion, namely the replete image of its inclusion. T: A --+ 8

fully-faithful inclusion

A" --+ 8

A'

--+

Clearly a

has a left adjoint if and only if the

B has one, and if and only if the inclusion

has one.

A full subcategory

T: A--+ 8

A

has a left adjoint.

reflective in

8 , 0

is called reflective if the inclusion

8

of

This implies, of course, that

but is in general strictly stronger.

A

is

0

It follows -

since it is trivially true for ordinary categories - that every retract

8 )

(in

of an object of the reflective

0

When

A

is reflective we may so choose the left adjoint

£: ST--+ 1

that

satisfies

R2

lies in the repletion of A.

A

=

R,

idempotent monad on

is the identity. nR = Rn

B.

=

1.

Then

Such an

(R,n)

is called an

Conversely, for any idempotent monad

is formed as above, the full image is

uniquely determined by of

A·,

A.

8 --+ A

n: 1--+ R = TS: B--+ B

both its full image and its replete image are reflective in R

S:

A

again; but

The replete image of

(R,n),

B.

(R,n)

If is not

R is the repletion

and there is an evident bijection between full replete

reflective subcategories of monads on

B.

8

and isomorphism classes of idempotent

2. I

54

2.

Functor categories

2.1

So far we have supposed that

V

is symmetric monoidal closed,

and that the underlying ordinary category

v0

is locally small.

v

Henceforth we add the further assumption that

is complete, in the

0

sense that it admits all small limits; and from 2.5 on, we shall

v

suppose as well that Consider a

universal every

is cocomplete. 0

V-functor

T: A p ®A--+

V-natural family

V-natural

unique

0

V.

If there exists a

AA: K--+ T(A,A),

aA: X--+ T(A,A)

in the sense that

is given by

T·

we call (K,A) the end of

f: X---+ K,

= A~f

aA

'

it is clearly

unique to within a unique isomorphism when it exists.

JAEA T(A,A)

for the object

The

T:

then the universal

K

V-natural

may be called the counit of this end.

V-naturality condition (1.40), in the case

under the adjunction

We write

K, and by the usual abuse of language

also call this object the end of AA: fA T(A,A)--+ T(A,A)

for a

V0 (X,[Y,Z]) : V0 (X®Y,Z) AA

------=-=-----

~

B = V,

V0 (Y,[X,Z])

transforms into

T(A,A)

j PAB [A(A,B),T(A,B)]

where

and

It follows that, when

are the transforms of A is small, the end

exists, and is given as an equalizer

T(A,-)AB

( 2. 1)

and

JAcA T(A,A)

T(-,B)BA" certainly

(2.1)-(2.4)

JAEA

55 p

T(A,A) ~ lljAEA T(A,A) ~ ~A,BEA [A(A,B),T(A,B)] , If

A,

(2.2)

while not necessarily small, is equivalent to a small

V-category, then the full image in

A'

subcategory

A of the latter is a small full

for which the inclusion

A' --+A

is an equivalence.

A,B

Since it clearly suffices to impose (2.1) only for

JAEA T(A,A)

follows that

still exists, being

JAEA'

consequence, !Je henceforth change the meaning of

to include those

A',

it

T(A,A),

In

E

smaU", extending it

11

V-categories equivalent to small ones in the sense

of 1.2. The universal property of the end - expressed, as it stands, by a bijection of sets - in fact lifts to an isomorphism in sense that

[X,AA]: [x,JA T(A,A)]--+ [X,T(A,A)]

V0 ; in the

expresses its domain

as the end of its codomain:

[X,jA T(A,A)] ~fA [X,T(A,A)] For

Y--+ [X,T(A,A)]

is, by 1.8(m). if

JA

T(A,A)

on taking If

X

is

(2.3)

(with counit [X,AA]).

V-natural if and only if

Y ®X--+ T(A,A)

Note that the right side of (2.3) exists for all

X

exists; the converse of this is also true, as we see

=

I.

a: T--+ T':

A0 p ®A--+ V,

and if the ends of

T and

T'

exist, the top leg of the diagram !,A _ _ _..:.;___ _-+

fA

T' (A,A)

;.'

T(A,A)

T' (A,A) A

(2.4)

2. l

56

is

V-natural by (1.42); so that, by the universal property of

JA

there is a unique

aAA

A',

rendering the diagram commutative.

This "functoriality" of ends lifts, in the appropriate circumstances, to a

V-functoriality.

the end

HB

=fA

Suppose namely that, for exists for each

T(A,A,B)

one way of making H into a V-functor AAB: HB ->- T(A,A,B)

V-naturaZ in

For, by (1.39), the

T:

A0 P ® A® B --+ V,

Then there is exaatZy

B.

H: B--+

v

that renders

B as well as in

V-naturality in

B

A.

is expressed by the

commutativity of

B(B, C)

[HB,HC]

T(A,A,-)j

![l,AAC~ t'

[T(A,A,B), T(A,A,C)]

The bottom leg is

- - 7 r . , - - -,-.,],.------>

') AB' 1

V-natural in

then follow from the

The

(2.5)

A by 1.8(c), 1.8(e), and (1.43).

The right edge is itself an end by (2.3), the diagram commutative.

[HB,T(A,A,C)]

Hence a unique

HBC

renders

V-functor axioms (1.5) and (1.6) for

V-naturality of

M and

j

H

(1.8(g) and (h)),

the composition-calculus of 1.7, and the universal property of the counit

ll,,\J.

In these circumstances, if

fA

T(A,A,B)

is

V-naturaZ in

R: B--+ V, a family

s

8

:

RB--+

B if and pnZy if the composite

RB - -- > - fA T(A,A,B) 8

T(A,A,B)

B

ir. ~o.

One direction is trivial, while the other follows easily

(2.6)

(2.5)-(2.9)

57

from (2.5) and the universal property of the counit

[l,A].

In fact

this result can also be seen- using l.B(m) to view (2.6) as a V-natural family

I--+ [RB,T(A,A,B)]

and using (2.3) -as a special

case of the following. T: (A~ 8) p ~ (A~ 8) 0

Let 0

0

(A~ 6) p

=Ap

fAEA T(A,B,A,C) 8°p ~ in

B--+ V.

A

~ 6°P.

V, recalling that

--+

Suppose that, for each

B,C

6, the end

E

exists; then, as above, it is the value of a Every family

aAB: X--+ T(A,B,A,B)

that is

V-functor V-natural

factorizes uniquely as X ---~--+

68

fAEA

T(A,B,A,B)

T(A,B,A,B)

-------+

(2.7)

AABB

and it follows again from (2.5) and the universal property of that the

in

B

V-naturaZity in

of

B of aAB

is equivalent to the

[l,A]

V-naturaZity

s8 •

Since, by 1.7, coincides with its

Fubini Theorem:

V-naturality of V-naturality in

if every

fA

aAB

(A,B) E A

T(A,B,A,C)

f(A,B)EA~B T(A,B,A,B)

in

A and ~

B,

B separately we deduce the

exists, then

f BE B f AE A T(A,B,A,B),

(2.8)

either side existing if the other does. This in turn gives, since

A® B ~

6 ®A, the following

interchange of ends theorem, often itself called the Fubini Theorem: if every

fA

T(A,B,A,C)

and every

f AE A f BE B T(A,B,A,B)

fs

T(A,B,D,B)

fscB fAcA

either side existing if the other does.

exists, then

T(A,B,A,B)

(2.9)

2.2

58

2.2

For

V-functors

T,S: A--+

B we introduce the notation

fA

[A ,B](T ,S)

(2.10)

8 (TA,SA)

for the end on the right, whenever it exists; and we write the counit as

EA,TS We write

[A,BJ (T,S)

for the set

0

a: I--+ [A,B](T,S)

(2.11)

[A,BJ(T,S)--+ B(TA,SA)

of this end.

V[A,BJ(T,S)

of eZements

These of course correspond to

V-natural families EAr:~.:

( 2. 12)

I--+ B(TA,SA)

which by l.B(a) are precisely the

V-natural transformations

a: T-+S:A--+8.

Note that, by (2.3) and the adjunction

[X,[Y,Z]]-

[Y,~X,Z]],

we have [X,! A, VJ(T,S)]

either side When

existing for all [A,BJ(T,S)

X

E

V if the other does.

exists for all

T,S: A--+ B, as it surely does

by 2.1 i f A is small, i t is the hom-object of a with all the

T: A

--+

(2.13)

[A,VJ(T,[X,S-])

"¥

B as objects.

V-category

The composition law of

is, by 1.8(g), (1.43), and the universal property of uniquely determined by the commutativity of

.

EA TR,

[A,BJ [A,BJ

(2.10)-(2 .17)

59 M

[A,B](S,R)®[A,BJ(T,S)

[A,Bl(T ,R)

j

EA®E A

B(SA,RA)08(TA,SA) - - - - M ; - ; - - - - - > B(TA, RA)

(2.14)

and its identity elements are similarly determined by

(2.15) the axioms (1.3) and (1.4) follow at once from the corresponding axioms for

B and the universal property of We call

[A,B]

0

[A,BJ

EA.

the functor category; its underlying category

is by (2.12) the ordinary category of all

and all

V-functors

A--+ B

V-natural transformations between them - that is,

(2.16)

V-CAT(A,B) • Even when use

[A,BJ

[A,BJ 0

When

does not exist, it may sometimes be convenient to

as an abbreviation for [A,BJ

V-CAT(A,S). T ~ TA

exists, the function

of (2.11) constitute, by (2.14) and (2.15), a called evaluation at

EA: [A,BJ--+ B, functor underlying component

EA

sends a map

a

A. €

and the maps

EA,TS

V-functor

By (2.12), the ordinary

[A,B] 0 (T,S)

to its

aA·

There is a

V-functor

E: [A,BJ ® A --+ B, simply called

evaluation, whose partial functors are given by E(-,A)

EA: [A,BJ --+ B,

E(T,-)

T: A --+ B

(2.17)

2.2

60 To show this we must verify the appropriate instance of (1.21); but, by the proof of 1.8(c) read backwards, this instance reduces via (1.29) to the instance of (1.40) expressing the

More generally,

[A,BJ(T,S)

V-naturality in

may exist, not for

all those in some subset of the set of example would be the subset of those

V-functors A

~

B

A of

aZZ T,S, A

~

but for

B;

a typical

which are left Kan

extensions, in the sense of 4.1 below, of their restrictions to some fixed small full subcategory of

A.

Then we get as above a

V-category

[A,BJ'

with these functors as objects, whose underlying category

[A,B]~

is the corresponding full subcategory of

have the evaluation functor

E: [A,B]' ® A

--+

V-CAT(A,3);

B.

Still more generally, we may wish to consider for certain particular pairs

T,S

Yoneda isomorphism of 2.4 below.

and

V

®A~

D.

B such that

[A,B](T,S)

just

for which it exists - as in the To discuss its functoriality in this

generality, suppose we have functors Q:

and we

P: C ®

A~

[A,B](P(C,-) ,Q(D,-))

Then there is, by 2.1, a unique

B and

exists for each

V-functor

C

H: C0 p ® V ~ V

having H(C,D) with respect to which is

V-natural in

[A,BJ(P(C,-),Q(D,-))

(2.18)

EA: [A,B](P(C,-),Q(D,-)--+ B(P(C,A),Q(D,A))

C and

D as well as

A-

Moreover by (2.6)

(2.7), in the situations (the second of which has

V =C)

and

(2.18)-(2.22)

a

is

61

V-natural in

C

When

or even some

[A,BJ,

or

if and only if B is.

D

[A,BJ', exists, the

H of (2.18)

can be given more explicitly - see 2.3 below.

2.3 We return to the case where E: [A,8] ® A ~

[A,BJ

exists.

The

V-functor

B induces, for each V-category C, an ordinary

functor

V-CATCC,[A,BJ) sending

G: C

~

C and sending

[A,BJ

®

~

V-CATCC

®

A,B)

(2.20)

to the composite

A --G®::c-:--+ [A,BJ 1

®

8: G--+ G': C--+ [A,BJ

Clearly the map (2.20) is

2-natural in

A

--E-~

B

to the composite

C·,

(2.21)

E(B ® 1A).

and in fact it is an

isomorphism of categories. To prove this we must first show that every the form (2.21) for a unique

G.

P: C ® A

---+

B

Equating the partial functors of

has P

and (2. 21) gives

P(C,-)

GC,

The first of these determines asserts the commutativity of

(2.22)

G on objects, whereupon the second

2.3

62

rA,B](P(C,-),P(D,-))

C(C,!"l) P(- ,A) Cll

""~ B(P(C,A),P(D,A)) Since GCD

P(-,A)CD

is

V-natural in

A by 1.8(c), this determines a unique

by the universal property of

EA.

By this same universal property,

together with the composition-calculus of 1.7, the (1.5) and (1.6) for

V-functor axioms

follow at once from those for

G

(2.23)

P(-,A).

Before showing that (2.20) is bijective on maps as well as on objects, let us analyze the If

P: C

®

A

--+

G: C--+ [A,B]

B

and

Q:

V-functoriality (2.18) when

0

CA,BJ P

we have the composite

®

[A,BJ

---:::--------+

Hom[A,BJ and with respect to this, V-natural in

C and

D,

exists.

V ® A --+ B correspond by (2.20) to

F: V --+"[A,BJ,

and

[A,B]

V-functor

v

EA: [A,B](GC,FD) --+ B(EAGC,EAFD) by 1.7 and 1.8(b).

By (2.22), this

EA: [A,B](P(C,-),Q(D,-))--+ B(P(C,A) ,Q(D,A));

hence the

(2.24)

is EA

is

H of (2.18),

by its uniqueness, must be (2.24). To complete the proof that (2.20) is an isomorphism of categories, it remains to verify that, if of

G

and

S: G --+ G'. -form

G'

'

every

a: P

P and --+-

P'

P' is

By (2.22) the equation

aCA = EASe•

are the images under (2.20)

a

for a unique

= E(S

®

lA)

has the component-

which by (2.12) is

(2.25)

(2 .23)-(2 .25)

63

Be: GC--+ G'C

Thus

must be the

cxc_:

P(C,-)--+ P'(C,-);

that

Be· so defined, is

map

V-natural transformation

and the proof will be complete when we show

H

as a

Sc

is identified as (2.24).

More generally, [A,B]'

if

[A,BJ

does not exist but some full subcategory

does, in the sense of 2.2, we have by the same arguments an

isomorphism between the category subcategory of

V-CAT(C ® A,B)

for which each

P(C,-)

(2.24) with The

[A,BJ'

V-CATCC.[A,BJ')

lies in

[A,B]';

replacing

then

[A,BJ

replaced by --+

and

C(0,1)

V(T,S)

B;

V,

0.

l,

with

C

when it exists,

V-CAT(-®A,B).

It is

C

to be

V - at least if

I

we identify the objects of to be the

X of

V

0

in (2.20), with

V

with the

V,

has an [A,B]

V-functors

V-category with two objects

C(O,O) = C(l,l) =I,

an arbitrary object

=fA

2-functor

[A,BJ,

2-functor is representable by some

For taking

while taking

A --+ B

®

[A,BJ.

exists and is isomorphic to

initial object

P: C

for we have the analogue of

2-natural isomorphism (2.20) exhibits

conversely true that, if this

and the full

determined by those

as a representing object for the

0

I f we look on

I--+ [A,BJ(GC,G'C), this follows from 1. 8 (m) and (2.19), now

that

A

c.

V-natural in

V,

with

C(l,O) = 0,

and with

we easily see that

B(TA,SA).

It follows from 1.10 that, so far as it exists, 2-functorial in

A and

B

in such a way that (2.20) is

every variable; which by 1.9 is equivalent to the each variable of

[A,BJ

E: [A,BJ ® A

--+

B.

is uniquely

2-natural in

2-naturality in

Using this latter criterion

2.3

64

we easily see that, for [M,N]: [A,BJ --~ [A',B'J

M: A '

A an d

--+

sends

T

to

N··

NTM

B --+ B'

,

the

V-functor

and is determined on

hom-objects by [M,N]TS

[A' ,B'](NTM,NSM)

[A,B](T,S)

i E'A

l

while the

(2.26)

B' (NTMA,NSMA)

B(TMA,SMA)

V-natural transformation

[u,v]: [M,N]--+ [M' ,N'J

is

given by '· Tu: NTI1 -~ :,; 'T'1'

As for the interplay of functor have a

cat~gories

( 2. 2 7)

.

with duality, we clearly

2-natural isomorphism (2.28)

either side existing if the other does. The when

2-functor -®A: V-CAT--+ V-CAT

[A,B]

exists for

has an initial object.

a~~

B;

has a right adjoint

This right adjoint surely exists when

For instance, a result of Freyd

V = Set,

A is necessarily small if

an ordered set, as in the cases for every

A and

B.

v

and by the above, only then, if

small, in the extended sense of 2.1; and for many then.

[A,-]

(see [A, lfJ

v=2

and

V

[73]) exists.

v = R+'

0

A

is

it exists only shows that, when Yet when [A,BJ

v0

exists

is

(2 .26)- (2. 31)

65

In this sense the symmetric monoidal

2-category

"partially closed"; while the synunetric monoidal of small

V-CAT is

2-category

V-Cat

V-categories is actually closed.

The partial closedness is sufficient to imply the isomorphisms (1.26) and (1.27) of 1.5, which here read: [I ,BJ ~

B ,

[C ® A,BJ

~

[C,[A,BJJ ,

(2.29)

the latter in the sense that either side exists if the other does, provided

[A,BJ

exists - which follows alternatively from the

Fubini Theorem (2.8).

2.4 We can now give a stronger version of the Yoneda Lemma of 1.9, no longer expressed by a mere bijection of sets, but by a Yoneda

isomorphism in Given a

v. 0

V-functor

1.9, we have the map in

A by 1.8(b).

FKA

in

A by 1.8(m).

expresses

V-natural

(2.30)

FK----+ [A(K,A),FA]

V (X,[Y,Z]) 0

~

V (Y,[X,Z]) 0

is

V-natural

The stronger Yoneda Lemma is the assertion that (2.30)

as the end ~:

FK

JA [A(K,A),FA], so that we have an isomorphism ~

(2. 31)

[A,VJ(A(K,-),F) •

For the proof, consider any transform

which is

The transform

under the adjunction

FK

V and an object K of A as in

FKA: A(K,A) --+ [FK,FA],

~A:

of

F: A --+

aA: A(K,A) --+ [X,FA]

V-natural

aA: X--+ [A(K,A),FA];

under the adjunction, being

its

2.4

66 V-natural in

for a unique (1.48). unique

A by

1.8(m), is the composite

A(K,A)

-----+

FKA

n: X--+ FK,

[FK,FA]

by the Yoneda Lemma of 1.9 in the form

This is equivalent to the assertion that n;

V = V (I,-): V0

A(K,-)--+ F,

--+

0

V0 (I,FK)--+ [A,V] 0 (A(K,-),F).

or

aA =¢An

for a

which completes the proof.

The image under

~n:

[n,l]' [X,FA]

This sends

whose component

n: I--+ FK

A(K,A)--+ FA

By the definition (2.30) of

the composite (1.47).

Set of (2.31) is a bijection

is

to EA~Tl

by (2.12),

¢A, this is at once seen to be

Hence the weak Yoneda Lemma of 1.9, which we

have used to prove the present strong Yoneda Lemma, is subsumed under the latter as its underlying bijection. The special case

F = A(L,-): A--+ V of the present Yoneda Lemma

strengthens the last assertion of 1.9 to

A ( L. K) -

I A.

v

I

(A ( K,-) 'A I L'-) I

(

Since (2.31) is true without any smallness restriction 0n

V-naturality must in general be discussed (cf. 2.2) in terms of

the

V-functor

P: C 0 A--+

0

V.

0

A--+ V

and some

V-functor

We_assert that ¢CK: P(C,K) - [A,V1(A(K,-),P(C,-))

ts

V-natural -:n

as in (2.18).

32)

A,

its

HomA: A p

~.

C and

K,

when the right side is made

This follows form (2.19); for

(2.33) V-functorial

(2.32)-(2.38)

67

EA¢ =¢A: P(C,K) l.B(m), since When

[A(K,.:\) ,P(C,A)] is V-natural in

A(K,A)--+ [P(C,K),P(C,A)]

[A,VJ

(2.20) to a

--+

exists,

0

HomA: A

p

C and

K by

is so by l.B(b) and l.B(c).

®A--+ V

corresponds under

V-functor

(2. 34)

sending

K to

YK

A(K,-)

(2.35)

Y is called the Yoneda embedding.

being fully faithful by (2.32), In these circumstances the

V-naturality (2.33) can be more globally

expressed by taking

for

[A,VJ

C and E: [A,V] ® A --+ V for P;

the right side of (2.33) is then by (2.24) the value of the V-functor

v

A® [A,VJ

(2.36)

so that, with respect to this functoriality,

the Yoneda isomorphism (2.31) is

V-natural in K and

F.

(2.37)

The extra-variable result of 1.9 also has a lifting to the present context.

Let

0

F: B p ® A

we have an isomorphism C and

B.

F(C,KB)

~

--+

V and

K: B

--+

A.

By (2.33)

[A,V](A(KB,-),F(C,-)), V-natural in

The Fubini Theorem (2.8) now gives

jB

F(B,KB)

~

[8°P ® A,VJ(A(K?,-),F(?,-))

either side existing if the other does.

(2.38)

2.5

68

2.5 We henceforth suppose, in addition to all our former assumptions V

V, that the underlying category

on

Then the ordinary functor

• I,

adjoint

Set

The

X

0

V0

--+

E to the coproduct

E· I

of

E

This functor sends the cartesian product in

F) • I

-

2-functor

: 0

,

®

(F • I)'

0

(E • I)

)

V

in that we clearly have 1 • I

has the same objects as L,

=

( 2. 39)

I.

now has a left adjoint

V-CAT--+ CAT

>v . The free V-category LV on the ordinary L

Set has a left

0

to the tensor product in (E

V (I,-):

=

sending the set

V •

in

I

copies of

V

ie cocompZete.

0

ZocaZZy-s~aZZ

and has hom-objects

category

Lv(K,L) = L(K,L) ·I;

its composition Zaw and its identities are induced from those of

L

using the isomorphisms (2.39), For the proof, we have first to verify that satisfies the

LV, as defined,

V-category axioms (1.3) and (1.4); this is trivial

from the corresponding axioms for

L.

There is an evident functor \jJ : L--+ (l )

v

0

which is the identity

on objects, and which is defined on morphisms by the obvious map L(K,L)--+ V (I,L(K,L) • I). 0

This

\jJ

is the unit of the

2-adjunction

in question. In fact it is easy to see that, for a functor with

T: L--+B

To\jJ = T:

o'

there is exactly one

on objects we have

TK

~

TK,

V-category V-functor while

B

and a

T: Lv

--+

B

(2.39)-(2.40) TKL: L(K, L) • I TKL: L(K,L)

69 ----+-

--+

B(TK, TL)

B (TK,TL)

is the transform of under the adjunction

0

Verification of (1.5) and (1.6) for Finally, if a.: T-+-

a.K

E

T

a.: T-+- S: L-+- 8

0

S: LV--+ B

with

is easy.

,

there is a unique

a. 1/J = a.; the component 0

80 (TK,SK), and again verification of (1.7) for

V-natural

a.K

must be

a.

is easy.

This completes the proof. If

L is a small category, LV is clearly a small

and we can form the functor category ordinary category

CLV,B]

0

[LV,B].

Since the underlying

V-CAT(LV,B)

of this is

V-category,

by (2.16),

the

2-adjunction above gives (2.40)

2.6 The possible non-existence of a

V-functor-category

[A,BJ

when

A is large (= non-small) is somewhat tedious, forcing us into such circumlocations as using (2.18) instead of (2.24), or (2.33) instead of (2.37) - which then propagate themselves throughout the applications below. Yet when

V = Set,

albeit not in general a when

A and

B are

however be seen as a

we always do have

as a category;

Set-category (= locally-small category)

Set-categories, unless Set'-category, where

sets in some larger universe containing are taking the view that

[A,BJ

obA

obA

A is small. Set'

It can

is the category of

as an element - we

is always an honest set, and supposing

that every set belongs to some universe.

2.6

70

We may ask whether such a view can be imitated for a general Consider first the various examples of a symmetric monoidal closed

2

ordered set, such as A;

V

or

0

R+'

or

V0

either

These are of two types:

given in 1.1.

v. v

is a particular

[A, m exists for all

in which case

consists of the smaU models of some theory - small sets,

small categories, small abelian groups, small differential graded R-modules, and so on.

V' 0

category

In the latter case we can clearly consider the

of models in a larger

monoidal closed category

V ',

Set',

which gives a symmetric

V

in which

closed structure is faithfully embedded.

with its symmetric monoidal

Moreover

V ' admits all

limits and colimits which are small by reference to

V0

inclusion

V0 •

--+

V0 '

0

Set';

and the

preserves all limits and colimits that exist in

(We are supposing that our

origina~

universe contains infinite

sets; otherwise we do not have the colimit-preservation above.) Then, for [A,B],

V0

--+

where

V0

'

V-categories

Set'

A

and

B, we always have a V'-category

is taken large enough to include

preserves limits, to say that

[A.B](T,S)

sense of 2.2 is precisely to say that the object actually lies in

V;

2.2 - that is, as a all

and to say that

obA.

[A,B]

exists in the

[A,BJ(T,S)

of

V'

exists in the sense of

V-category- is to say that

[A,B](T,S)

E

V

for

T ,S.

That a similar thing is possible for a perfectly general follows from Day

[17].

V

Using his construction, one can show that

there is always a suitable II0 '

Since

II'

whose underlying ordinary category

is a full reflective subcategory of

[ V op Set' J 0

'

'

is embedded by Yoneda.

Since the monoidal structure of

by eornds (relative to

Set'),

into which

V'

V

0

is defined

we defer a description of it until 3.11

and 3.12 below, after these have been introduced.

( 3. 1)

3.

71

Indexed limits and colimits

3.1

For a discussion of completeness, or of Kan extensions, or of

density, in the context of

V-categories, ordinary limits - defined

in terms of the representability of cones - do not suffice; we need the wider notion of indexed limit.

These latter fully deserve the name

of "limit", since they enjoy all the formal properties expected of a limit-notion, and include the classical cone-type limits as a special case.

When V

= Set,

V,

in contrast to the situation for a general

an indexed limit may be expressed as an ordinary limit; yet even here, the indexed limit is a valuable concept in its own right - its meaning is scarcely clarified by its construction as an ordinary limit.

Since

the cone-type limits have no special position of dominance in the general case, we go so far as to call indexed limits simply "limits", where confusion seems unlikely. We may think of a a small one when

V-functor

F: K --+ V as an indexing type -

K is small, which will be the usual case, but not

the only one.

Then we may think of a

diagram in

of type

V-functor

B

B(B,G-):

of the object

For each

K --+ V;

B

E

B,

G:

K --+ B as a

we get from

G a

and we can consider the existence in

[K,VJ(F,B(B,G-)).

surely does when V-functor

F.

V-functor

If this exists for all

B

V

as it

K is small- it is as in (2.18) the value of a

B0 p--+

V.

If this

V-functor not only exists but admits a

representation B(B,{F,Gj) - [K,V](F,B(B,G-))

with counit say

( 3. 1)

J. 1

)1:

(3 .2)

F __,. B({F,G},G-) , G indexed by

({F,G},Jl) the limit of

we call the representation

An evidently equivalent formulation not mentioning is the following:

each component B(B,{F,G})

V-natural in

B

and

(3.3) is an end over

K·'

JlK

----+

of

K for each

B.

induces by Yoneda a family (3.3)

is the limit precisely when

In practice, by the usual abuse

of notation, we call the representing object to the background the counit

explicitly

[FK,B(B,GK)]

({F,G} ,Jl)

and

Jl

[K,VJ

F.

{F,G}

the limit, consigning

or equivalently the choice of the

Jl

V-natural isomorphism (3.1). If we take the view of 2.6, that a

V-category, then as a

[K,VJ

always exists, if not as

V',

V'-category for some larger

be concerned with the existence of

[K,VJ(F,B(B,G-)),

representability in the form (3.1) for some

{F,G}

E

but only with its

B.

representable, the right side of (3.1) in fact lies in left side does.

we need not

If it is so

V,

since the

We shall in future take this view whenever it is

convenient - usually without bothering to change "V-functor" to

"V '-functor", and so on. Applying

V

to the isomorphism (3.1) gives a bijection of sets

B0 (B,{F,G}) but the requirement that

~

[K,VJ (F,B(B,G-)) ; 0

( 3. 4)

({F,G},Jl) induce·a bijection (3.4) is, except

requirement that i t induce an isomorphism (3.1), and doPs not suffice to

make

({F,G},IJ)

t11f? limit; see 3.7 below.

However i t does suffice to

(3.2)-(3.8)

detect

73

({F,G},~)

as the Zimit if the ~imit is known to exist; for

the right side of (3.4) admits at most one representation to within isomorphism.

a: F --+ B(B,G-)

An element

may be called an

(F,B)-cyZinder ove~

G;

of the right side of (3.4) then the bijection (3.4)

is the assertion that every such cylinder factorizes through the counit-cylinder

An

u

as

B( f, 1)

indexed coZimit

in

but it is usual to replace

G: K --+ B,

still have

of G indexed by

F

\.1

for a unique

f: B

---->- { F,

G}.

B is nothing but an indexed limit in

K

by

K0 p

but now have

B0 p;

in the definition, so that we

F: K0 p

--+

V. F

is the representing object

Then the co~imit

* G

in

(3.5)

with unit

(3.6) When

B

V,

we have from (2.13) and (3.1)

{F,G}

= [K,V](F,G)

either side existing if the other does.

for

F,G:

K--+ V

(3. 7)

Hence (3.1) and (3.5) can be

written

B(B,{F,G}) - {F,B(B,G-)},

B(F*G,B)

~

{F,B(G-,B)}

( 3. 8)

Thus, just as classical limits and colimits in an ordinary category

B

are defined, via representability, in terms of the primitive limits sets of cones - in

Set,

so indexed limits and colimits in a

V-category

B are defined, via representability, in terms of indexed limits in

V;

and thus ultimately, by 2.1 and 2.2, in terms of ordinary limits in

v0 .

3. l

74

Again, in the special case ['FK,[GL,B]] ~ [GL,[FK,B]]

the

V-natural isomorphism

gives by (3.5)

for The

B = V,

F:

0

K P --+ V and

(3.9)

K--+ V

G:

V-natural Yoneda isomorphism (2.33) immediately gives the

values of

{F,G} and

F

* G when F is representable, for any

K--+ B; namely

G:

{K(K,-),G}

~

More precisely, the cylinder exhibits

GK

as

K(-, K)

GK,

*

GK_: K(K,-) --+ B(GK,G-)

{K(K,-),G}.

(3. 10)

G ;; GK •

~s

counit

We refer to (3.10) as Yoneda isomorphisms;

by (3.7), the first of them includes as a special case the original Yoneda isomorphism (2.31). When

K

is small, the maps

(F,G)

are by 1.10 the object-functions of 0

{,}: ([K,VJ P ® [K,BJ)'--+

r---+

(F,G) f---+ F

B,

0

*: ([K P ,V] (F,G)

®

(F,G).

for which

We call the limit

V-natural in

or

V-natural

if

K and

G.

(3.12)

{F,G} a sma~~ limit if the indexing-type

K--+ V is small; that is, if K is small.

B eomplete

{F,G}

(3.11)

Then (2.37) gives:

The Yoneda isomorphisms (3.10) are

3.2

* G

[K,BJ)' --.. B ,

* G exists; and in such a way that (3.1) and (3.5) are

in

F:

and

V-functors

where the domain consists of those pairs F

{F,G}

We call the

V-category

it admits all small limits; and, dually, cocomp~ete i f

it admits all small colimits.

By (3.7) and 2.2,

V is complete;

shall see in 3. 10 below that V is also cocomplete.

we

(3.9)-(3.15)

75 T: B--->- C

A V-functor

is said to prese!'Ve the limit

{F,G}

if

the composite cylinder

F ~ 8({F,G},G-) ~ C(T{F,G},TG-) ~

where

is the counit of

in other words, if

{F,G},

exhibits

T{F,G}

(3.13)

as

{F,TG};

exists and if the evident canonical map

{F,TG}

T{F,G}---->- {F,TG}

(3.14)

is an isomorphism.

We call preserves all

T: B---->- C continuous (the dual is cocontinuous) if it

smaZZ limits that exist; of course this condition is of

interest particularly when The representables

B is complete.

B(B,-): B--+ V are more than continuous:

by (3.8) they preserve every limit that exists; while in their totality, by definition, they "detect" the existence of

{ F, G}.

For a very general result on the preservation of limits, consider a functor those

Q: Cop B

0

B--+ V, and let B' c B be the full subcategory of

for which

Q(-,B)

is representable, say as

we shall refer to the objects of Let

{F,G}

for each

be a limit in C.

its values in

Finally, let

B'.

B

~

C(C,TB);

B' as "those B for which TB exists".

which is preserved by

TG

Q(C,B)

Q(C,-):

exist; that is, suppose that

B--+ V G takes

Then we clearly have T{ F, G} - { F, TG} ,

(3.15)

3.2

76 either side existing if the other does; for, using (3.7), the two sides of (3.15) represent the isomorphic ~

[K,VJ(F,C(C,TG)) If then

T:

~

[K,VJ(F,Q(C,G-))

Q here is defined by

B'

V-functors

exist.

and

{F,Q(C,G-)}.

Q(C,B)

= B(SC,B)

S: C--+

where

--+ C is the partial right adjoint of

preserves all limits, we have (3.15) whenever TG

Q(C,{F,G})

Since

S.

{F,G},

T{F,G},

B,

B(SC,-) and

partial right adjoints,

This conclusion may be expressed as:

in so far as they are defined, preserve limits. In the special case where

S

right adjoints preserve aZZ Zimits that exist.

the conclusion becomes:

3.3 We now consider limits in a functor category G: --+

K--+ [A,BJ

B.

B--+ C,

in fact has a right adjoint T:

correspond under

th~

[A,BJ.

adjunction (2.20) to

Let

P: K ®A

In order to avoid formulae containing various letters whose

inter-relation must be explained in a subordinate clause, let us write

(2.22) with cyphers for variables, in the form in particular, Then, if [A,BJ

(G-)A

denotes

{F,(G-)A}

P(-,A):

exists for

P(-,?) =(G-)?;

K--+ B.

e~oh

A,

Let

F:

the limit

so that,

K--+ V. in

{F,G}

exists, and we have {F,G}A

For, if we define

{F,G}

given by (3.16), we have

to be the

{F,(G-)A} . V-functor whose value at

(3.16) A

is

( 3. 16)

77

fA

[A,8](H, {F ,G})

-fA

B(HA,{F,(G-)A})

by (3.16)

[K,V](F,B(HA,(G-)A))

by (3.1)

fA

- [K,VJ(F,

B(HA,(G-)A)) by (2.3)

[K,VJ(F,[A,BJ(H,G-))

by (2.10) .

In these circumstances we say that the limit

{F,G}

exists pointwise; it comes to the same thing to say that and is preserved by each

[resp. cocomplete]

existing pointwise in When in

[A,BJ,

B

all small limits

[A,BJ,

[A,BJ

exists

{F,G}

Clearly if B is complete

EA: [A,BJ --+B.

so is

in

[resp. colimits]

[A,BJ.

is not complete, fortuitous non-pointwise limits may exist

even when

V =Set.

Take

A= 2

and let

B be the category

generated by the graph

with the relation

hf • hg.

Then the map _ _ _f=-_

(f,h): g--+ h

given by

_,.

h

g

------+

h

in x

[2,BJ

is monomorphic, although

h

is not monomorphic in

is monomorphic in any category precisely when

B.

Since

3.3

78 . __!__ _ _,.

X

is a pullback, the pullback of -pointwise limit.

[A,BJ

B·,

by itself in

is a non-

[2,8]

The existence of a non-pointwise limit in

[A,BJ

says something about

A and

(f,h)

[A,BJ

as a category, ignoring its relation to

while the existence of a pointwise limit says something about

seen precisely as the functor category, and is really a

completeness assertion about

E.

[K,V]

Applying the above, in the colimit case, to

where

K is

small, we get from (3.9) and (3.10) a pointwise colimit

F

K--+ V,

*

y

(3.17)

F '

for any

F:

(2.34).

This gives a canonical expression of any

(indexed) colimit of representables. limits exist in for

K0 p--+ [K,V=

Y being the Yoneda embedding

E Y ~ K(-,K): K P K

--+

poin~ise

K--+ Vas an

F:

Y preserves ~hatever

In contrast,

sending them to 0

of

limits

i~

V.

We can now give the general Fubini Theorem relating repeated limits to double limits.

Suppose that

latter corresponding.to

P: K ®A--+ B,

exists pointwise; and let

for the functor sending

F: K --+ V and

H: A --+ V. (K,A)

to

G: K

--+ ~A,B~,

are such that the limit Writing

FK

® HA~

{F

®

F

®

H: K

®

the {F,G}

A --+ V

we have a canonical

isomorphism (H,{F,G}}

either side r.xisting if

th~

=

other does.

H,P\

(3.18)

(3.17)-(3.19)

79

For by (3.1) the left side of (3.18) represents [A,VJ(H?,B(B,{F,G}?)),

= [A,V](H?,B(B,{F-,(G-)?}))

by (3.16)

= [A,VJ(H?,[K,V](F-,B(B,(G-)?)))

by (3.1)

I A [HA, f [ FK,B(B, (GK)A)]] K

by (2.10)

-

I A f K [HA,[FK,B(B,(GK)A)]]

by (2.3)

-

I I

by ( 1.27)

[HA

®

FK,B(B,P(K,A))]

A K

I I

and by the Fubini Theorem (2.8) we can replace

here by

A K

f

(A,K)'

giving by (2.10) and (3.1) the object represented by the right side of (3.18). The use of the cyphers "?" and "-" here to keep track of unexpressed variables is presumably self-explanatory. it by allowing

{F-,G-}

If we extend

as an alternative notation for

can write (3.18), suppressing the letter (H?, (F-,P(-, ?) I 1

(H?

®

G,

as

F-,

P(-,?))

{F,G},

we

(3.19)

Then there is no need for explicit mention of the functor category [A,BJ,

which may exist only as a

V'-category in the sense of 2.6; nor

for explicit mention of the pointwise existence of just the existence of

{F-,P(-,A)}

{F,G},

for each value of

A.

which is

3.3

80

In this notation the general Fubini Theorem gives at once the

generaZ interchange of limits theorem: and

P: K ®A~ B;

A and that

and suppose that

Let

{F-,{H?,P(-,?)}}

K.

H:

A-

V,

exists for each

{F-,P(-,A)}

exists for each

{H?,P(K,?)}

K --+ V,

F:

Then (3.20)

{H?,{F-,P(-,?)}},

=

either side existing if the other does. 0

F: K p ~

For the colimit form of (3.19) and (3.20), let H:

A0 p

~

V,

P: K

and

®A--+ B;

F- * (H? * P(-,?))

then, under similar hypotheses,

(F ®H)* P =

=

V,

H?

*

(3.21)

(F- * P(-,?))

In fact (3.20) also follows alternatively from the general principle of (3.15); for the right side of (3.1) clearly preserves rointwise limits in the variable

G

E

[K,B].

FE [K,VJ,

In the variable

the right side of (3.1) sends

arbitrary colimits, by (3.8), to limits; of course all smal: colimits in

[K,VJ are pointwise, since we have remarked that

Hence we get from the principle (3.15) the general

to be cocomplete.

theorem of the continuity of a limit in its in~qx: G: K--+ [A,VJ, tJ~t

each

T: A--+ B;

and

{GK,T}

exists.

and supp0se that

=

and

T

F: K0 P

--+

V,

F * G exists and

{F,{G-,T}}

eithf'r eide existing if thq other do•:s. F

Let

Then

{F * G,T}

culimits, with

V will be shown

(3. 22)

The corresponding form for

as above but now with

0

G: K--+ [A P,ll], is

(3.20)-(3.27)

81 (F

*

*

G)

T

F

* ( G- *

T) •

(3. 2 3)

In future, when we equate limits in a general theorem, the phrase "either side existing if the other does" is to be understood.

3.4 We now consider,in the special case V

Set, the relation of indexed

limits to the ordinary classical limits. Writing

6B:

K --~ B for the constant functor at BE B, we have a

natural bijection [K,B](t1B,G)

=

[K,Set](LI1,B(B,G-))

between the set of (projective) cones over set of cones over

B(B,G-)

G with vertex

with vertex the one-point set 1.

or classical, or conical limit representing object

(3.24)

[resp. colimit]

lim G [resp. colim G]

of

B and the The ordinary,

G: K --~ B is the

in

B(colim G, B) - [ K,BJ (G,11B),

B(B,lim G) - [K,B](AB,G),

existing when the right side here is representable.

(3.25)

Using (3.24) we

conclude from (3.1) and (3.5) that lim G

{61 ,G},

colim G

K --> B, often called

Thus the classical limit of

G:

K,

G indexed by

is in fact the limit of

111

Al:

*

( 'J. 26)

G. a limit indexed by

K--+ Set.

Of course the

counit (3.2) now corresponds to the limit-cone lim G--+ GK .

~k:

Set,

taking

t

=

(3.27)

in the first cqualion of (J.25) gives

3.4

82 lim r.

!K ,Set }(I'll, G)

Z'

for

G: K

--+

(3.28)

Set ,

so that here the limit ie the set of cones with vertex l.

Then, as we said

in 3.1, all conical limits and colimits are defined representably in terms of these primitive ones in

Set;

for (3.24) and (3.28) allow us to re-

-write (3.25) in the form (analogous to(3.8)) B(B,lim G) = lim B(B,C-), Now consider an arbitrary

eZ

category K

E

F

K and x

f: K--+ K'

of elements of

E

in

functor sending

FK,

B(colim G,B) =lim

F: K F,

(K,x)

to

K,

l~e

Set.

(K,x)----> (K',x')

(Ff)x = x'.

Write

G: K

and let

---->

(3.29)

have as in 1.10 the

whose objects are pairs

and whose maps

K for which

---->

B(c~,B).

Set

with

are the e~

d:

(K,x)

F--+

K for the

be arbitrary.

Then

we have

lim (

eZ

F~

K ~Set)

~

tK,Set](F,C) .

For an element of the left side is by (3.28) a cone give such a cone is to give for each of

GK;

and the naturality of

naturality of

Cl

X

E

FK

(3. 30)

a(K,x): 1

an element

(:

X

K

---->

GK;

to

= a (K, x)

as a cone translates exactly into the

SK: FK----> GK.

Since a map of a representable Yoneda to an element

x

E

FK,

K(K,-)

into

F

corresponds by

we have a canonical inductive cone

(3.31)

in

LK,SetJ

whose base is indexed by

the expressibility of an!!

F: K----> Set

(al F) 0 P.

The classical result on

as a aanonicaZ (conicat) aolimit

of representabZes is the assertion that this is a colimit-cone; more precisely,

(3. 2 8)- (3. 34)

83

=

F

colim ((e~ F) p----+ ~P ----+Y [K,Set]) , dop 0

with unit (3.31), where

Y

is the Yoneda embedding.

(3.32)

This follows from

(3.30) and the characterization (3.29) of the colimit, since the left side of (3.30) is also

lim[K,Set](Yd 0 P-,G)

by Yoneda.

Because indexed limits are continuous in their index in the sense F ~ colim K(K,-)

of (3.22), the colimit-expression for any

G:

of (3.32) gives,

K--+ B, a limit-expression {F,G} =lim {K(K,-),G};

so that

by the Yoneda isomorphism (3.10) we have {F,G}

eZ

lim (

F ~ K ~ 8)

(3.33)

,

of which (3.30) is a special case by (3.7).

V = Set,

We conclude that, when

the classical conical limits are

special cases of indexed limits by (3.26), while the general indexed limit can be expressed as a conical limit by (3.33).

K is small, a Set-category

Since

eZ F is small when

B is complete in our sense of 3.2 above

exactly when it admits aZZ smalZ conical Zimits - that is, when it is complete in the classical sense.

functor

T:

It follows in the same way that the

B-+ C is continuous in our sense of

3.2 above exactly when it

preserves aZl emaZl conical limite. For

F: ~p

--+

Set

and

G: K

--+

B,

the corresponding colimit-

-form of (3.33) is

F * G We had in (3.17), for any as a colimit

F

0

colim (( el F) p

V,

(3.34)

K---+ 8) • G

the general expression of any

F:

K --+ V

* Y of representables; observe that when we use (3.34)

3.4

84 to evaluate that, when

*

F

Y in the case

B =Set,

V =Set,

~1:

components of

K--+ Set

K,

Note too

there is a second expression (3.34) for

since this is then isomorphic to colimit of

were-find (3.32).

G

* F by (3.9).

is clearly the set

taking

G

= ~1

F

*

G,

Finally, since the

n(K)

of connected

in (3.34) and using (3.26) and (3.9)

gives TI(

colim F

el

for

F)

F:

(3.35)

K--+ Set .

We conclude this section with some remarks about conservativeness and faithfulness for ordinary functors

equalizers and T preserves equalizers, conservative; for if T

Tf

=

Tg,

B.

n,e: S

First, if

A has

T is faithful if it is

the equalizer of

Next, if

to an isomorphism.

T: A --+

f

and

g

is sent by

--1 T: A--+ B is an adJunction,

T is faithful if and only if each

eA: STA--+ A is an epimorphism;

this is immediate from (1.53), for

!AA'

each

A(eA,A')

is.

Finally, if T

is a monomorphism exactly when

is a right adjoint as above, and if

A admits equalizers, then T is conservative if

and only if each

£A

is an extremal epimorphism (one that factorizes through no proper subobject of its codomain). To prove this, first note that any through no proper subobject of exist.

Suppose then that

factorizes through a 1: TA--+ TA

T

f:

A~

B is an epimorphism, because equalizers is conservative.

~onomorphism

i: B--+ A,

If

i

is.

Conversely, let each

epimorphism; then, by the remark above,

T

f: A--+ B

Tf

is surely a monomorphism if

£A: STA--+ A

then (by the adjunction)

factorizes through the monomorphism

isomorphism, whence

B in A which factorizes

Ti;

hence

Ti

is an

be an extremal

is faithful,

so that

is an isomorphism.

But

(3.35)

85

in this case, by the naturality of that

f

E, we have

EB = f.EA.(STf)-1;

so

is an isomorphism.

V, we now consider some properties

3.5 Returning to the case of a general

of (indexed) limits in relation to fully faithful functors.

A fully faithful for F:

J: A --+ B refleats limits; in the sense that,

K --+ V and G: K --+ A,

A as the limit

{F,G}

in

F --+ A(A,G-) --+ B(JA,JG-)

a cylinder

expresses

it is the assertion that

{F,JG},

If the full subcategory

JA

as the limit

if it lies in

~:

B(JS{F,JG},JG-)

By 3.4, the cylinder f: JS{F,JG} we have of

A,

--+

fn = 1.

~

{F,JG}; Hence

A,

~

factorizes as

exists,

B

and

By (1.48) the

B({F,JG},JG-) B(f,1) ~

is

B(n,1).

for a unique

and by the uniqueness part of the same result {F,JG}

is a retract of the object

and so by 1.11 lies in the repletion of

fuZZ subaategory

{F,JG}

which is therefore

F--+ B({F,JG},JG-).

A(S{F,JG},G-)

B(n,1)- 1 ~

B.

in

is also

n,1: S --4 J: A--+

To see this, let the adjunction be

isomorphism

A,

A here is reflective, and

let the counit of the limit be

{F,JG}

A is a full subcategory of

then the latter is isomorphic to an object of {F,G}.

expresses

A whenever the composite cylinder

This is immediate; and in the case where

B,

F--+ A(A,G-)

A.

A of B is alosed under limits,

JS{F,JG}

Thus a refZeative

and in partiaular is

aompZete if B is so. As for colimits in this reflective situation, if the colimit exists,

S

preserves it by 3.2, giving

S(F

particular, therej1eative fuZZ subaategory

B os.

*

JG) = F

A of

*

SJG

=F *

F

* JG

G. In

B is aoaompZete if

3.5

86 Consider now a family in a

B,

subcategory of

such that each

B,

V-category

= (F y :Ky op

¢

with the property that, if any

Fy

*

A

called the aZosuroe of

c.

lies in

¢

-type in

F; then

type

A

If

C containing

A

c,

F=

lF 0

in the

v to be conservative,

be a strong generator for

It is of course true for

vo'

V

= Set,

v0;

i t is

and for

for instance, sets on which a commutative monoid

acts, with a tensor product analogous to that for modules over a commutative ring, Hhat is true when any

X

E

V , 0

V is conservative, by Proposition 3.41, is that

although not the canonical colimit of the maps from .I,

3.9

98

f I}

is in all reasonable cases in the closure of it is, so to speak, an

iterated colimit of

I.

under small colimi ts;

In this case an

F: K ~ V is an iterated colimit of representables; and so

arbitrary

B does imply the existence

the existence of all small conical limits in

{F,G},

of all indexed limits

as we shall show formally in 3.10 below-

{F,G} cannot be expressed as a single conical limit.

even though

In the study o: enriched universal algebra, it is an important

Cat

observation that, for many

V

isomorphism whenever

left exact; see

F

is

including

Gpd,

and

(3.56) is

an

[45].

Before ending this section, we look at the possible existence in certain cases of more general conical limits.

The notion of a cone

involves the notion of a constant functor; yet in general there is no such thing as "the object

B

B may be seen as a

E

canonical

K--+ I;

V-functor

There is certainly a unique terminal

1(0,0), --+

V

I

K

V-functor

~~.

I

CGTop*,

S,

K--+

of

0

E

B ".

The

but there is no

V

0

where

but there is no

;

V

in

is the

V-functor

which is false for

0

and so on.

conical limits as above, taking for

1,

and with, for hom-object

is a retract of

DG-R-~~od,

= R-Mod,

B

in fact there may well be none at all.

V-functor

the terminal object

I

V-functor

V-category with one object

unless

B constant at

This is why we have defined

K a free

there are always constant ordinary functors

V-category

L

--+

LV;

for

B • 0

Yet in the special case of

cart£?$ian closed

other cases as well), it happens that V-categories 11

l'eser>ve this right Kan extension if

P~·

P: B

exhibits

~

PT

V as

is said Ra~PG.

Obviously:

Propositi on 4.14

P

pr>eserves thr riaht i:a;~ cxk"lsio;z

only if P preserves the limit

{KC,G}

for a?"

c.

Ra~G

if anC:

0

In particular, therefore, a r>ight adjoint preserves any right Kan

extension that exists; and the representables

B(B,-):

B--+ V not

( 4. 13)

(4 .10)-(4.19)

on"ly presePVe

115

RanKG

but, in their total-ity, detect it (which by (3.7)

and (4.9) is a re-statement of Theorem 4.6(iii)). For the dual notion we observe that a diagram

A -----G-=------->- B induces, for any

H: C0 p

--+

V,

a map

0

(K,¢)*: HK p we say that

along

¢ exhibits

S

*

G----+ H

¢

*

S

(4.16)

as the "left Kan extension

K if (4.16) is an isomorphism for each

exists; equivalently, if

( 4 .15)

of G

LanKG

H whenever either side

induces an isomorphism of

S with the

functor

(K-) * G ,

( 4. 17)

which exists when the colimit

KC * G does so for each cocomplete.

C:

C(K-,C)

as it surely does if

(4.18)

A is small and

B

is

Then (4.16) becomes

for aU Any left adjoint

H:

C0 P

-+

V ,

Q: B --+ V preserves the left Kan extension LanKG;

and a representable extension in

* G

8(-,B): 8°p--+

V - giving

V turns it into a right Kan

(4.19)

4.1

116 0

[A P,VJ(C(K-,C),B(G-,B))

B((La~G)C,Il)

as a further equivalent characterization of

La~G.

(4.20)

Since left Kan

extensions occur more than right ones in our applications below, we shall give our further results in terms of these; of course the two notions are precisely dual,

)

0

p

turning a left Kan extension (4.15) into a

right one. For the genesis of these ideas, see Kan

[38J.

4.2 We continue with a collection of elementary results, formulae, and examples.

The accepted term "extension" is somewhat misleading here, for in the diagram

(4.21) (LanKG)K~ G,

it is not in general the case that The fact is that

Kx:

0

A--+ [A P,VJ

we have a canonical map

sends

A

as (4.34) below shows.

to

K_A: A(-,A) --+ C(K-,KA),

A-component of a canonical map

K:

KKA

~ C(K-,KA),

and

which is the

Y--+ KK: A--+ [A 0 P,Vl

where

y

is the Yoneda embedding: and clearly v: Y ~ KK ir; an iBomopz,hism ~·f m1C: rov:?:J if K is f'A.lZ:· f7.ithfuZ.

(4.22)

We now have:

Proposition 4. 23 fr>.ithfuZ; u!hac·

1-: G -·· (LanKG)K r!MllFr•.-;dy

K

z.r· an icomory·hism if

i~ fuZZy faiihrul if

rp

K 7-s fuUy

in an isomor>phism

(4.20)-(4.27) for all

B,G

I I7 ~.Jith

Lan G ;:J-:~st-in . :>. K

*

(LanKG)K = (KK-)

Proof

G by (4 .17), while

Yoneda isomorphism (3.10); clearly

~:

(LanKG)K--+ G is

hence it is by (4.22) an isomorphism if

8

converse take isomorphism and

K

=V

¢A=

and

G

* A(A,-)

(K-)

is

G by the

(K-) * G;

K is fully faithful.

For the

then by (3.9) and (3.10) the so that

KA'

K

is an isomorphism

C

is fully faithful.

When

A(A,-);

=

*

G ;; (Y-)

B has cotensor products [resp. tensor products]

the

formulae (4.10) and (4.17) have by (3.69) and (3.70) the more explicit forms Ra~G

-

fA

C(-,KA) ~ GA,

(4.24)

(4.25) either side existing if the other does.

V

In the classical case

=

Set we can use (3.33) and (3.34) to

express the Kan extensions by conical limits and colimits. that

el C(C,K-),

where

C/K;

an object is

(A

is some has

g: A--+ A'

el C(K-,C) (Ra~G)

E

C(C,K-): A--+ Set,

A,

with

= (K/C) 0 P, C =

(LanKG)C

=

f: C--+ KA), Kg.f

= f'.

Observe

is the comma-category

while a map

Similarly

(A,f)--+ (A' ,f') 0

C(K-,C): A p--+ Set

Hence (3.33) and (3.34) give

--c-+

B)

for

v

Set

colim(K/C ~ A --c;+ B)

for

v

Set

lim(C/K ~ A

again either side existing if the other does.

'

(4.26)

(4. 27)

4.2 118

Clearly (4.18) simplifies if for then

(La~G)C

= C(K-,C) *

Yoneda isomorphism (3.10).

K: A--+ C has a right adjoint

G ~ A(-,LC)

* G,

GLC

which is

L,

by the

Thus GL

e-

(4.28)

K-1 L •

if

Note that obviously:

Any functor Q: B--+ V preserves the LanKG cf (4.28) . We pass to some particular examples. Y: A--+ [A 0 P,vJ is

be the Yoneda embedding.

Let

G: A --+ B,

Since

YF

(4.29)

and let

0

= fA P,v~(Y-,F)

F by the Yoneda isomorphism (2.31), we have

-

(4.30)

y

Hence (4.17) gives (4.31)

On the other hand, (4.17) also gives

LanGY

(G-)

* Y,

that by the

S

A

in any commutative diagram

.\

where

K

that

A =

is initial and

d

iH a dof.

eZ

F:

3

F

for some

--+

Set.

We may as vJell suppose at once Then

P

corresponds as above

(4. 78)

139

to a functor FdP

=

FQK.

the form

Since a

B with

S: C --+

=

K

SK

dP

R: V

A

--+

If we write

a: ~1

and a cone

S: 1 --+ FQ.

with

R.

dR

Q,

R': D

It is evident that any

must coincide with

S,

A

Now

Q

and

J:

dR' = Q and

with

gives a reflexion

Cat

for the full subcategory of

0

/B

--+

dof/B.

Cat /B

[B,Set].

On the other hand

determined

--+

B,

fully faithful.

Here

and nay be called

left adjoint

~

sends

0

sends

S

V

is

F:

0

!B ,

B--+

Set

( 4. 78) ·to

d:

el

F

the discrete Grothendieck construction; its to the

F

of (4.77).

While this is special to the case to the important cases

dof/5

Thus we get an adjunction

Q ~ 0: CB,SetJ ----+ Cat

0

=

Gpd

and

V

V

=

=

Cat,

Set,

it has generalizations the latter being the

setting for the original Grothendieck construction

[34].

We do not give

these here, for they transcend the context of. enriched category theory as developed in this account, and involve "transformations" more general than the

Gpd-natural or

Cat-natural ones; we shall treat them in

l46). Note finally that

Q8S

means by (4.76) that, for any extension

p

that are dofs, it is clear that this factorization

evidently equivalent to

with

R'K

This completes the proof.

dof/B

A --+ B

S

and clearly with

0

by those

=

--+ FS

is initial, this cone is by Theorem 4.67(ii) of

for a unique cone

constitute a functor RK = P.

=

~

S, F:

since

B

S

is fully faithful.

--~Set,

we have a l~ft Kan

This

4.7

140

el F

(4.79)

- - - - - f 1 , -, - - - - - - +

1

since this asserts by (4.77) that

FB- colim B(d-,B),

it is precisely

a restatement of (3.32).

4.8

In this section and the next we look at conditions ensuring the

representability of a

V-functor

simplest case of a complete T: C--+

B

C.

F: C

~

V,

especially in the

Since by 1.11 the

has a left adjoint precisely when each

V-functor B(B,T-)

is

representable, the same considerations yield conditions for a left adjoint to exist. Since a colimit in functor

C--+ V,

C

is itself the representing object for a

the representing object for

F: C--+ V,

exists, is in some sense a "generalized colimit".

if it

Representing objects

are, in fact, very commonly constructed as colimits: consider some examples.

First, in the case

V = Set,

a representation for

corresponds, as we observed in 1.10, to an initial object in that is, to the colimit of the empty diagram in Proposition 3.36, a functor

el F.

F

elF;

Next, in

F: C --+ V was in effect given as an

iterated limit of representables; and we constructed its representing object as the corresponding iterated colimit of the representing objects for these.

Again, the left Kan adjoint

constructed via the colirnits

KC *

G.

LanK

of Theorem 4.50 was

A free abelian group is

constructed as a coproduct of infinite cyclic groups; the tensor

(4. 79)

141

product of two abelian groups is constructed as a cokernel of a free abelian group; and the free monoid on a set

In~O Xn.

X is the coproduct

More generally, numerous constructions of free algebras, free

monoids, and free monads, as iterated colimits, under cocompleteness but no completeness hypotheses, are given in

[44].

Yet the only general necessary condition that we so far have for the representability of

F

is expressed in terms of limits: by 3.2,

F

must preserve whatever limits exist.

F

must be continuous (and similarly for a

a left adjoint).

Hence if

C is complete,

T: C --+ B

that is to have

We here consider what must be added to this continuity

to ensure representability (or the existence of an adjoint).

Since

the desired representing object is a generalized colimit, this is analogous to the question of what must be added to completeness to ensure cocompleteness - or rather, to ensure the existence of a particular colimit. A smaU complete set

-

for if

at least

2n ,

cardinals

n.

C(A,B) making

Set-category

c

had at least two elements,

c

large if

Bn

C(A,Bn)

would have

existed for all small

Hence, by the classical argument on complete lattices,

a small complete

C is also cocomplete:

the infimum of its set of upper bounds. even if

is of necessity a preordered

C is a preordered set:

cocomplete but not complete.

the supremum of a subset is Not so, if

C is large -

the ordered set of small ordinals is

What must be added is a condition ensuring

that the colimit we need can be manufactured from smaZZ limits. Similarly in the case of representability of condition needed is a "smallness" one on

F.

F: C--+ V;

the extra

4.8

142 We begin with a general result expressing a representing object as a limit:

a

Theorem 4.80

C

~;

and the counit

{F,1},

happens to be small.

F: C--+ V is representatle if and only if the limit

exists and is preserved by

{F,lC}

is

large one, unless

Then the representing object

F.

of this limit is an

F--+ C({F,1},-)

isomorphism giving the representation of F.

Proof

One direction is immediate:

then by (3.10) the limit --+

C(D,-);

isomorphic

if

{C(D,-) ,1}

F is

is the representable D,

similarly, with appropriate changes, when to

C(D,-);

u =

with counit

and the representable

F

F

C(D,-), 1: C(D,-)

is only

preserves this limit.

For the other direction we have by hypothesis that

F

~

is the counit of the limit over

F

{F,F}.

foraunique

--+ F

for the composite

prove

u

C({F,l},-)

i

[g,1]F,

is

Since

through itself as =

I.

has the form

we therefore have

J.jJ = 1;

and to

UA: C({F,1},-) --+

l.

AU= 1

UA

is

C(f,1)u

for a

C(f,1) UA~

we have

However, as the counit of the limit

J.!.

(F,I)-cylinder

\hilin,;

an isomorphism it remains to show that

But by Yoneda, this map --+ {F,1}.

i; F--+ [I,F-J

g:l->F{F,lt. - l

[F{F,l},F-]

Hence by (3.4) the

given by the isomorphism

i=[g,l]FJ-

~

C({F,1},-)

=

u,

{F, 1},

unique

f;

for a unique

f: {F,1}

or

u, = C(l,l)u.

C(f,l)u

the cylinder whence

f

=

u 1

factorizes and

(4.80)-(4.83) There is a corresponding adjunction theorem:

( l.i)

r\:J GS

Theorem 4.81

1

(' Ii)

T: C ---+ B has a left adjoint if and only if Ran 1

exists and is preserved by

and the counit

T.

Then the left adjoint

-

f

g,h: G

of maps

"I

M

the inclusion.

g

and

E

--+

X

h.

D

-->-

be a small

A be the full

and let

-->- D

c

For any

€

c,

is

X

f.

Write

Eg = Eh,

such that

M

X

Gx'

let

c be the canonical map whose

C(G ,C),

X

• G

X

X XEX

de terrnined by the small coproducts of the

E: D = LxEX C(G , C) • G X X -component, for

{G }

let

c,

set of objects in the cocomplete category

~X

= KC

F

(x,f)-

for the set of pairs

and let

,\j!:

E =

(x,g,h)-components are

be the maps whose respective

The commutative diagram (4. 92)

E -----'------+ D ----+ C -----+

E

is not in general a coequalizer diagram; but for each

x

the diagram

C(l ')

( 4. 9 3)

is a split coequaZizer diagram. and C(1,E)t = 1,

C(l,)s = 1,

That is to say, there are maps

s: C(G ,D) --+ C(G ,E) X

and

C(l,\j!)s

implies that (4.93) is a coequalizer. the copro.iection

ix,f

X

= tC(1,E);

To wit,

of the coproduct, while

the coprojection corresponding to the index

for which which easily

t(f): Gx--+ D is s(u): Gx--+ E

(x,u,i

X, EU

).

Any

is

4.10

152

product of the diagrams (4.93) over a family

(xj)

of indices

clearly again a split coequalizer; whence it follows that

x

is

K sends

(4.92) to a coequalizer diagram

A(-,E) ----t A(-,D) ~ KC

in

[A 0 P,setJ.

along

K:

G: A--+ B

We conclude that the left Kan extension of

A --+ C

exists whenever

B admits coequalizers, and is

given as the coequalizer of the two maps

G~,

G~:

--+

GD.

remark that, when (4.92) is a coequalizer diagram for each

C,

C·,

is said to be a regurar generator of

GE

(We

{G } X

it is easily verifi.ed to be

a fortiori a strong generator and a generator.) Clearly Theorem 4.91 could be so modified as to cover the case

KC

where

was expressed as an iterated

colimit of representables.

There is a particularly neat and useful result of this kind when

K

is fully faithful; which we now suppose. Let let

C

E

F: C,

L0 p--+ V, and let

let

P: L--+ C be a diagram of type

a: F--+ C(P-,C)

in the language of 3.1.

Let us say that

be an a

F,

(F,C)-cylinder over is a

P,

K-cyZinder if the

composite cylinder

F

-a--+

C(P- ,C)

is a colimit-cylinder, giving

KC

F * KP.

For this it certainly suffices that, for each

F

~

C(P-,C)

C(KA,-)

(4.94)

A c A,

the cylinder

[C(KA,P-),C(KA,C)]

(4.94)-(4.97)

~3

be a colimit-cylinder, so that C(~~.C)

F * C(KA,P-)

-

(4.9~)

this is just the pointwise existence of the colimit (4.94); and the pointwiseness is automatic if An expression of

KC

by 3.3

L is small.

as a colimit

F

* YQ of representables,

as in Theorem 4.91, has as unit a cylinder

S: F--+ [A0 P,V](YQ-,KC).

The codomain of

S

moveover, since

K is fully faithful, we have

It follows that

S

(KC)Q 0 p

is isomorphic by Yoneda to

Y

~

KK

=

C(KQ-,C);

by (4.22).

has the form

F ~ C(KQ-,C)

(4.96)

in which the second map is in fact an isomorphism. expression of as a

KC

Thus such an

as a colimit of representables is the same thing

K-cylinder in which

P

factorizes through

A

as

KQ.

In this fully-faithful case we have a characterization of LanKG

as follows:

Theorem 4.97 An extension S: C--+ B of G: A--+ B along the fuZZy-faithful

K:

isomorphic to

A --+ C is

La~G

(that is,

S

LanKSK) if and only if it sends each

colimit-cylinder.

If

is canonically K-cyZinder to a

A is small, it suffices to speak of

small K-cyZindePs.

Proof The canonical expression of KC as a colimit KC * Y of representables corresponds as in (4.96) to a C(K-,C).

If

which is

KC

S

K-cylinder

sends this to a colimit-cylinder we have

* G since S is an extension of G;

thus

1: KC--+ SC ~ KC * SK, S ~ LanKG,

4.10

15 4

Conversely, if

LanKG

exists, any

sent to a colimit-cylinder by

K-cylinder

K;

= K- *

~

G sends

Still supposing

(~ : F

=

Y

and P : i

--+-

Y

fully faithful, consider a family

Y

of

K-cylinders, where

YEf

F : L up Y

l'

Y

C is the closure of A under the family

of colimits, we say that

if there is no proper replete full subcategory

A,

~

Generalizing 3.5, where it was a question of a family

--+C.

y

- *G

KPA * G exists; hence

to a colimit-cylinder.

C (P - ,C ) )

Y

L

K

is, by definition,

by (3.23) the functor

preserves the colimit (4.94), since each LanKG

a

and such that

V

C E y

whenever

P y

V

of

C,

containing

V.

takes its values in

Theorem 4.98 Let K: A--+- C be fully faithful, and let C be the closure of A for a family

s: C --+ B is exhibited by

functor

only if s LanKG

of K-cylinders as above.

sends each cylinder of

exists for any

1: SK

--+

as

SK

Then a

La~SK

to a colimit-cylinder.

if and Moreover

A --+ B if B admits FY-indexed colinits

G:

for each y " r.

Proof Theorem 4.97 gives the necessity in the first assertion. the sufficiency, let consisting of those exhibits

SC

as

isomorphisms

V C

be the replete full subcategory of for which

KC * SK.

Then

we have a canonical isomorphism

of

V· '

F

*

SP

c·r V·' (

y

~

KC KC '(

thus

Ac V

SKA ;;:: YA * SK = KKA * SK.

v,

and

S: C(K-,C)

y

~

F * KP y y

* SK.

v

Since

is all of

c,

* SP

-

B(SK-,SC)

C as unit

since

we have canonical

p

takes its values in

y

KP - * SK = y

by (4.90.

F

If

--+

For

SP

y

by the definition

Hence (3. 2 3) gives

c

y

as desired.

by hypothesis, we have The second assertion is

(4.98)-(4.99)

155

proved by a similar argument, taking for which

KC

*

V

c

now to consist of those

D

G exists.

For an example of this, let us modify the example following Theorem 4.91.

With the notation as there, write

c

subcategory of

A'

for the full

determined by the finite coproducts of the

G. X

Since every coproduct is the filtered colimit of the finite coproducts of its summands, every object of of

A'.

Suppose now that each

that each since

C(Gx,-): C--+ Set

A

is a filtered colimit of objects

Gx

is finitely presentable- meaning

preserves filtered colimits.

filtered colimits commute with finite limits in

by Theorem 4.73, each object of

A'

to

Z

is the inclusion

Z-cylinders.

Set

is finitely presentable; and thus

Z:

the filtered colimits in question are preserved by where

Then,

A' --+A--+ C;

C--+ [A' 0 P,setJ,

that is, they correspond

It now follows from Theorem 4.98 that any

--+ B admits a left Kan extension along

Z: A' --+ C if

G: A'

B

admits

coequalizers and filtered colimits. We end with a result, a special case of which we have already used in Theorem 4.51:

Theorem 4.99 Let K: A - - + C be fullw faithful, and denote by

[A,BJ' which

the full subcategory of [A,BJ LanKG

exists.

Then

LanK:

determined by thooe

[A,BJ'

--+

G for

[C,BJ is fuZZy faithful;

so that we have an equivalence

[A,BJ' ~ [C,BJ~,

the fuZZ subcategory of [C,BJ

determined by the functors isomorphic

to some K.

LanKG;

where the latter is

the inverse to this equivalence is composition with

If A is smaU, it follows that

[C,BJ~

is a

V-category.

4.10

156

Proof we have by Theorem 4.38 that [A,BJ(G,(LanKG')K), [A,BJ(G,G').

=

[C,B](LanKG,LanKG')

which by Proposition 4.23 is isomorphic to

II

of course this fully-faithfulness is just a special case of the observation in 1.11 following (1.53), to which we could have appealed: for we are dealing with an adjunction ~

is an isomorphism.

[A,BJ

I

:::

[C,Blt

LanK --i

[K, l]

In practice we use the equivalence

chiefly when

[A,BJ'

rA, B),

is all of

instance under the hypotheses of Theorem 4.98

or when

as for

A is small

B is cocomplete.

and

5.

whose unit

Density

5.1

A function

k: A--+ C,

not necessarily continuous, between

hausdorff spaces is dense (in the sense that its image is a dense subset) precisely when any continuous map from space when

B

into another hausdorff

is uniquely determined by its composite with

Set(k,l): Haus(C,B)--+ Set(A,B)

analogous definition of density for a

For a as a

C

V-category

B,

is injective. V-functor

V'

V'-category for some suitable extension

Consider those

S: C--+

whose indexing-type

that is,

We give an

of

exists at least

V

as in 3.12.

B which preserve every colimit existing in

F: A --+

V

full subcategory

A-Cocts[C,B],

full subcategory

Cocts[C,BJ

that exist.

'

K: A --+ C.

[C,B]

the functor category

k·

has domain which if

of those

S

A; A

C

these constitute a

is small contains the

preserving all small colimits

(5 .1)

157

Theorem 5.1 (i)

FoP a

V-functop

K: A--+ C the following aPe equivalent:

For any

B,

functor

[K,l]: [C,BJ--+ [A,BJ

(ii)

The funator

(iii)

For eaah

the restriction to

K:

C,D

€

0

C--+ [A P ,VJ C the map

K:

of the

A-Cocts[C,BJ

is fully faithfUl. is fuZZy faithful. C(C,D) --+[A 0 P,VJ(C(K-,C),C(K-,D))

is an isomorphism. (iv)

For eaah

C

€

C,

as unit exhibits (v)

The identity

(vi)

Some isomorphism

the identity aylinder

1: C(K-,C) --+ C(K-,C)

c as KC * K.

1: K --+ lcK

exhibits

¢:K--+ lcK

lc

as

LanKK.

exhibits

lc

as

LanKK.

Proof Because a representable C0 p --+ V preserves all limits that exist, its dual

C --+ ~p

preserves all colimits that exist in

Since the Yoneda embedding is fully faithful, and since

K

C.

is the

composite

c

y

it follows that (i) implies (ii).

The direct translation (iii) of (ii)

is equivalent by (3.5) to (iv), which is in turn

equival~nt

by (4.18)

to (v), itself a special case of (vi); so that it remains only to show that (vi) implies (i).

Suppose then that

it thus preserves the colimit in (vi), so that

KC

S¢: SK--+ SK

4.38 we therefore have for any [C,BJ(S,S')

~

[A,BJ(SK,S'K),

* K,

S: C--+

B

is

A-cocontinuous;

and hence the left Kan extension

exhibits

S

as LanK(SK).

By Theorem

S': C--+ B a representation

whose unit is

S¢.

By (1.48), this

5.1

158

isomorphism is the composite of and

r:K,l]: [C,BJ(S,S') ---+- [A,BJ(SK,S'K)

isomorphism since A

V-functor

cp

is,

K:

A ---+-

is said to be dense. the concept for

so is the former- which proves (i).

0

(The original term used by Isbell, who introduced

V =Set

0

0

K p: A P--+ C P

the dense

K:

A --+

in

was left adequate.)

[35],

is dense.

A--+ C

K:

is codense

In the important special case where

C,

leaving

is the one-object full subcategory

in

For the dual

C is the inclusion of a full subcategory, we merely

A is dense in

say that

D

C satisfying these equivalent conditions

notion (Isbell's right adequate) we use aodense; if

the latter being an

[A,BJ(S.P,l): CA,B](SK,S'K) ---+- [A,BJ(SK,S'K);

K

{A},

to be understood; and if we say that

A

A is dense

C.

It

is clear that the density of

class of

K in the

2-category of

K depends only on "the equivalence V-functors"; that is, if

K

is

dense, so is the composite

c where the outer functors are equivalences and where to

map

K'

is isomorphic

K.

When of

(5. 2)

C admits tensor products, the criterion (v) for the density

K: A--+ C becomes by (4.25) the requirement that the canonical

C(KA,C)

®

KA

--+

C express C

while when

V

= Set

= fA

C as the coend

C(KA,C) ® KA

it becomes by (4.27) the requirement that the

canonical cone express

C as the colimit

(5. 3)

(5 .2)-(5 .5)

159

c If

K:

colim (K/C

-A-C) d

K

A ~ C is dense, applying

V:

V

--+

0

(5.4)

'

Set to the isomorphism

in Theorem 5.l(iii) above gives a bijection of sets

= CA0 P,VJ 0 (C(K-,C),C(K-,D))

C (C,D) 0

whose meaning is that every C(K-,D)

is of the form

V-natural transformation

C(K-,f)

for a unique

f:

C

(5.5) ~:

_,.D.

clear that (5.5) is in turn equivalent to the density of conservative, and in particular in the classical case also equivalent to density if cotensor products

v0 :

genera tor of

XB

~

C

if

K

C for some set

{XB}

for if we write (5.5) with

constituting a strong X ~ D

in place of

0

2-category (3.54), and let {l};

isomorphism in

It is

C admits

0

0

is

V

= Set.

is cotensored; or even if

general (5.5) is strictly weaker than density:

subcategory

V

I t is

V (X,C(C,D)) ~ V (X,[A P,VJ(C(K-,C),C(K-,D))).

we easily get

C be the

C(K-,C) --+

take

V = Cat,

D,

Yet in let

K be the inclusion of the full

we have the bijection (5.5), but not the

V of Theorem 5.1(iii).

After first surveying various examples and elementary properties of density, we shall return in 5.11 below to the study of the important map

[K,l]: Cocts:C,BJ--+

[A,BJ of Theorem 5.1(i).

5.2 We begin by considering the composability and cancellation properties of dense functors.

The positive results below are for any

the counter-examples are all for

V = Set;

V, while

the assertions about the

density or the non-density of particular functors in the counterexamples are easily verified using (5.5).

We consider a diagram

5.2

160

/

~~

.

A

in which

""~.

,,

K

"· C

is an isomorphism, so that

L'


of diagrams in C such that each colimit and is

K-absolute,

K-absolute colimit of objects of A,

colimit of the form (v)

as unit

C(K-,C)

K-absolute, and such that

Y

F

y

*

P

y

YEf

exists

C is the closure of

A under this family of colimits. If A is small, the colimits in (iv) and (v) may be taken to be small.

Proof (i) implies (ii) since, when K is fully faithful, it reflects colimits by 3.5.

By the remark preceding the theorem, (ii) gives (iii)

as a special case.

Q

= 1;

if

A

family

Then (iii) gives (iv) by taking

and the domain of is.

F

here is

A,

F

= KC and

so that the colimit is small

From (iv) we get (v) by taking for

~

the one-object

(F,KQ). Finally, by Theorem 4.98 and Lemma 5.18, it follows

from (v) that Theorem 5.1(v).

1: K--+ K exhibits

0

1c

as

LanKK,

giving (i) by

5.4

172 The chief purpose of this theorem is to exhibit (v) as a practical criterion for the density of in

K.

C is presented (or exhibited) by the family

more loosely, by the family

F

y

A

We shall say that the density of

*

p

= (F ,P )

y

may call such a family a presentation of the density of or simple a density presentation.

y yE

r ; or, We

K-absolute colimits.

of

y

- C be the inclusion of the

full subcategory determined by the small set of objects in question. Since

K

Moreover so.

is strongly generating,

K:

0

C--+ [G P,VJ

is conservative.

K preserves all small colimits, since each

In particular it preserves for each

C

Since, by the remarks following Lemma 5.18,

E

C

C(KG,-)

the colimit

K sends

does

- *

KC

K.

1: KC--+ C(K-,C)

to a colimit cylinder, we conclude by the observations at the beginning of 3.6 that 1 is already a colimit cylinder, exhibiting Thus

K is dense by Theorem 5.1(iv); and consequently

C as

KC

* K.

K is fully

faithful. To prove

-

K an equivalence, it remains by 1.11 to prove it

essentially surjective.

Given

0

FE [A P,VJ,

the colimit

F

* K exists

5.5

176

because

A is small and

colimits we have

*

K(F

C is cocomplete; and since K) ~ F

* KK

=F * Y =F.

There may well be various inequivalent 0

[A P,VJ.

equivalent to [A 0 P,VJ

A·

'

A

CGTop; when V = Ab,

of representables; when

A

if

A,

A c

is small.

and for many

v

V • Set,

We shall show in 5.8 that, for

retracts of the representables.

If

it is small whenever

A consists of the V

The same is true for

to

A, A.

consists of the retracts of finite coproducts

V

A.

= R+,

Lawvere

The passage from

as the small-projectives in Then

A

[54]

A.

In all

A and

A ~ B. B· '

to

A

cXOP,vJ

is a closure operation; z [A

0

P,V],

is equivalent

A --+A

A and

B,

we have

is an

[A 0 p, VJ ~ [B 0 p, V]

This relation is called Morita equivalence

it was first studied by Morita

in the case· V

that is, rings

A

In these circumstances it follows at once from the

if and only if

= Ab

R and

the categories of right

s. '

with

A

and

[65]

B one-object

in which case

R-modules and right

situation has been studied by various authors [33]),

A to be

has shown

is Cauchy complete if the inclusion

facts above that, for small

[ 2])

or

V is such that this last is so, we may call A the Cauchy

equivalence.

between

= Cat

A is small when A is.

completion of for

and the

In general

the Cauchy completion of the (generalized) metric space these cases

C is

for which

denotes the full subcategory of

[A 0 P,VJ z [~P,V]

is strictly bigger than

A is.

A

D

determined by the small-projectives, then

proof above gives

A

If

A

K preserves small

(see also Ab-categories-

and

are

S-modules. ( [66],

The general

[68],

[27],

and may be expressed in terms of the profunctors or distributors

(5. 2 7)

177

introduced independently by Lawvere

A

profunctor from

[53]

and Benabou

[8 J;

B is a functor T: B ® A0 p--+ V,

to

these compose by the operation

S(C,-)

* T(-,A)

a

and

to form a biclosed

bicategory.

V,

For some

however,

v0

example is the category

A

A may be large when

separately.

A

x

X.

Here the Cauchy completion of the unit

For such

0

the relation between

V

V and any sma~Z

Theorem 5.27 For any

of the sma~~-projectives in

K:

the inc~usion, [A P,VJ

[A

0

V-category

[A P,VJ

[A 0 P,VJ.

[~P,VJ

and

is

[SS]:

A,

V-category Then if K:

P,VJ--+ c~P.v]

I

Px of subsets

X the lattice

more complicated, and has been given by Lindner

0

A ® B --+ C are

B --+ C which are sup-preserving in each variable

is not small, containing for each set of

An

of complete lattices and sup-preserving

maps, with the tensor product such that the maps the functions

is small.

~et

A

0

P,VJ

[A

A--+

consist is

induces an equiva~ence of

with the full subcategory Acc[~P,V]

determined by the

accessible functors; and the inverse to this equivalence is composition with the inclusion

Proof Since KZ

=

Z: Y:

A

--7

A.

A---+ [A 0 P,VJ

is dense by Theorem 5.13.

Hence

is dense, the fully-faithful

K: [A P,V]--+ [~P,VJ_ is fully

faithful, and it remains to determine its full image. F: is

A0 P--+ V

we have

rA 0 P,V](K-,F*Y).

KF

F? * A(-,Z?). ~p---+

V,

= rA 0 P,V](K-,F),

Since each

isomorphic to the colimit

F?

K

0

KB

For any

which by the Yoneda (3.17)

is small-projective, this is

0

0

* [A P,V](K-,Y?) = F? * [A P,V](K-,KZ?)

The last term here is a small colimit of representables

and hence is accessible by Proposition 4.83.

Moreover

g

5.5

178 composing it with

Z gives

that composition with

Z

F?

is the inverse.

K,

accessible is in the image of representable colimits. A(Z-,B):

A(-,B)

A p--+ V;

it suffices to show that every

A(Z-,B) =

=A(-,B),

verifying

To show finally that every

K preserves small

is in this image, since

but

=F,

~ F? *A(-,?)

Composing this representable with 0

= [A0 P,VJ(K-,KB)

KKB

* A(Z-,Z?)

[A

0

Z

gives

P,VJ(Y-,KB) ~ KB.

Now we have

D

as desired.

This gives at once:

Proposition 5.28 For any equivalence

[A 0 P,VJ

Of course, since

':>
J:em 5.35 is a special case of that of Theo~:em 5.31; what is new here- is the f.,.e/>"""s of the F-cocompletion, which allows us to replace F-Cocts[C,B].

K

Cocts [C,B]

by

(Often, using the word "completion" in a wider sense,

185 we may refer to the

under

F-cocompletion of

A as the completion of

A

F-ooZimits.) As the first example of such an

F-cocomp1etion, take

A

consist of aZZ small indexing types, so that

oooompZetion of A.

F

to

is simply the

From Proposition 4.83 and Proposition 5.34, this

cocompletion is precisely the

V-category

0

small, this reduces to

[A P,VJ,

Acc[A0 P,V].

and were-find

When

A is

(part of)

Theorem 4.51. Other examples with

V

~

Set will be found in the article [45]

on enriched essentially-algebraic structures.

V=

examples discussed below we take

Set.

For the remaining

The colimits we are

interested in will then be conical ones, so that

F

is in effect a

set of small categories. Before we proceed, however, we anticipate a possible misunderstanding that may arise from the very special nature of our main examplei. In the transfinite construction of 3.5 for the closure of

A under

F-colimits in

taking all the

0

[A P,V],

F-colimits in

0

[A P,VJ

the first step consists in of the objects of

important cases the full subcategory given by these itself closed under

F-colimits and hence constitutes

construction then terminates after one step. example above, where

A

A.

In some

F-colimits is A;

the

This was so in the

F consisted of all small indexing types; and

it will also be so in the three important examples of 5.8 - 5.10 below.

However, it is far from true in general.

taking

V = Set

henceforth, if

F

For instance,

consists of the two discrete

5. 7

186 categories 0

and 2,

coproducts, we reach

F

trivially, if

A

so that

A

w

only after

steps of the iteration.

consists of the single category

A

is the closure of

A

has two objects

A

A--+ P,

maps

0

==!

1,

Less

so that

under coequalizers, consider the case where

A, B

f,g,h: A--+ B.

and three non-identity maps P, Q, R

In the first step we add in the coequalizers respective pairs

A under finite

is the closure of

(f,g),

(f,h),

and

(g,h);

of the

but now there are two

and we must next add in the coequalizer of these, and

so on.

5.8

V

For our first main example with

= Set,

let

F consist of

the single category with one object and with one non-identity endomorphism

A

e

=

e.

Then a diagram of type

A with an idempotent endomorphism e.

is an object

has a colimit

e2

satisfying

r: A--+ B

in

A

splits; that is,

e

has the form

i: B--+ A with

ri

=

1.

F

This diagram

if and only if the idempotent

= ir

e

for maps

in

r: A--+ B

e and

Such a splitting is unique (to within

isomorphism) if it exists; so that the split idempotents correspond

A in

bijectively to the retracts of we may call the

In this case

idempotent-splitting completion of A,

of all the retracts in

0

[A P,Set]

it is evidently small if when

A.

A

is.

all idempot~nts split in

of the representables Moreover

B.

B

is

A,

which

clearly consists

A(-,A)

E

A;

F-complete precisely

In fact we have the result promised

in 5.5 above:

Theorem 5.36 For a small

A,

the small-projeatives in

[A 0 P,set]

are

precisely the retracts of the representables; so that the idempotent-

(5. 36)

187

-splitting completion of A is its Cauchy compZetion.

A

Moreover

A.

is Cauchy complete exactZy when all idempotents spZit in

Proof We saw in the proof of Theorem 5.26 that the representables in

[A

0

P,set]

are small-projective, whence so too are their retracts by

Proposition 5.25. -projective.

F: A0 p --+Set

Suppose conversely that 0

Then

[A P,Set](F,-)

preserves all small colimits, and

in particular the canonical expression (3.32) of representables. indexed by

el F) P, Set

0

[A P,set](F,F) with some

[A P,Set](F,F)

0

(

colimits in

0

Thus

of the sets

as a colimit of

0

[A P,set](F,A(-,A)).

is the composite of some A(-,A);

F

is the canonical colimit,

are calculated by (3.35),

i: F---+

is small-

that is,

Since

the element

IF

r: A(-,A)--+ F F

in

of

eZ F

is a retract of some

A(-,A).

To say that inclusion

A--+

A

A

isomorphic to one in

is Cauchy complete is, by 5.5, to say that the

A.

A.

Note that, in this example, the inclusion

A

are

F-colimits as exist in

A.

In fact

A--+

D

preserves

example is quite atypical, however: by no means preserves the

for a general

A(C,A +B)

A

This

F the inclusion

F-colimits that exist in

For instance, consider the completion of

F

F-colimits for this

absolute - they are preserved by any functor whatsoever.

A --+ A

is

Since idempotents split uniquely if at all,

this is exactly to say that all idempotents split in

such

A

is an equivalence, or that every object of

A.

under finite coproducts:

is in general quite different from

A(C,A) + A(C,B).

5.9

188

5.9

V = Set

Our next example with

A

is the completion of

under

finite colimits; since the number of finite categories is small,

the finite-colimit completion of A is small if A is.

Theorem 5.37 The completion of A under finite coZimits is the fuU suhcategor•y of [A 0 P,Set]

determined by aU finite co limits of

the representables.

Proof

A

Let

[A 0 P,Set]

c

consist of all finite colimits of the

representables; we are to show that Clearly

A

objects

F

and

S:

A

is closed under finite colimits.

contains the initial object of

=

colim YT

Q--+ A

and

P

with

G

with any vertex

YT

and

YS,

H

colim YS

Q

and

the functor with components YR

=

T

S.

0

P,Set].

A,

of

finite.

and

[A

Let

Consider

where R:

T:

P + Q--+

P --+ A

A

be

Since an inductive cone over

clearly consists of independent cones over

it follows that

colim YR

=

F

+ G.

A

Hence

is closed

under finite coproducts, and it remains to prove it closed under coequalizers.

With

F

and

G

(pp: A(-,TP)--+ F) a,8: F--+ G. satisfies each

P

€

ya

P.

as above, let the colimit-cones be and

(aQ: A(-,SQ)--+ G),

Since the

= y8 The

are jointly epimorphic, a map

if and only if it satsifies map

and consider

app: A(-,TP) --+ G

yapp

corresponds by Yoneda to

an element of

GTP.

Since

surjective in

Set,

this element is the image under

up e A(TP,SP')

for some

the

P'

E

aQ,TP: A(TP,SQ)

Q.

y: G--+ H

--+

GTP 0

are jointly

P', TP

of some

By the naturality of Yoneda we

then have commutativity in the left diagram of

(5. 3 7)

189 A(-,~)

A(-,TP)

A(-,vp)

A(-,SP')

A(-,TP)

,,j

j'p•

'Pl F

j'P" F

G •

a

A(-,SP")

G

B

while the corresponding right diagram comes from a similar argument applied to

The condition

Define a finite graph

now becomes

G whose object-set is

Q

and whose arrows are all those of xp: P->- P'

and

yp: P->- P"

category generated by

Q),

together with arrows

for each

P

E

P.

Let

S

be the

G subject to the composition and identity Q;

clearly

S is finite.

S--+ A to agree with S on Q,

to send

P

relations that already obtain in N:

(ob P) + (ob

send

and

colim YN

is the coequalizer of

to

and

respectively. a

and

E

P to

Define

TP,

and to

Then clearly

D

B.

(It may not be inferred from the truth of Proposition 5.34 and Theorem 5.37 that, for a generaZ cocomplete

C,

the construction of

the colimit-closure or of the finite-colimit-closure of a full subcategory

Cat 0

A

terminates after one step.

The ordinary category

is by Proposition 3.40 the colimit-closure of the full

subcategory in 5.3.

{2},

since

The category

2

is a strong generator, as we observed

A with one object and one non-identity idempotent

map is in fact easily seen to lie in the finite-colimit-closure of

{ 2 }.

Yet

{2};

through 2

to

A is not the colimit in

A

Cat

0

of any functor factorizing

for it is soon verified that no map from a copower of

is a coequalizer.)

5.10

190

5.10

For our final example with

of a small

Set

we consider the completion Let us write this as

and for the rest of this section keep

completion of

A

for the

A under finite colimits.

A functor F*

=

A under small filtered colimits.

[A 0 P,setJ,

Ac

V

F: A0 p --+ Set

is called j1at if the functor

-: [A,Set] --+Set is left exact - that is, finite-limit-preserving.

Every representable

A(-,A)

which is the evaluation limits.

is flat, for

A(-,A)

EA: CA,Set] --+Set,

Moreover, since

* ? = (?)A by Yoneda,

preserving all small

* is cocontinuous in both variables, it

follows from Theorem 4. 73 that a small filtered colimit of flat functors is flat.

Theorem 5.38

For

0

F: A P--+ Set

with

A small, the fol~ing

are equivalent: el

0

is filtered.

(i)

(

F) P

(ii)

The aanoniaal expression (3.32) of F as a aoZimit of representables is a filtered aolimit.

(iii)

F

is some small filtered colimit of representabies.

(iv)

F

lies in the completion

A of

A under smaZl filtered

aolimits. (v) Proof

F

(i) 0

( al F) p; trivial.

is f1at. implies (ii) since the colimit (3.32) is indexed by that (ii) implies (iii) and that (iii) implies (iv) are Since

A is by Theorem 5.35 the closure of

A

in

(5. 38)- (5. 39) [A

0

P,set]

191

under filtered colimits, (iv) implies (v) by the remarks

preceding the theorem.

Thus we have only to show that (v) implies

(i) .

Let

P--+ elF where

T:

corresponds to a functor By the flatness of

F

P is finite. 0

P --+

S:

we have

F

A p

*

As in 4.7,

and an element

a

lim FS.

E

*

li~A(SP,-) ~ li~(F

and by Yoneda the latter is limPFSP =lim FS.

T

A(SP,-));

By the formula (3.70)

and the dual of (3.68), we have therefore a canonical surjection

LAEA FA

li~A(SP,A) ----+lim FS.

x

Let some inverse image of (x

a

FA,

E

li~A(SP,A)).

E

inductive cone in maps

x

to 0

a

ap

( el F) p

over

filtered.

D

A

under this surjection be given by By (3.28),

over

S

with vertex

by construction, T

with vertex

8

a

we can regard A.

Since

as an

FBp: FA ---+ FSP

is equally an inductive cone in

(A,x).

Thus

(

eZ F) 0 p

is

Proposition 5.39 A f1at functor F: A--+ Set with A small preserves any finite limit that exists in f1at functors

F: A --+ Set

P--+

A.

If A is finitely conrplete, the

are precisely the left-exact ones.

P

Proof

If

of

gives,as in the proof above,that

F

S:

A with

finite has a limit in F

However the left side is now isomorphic to So a flat functor complete. exact,

F

A

F

*

A(-,lim S), ~ F(lim S). A

is finitely

is finitely complete and

F

is left

is finitely complete by Proposition 4.87; hence

is finitely cocomplete and therefore filtered, so that Theorem 5.38.

the flatness

li~A(-,SP) =lim FS.

is certainly left exact if

Conversely, if

eZ F

*

A,

0

F

(

0

el F) p

is fla.t by

5. 10

192 Proposition 5.40

finitely-presentab~e

The

-colimit-completion

A of a

smal~

A are the retracts of the

representabZes; so that they reduce to Proof

objects in the filtered-

Since filtered colimits in

A if idempotents

A are formed as in

sp~it 0

[A P,SetJ,

the proo.f of Theorem 5.36 applies here virtually unchanged. point is that, for a finitely-presentable

F,

in A.

the functor

The new A(F,-)

is

only known to preserve filtered colimits; but now the canonical F is filtered,

colimit of representables giving

Proposition 5.41

If A is the finite-coZimit-compZetion of the small

A--+

then the inclusion

A,

0

0

is, to within equivalence,

[A P,setJ A

the filtered-colimit-completion 0

F: A P --+ Set

A

of

A.

In consequence, every

is a filtered colimit of objects of A,

and A

consists of the finitely-presentable objects of [A 0 P,setJ. Proof

Let

K: A--+

A

be the inclusion, so that by Proposition 5.16 0

the inclusion A--+ [A P,Set]

is

K.

Using

Lex

we have by Theorem 5.38 and Proposition 5.39 that subcategory

Lex[~P,set]

we have an equivalence

of

[K P,l]

with the Yoneda

A

is the full

By the dual of Theorem 5.35

0

RanK 0 P: [A P,Set] ~ Lex[~P.setJ,

inverse is the restriction of 0

[~P,setJ.

for "left exact",

A--+

this gives the first assertion.

0

[K P,l]. cXOP,setJ

whose

:ance the composite of is

K: A--+

[A 0 P,setJ,

Then the second one comes from

Theorem 5.38 and the third from Proposition 5.40.

0

Finally we give the converse to Proposition 4.74 promised in

4.6:

(5 • 40)- (5 • 42)

19 3

Proposition 5.42 For a smaZZ A with finite-coZimit-compZetion

A,

the foZZowing are equivaZent: (i)

A is fiZtered.

(ii)

colim:

(iii)

The incZusion K: A --+

[A,Set] --+ Set is Zeft exact.

A

is finaZ.

Proof Since by (3.26) we have colim F 0

is the flatness of

nl: A P--+

Set.

equivalent to the filteredness of

~ 61

* F, what (ii) asserts

By Theorem 5.38, this is

eZ nl) 0 P,

which is

A;

thus

(ii) is equivalent to (i).

A

Since

is finitely cocomplete and hence filtered, what (iii)

asserts by Proposition 4.71 is the non-emptiness of each F

E

A.

F

E

A so arises.

Any finite diagram in

with vertex for some

5.11

A

A;

E

A has a colimit

F

in

A,

F/K

for

and every

To say that the diagram admits an inductive cone A is exactly to say that there is a map

thus (iii) is equivalent to (i).

F --+ KA

0

From the beginning of 5.4 until now we have dealt only with

fuZZy-faithfuL dense functors - exploiting the notion of a density presentation, which does not seem to generalize usefully to the non-fully-faithful case.

arbitrary dense

Now we drop this restriction, and consider an

K: A

full subcategory of

~

[C,BJ

C.

For any

B,

given by those

write

Ladj[C,BJ

for the

S: C--+ B which are

left adjoints (or which have right adjoints); since left adjoints are certainly

A-cocontinuous, it follows from Theorem 5.1 that the

restriction of

[ K, 11: [C,B~ -

LA,BJ

to

Ladj[C,B I

is fully

5.11

194 faithful, so that

Ladj[C,BJ

is a

equivalent to its replete image in

A

V-category if [A,BJ

under

is small, being

[K,l];

the main

goal of this section and the next is to determine this replete image. In this section, devoted mostly to examples, we suppose for simplicity that

A is small; if it were not, the expressions

Cocts K [C,BJ

would have to be modified.

Cocts[C,BJ

and

Then Proposition 5.11 and

Theorem 5.33 give Cocts[C,BJ

Ladj [C,BJ

if C

~s

(5.43)

cocompZete.

We also abbreviate fuZZ subcategory to subcategory, since we consider no others.

Finally, we loosely write

[K,l]

restrictions, specified by the context,

and

LanK

for various

of the functors properly so

named. The following remarks, to the end of this paragraph, apply to any

K: A --+ C,

dense or not, and do not need

A to be small.

In

Theorem 4.51, and in its generalizations Theorem 4.99 and Theorem 5.35, we have used the notation G:

A --+ B

for which

more than one functor

[A,B]'

La~G

K,

for the subcategory given by those

exists.

Since we have soon to deal with

we replace this notation by

[A,BJ(K).

Similarly we have used, in Theorem 4.99 and elsewhere, the subcategory given by those some

G;

we now change this to

subcategory of G---..

(La~G)K

subcategory of 1: SK--+ SK

[A,BJ(K)

[C,BJ9.K.

given by those S

as

LanKG

for

Write [A B] (K)

'

G

is an isomorphism; and write

exhibits

for

S: C--+ B of the form

given by those

[C,BJ9.K

[C,BJN

LanKSK.

s

*

for the

for which the unit [C,BJ!K

for the

for which the identity

It is immediate from Theorem 4.38

that we have an equivalence (with the evident unit and counit)

(5.43)-(5.48)

195 [A BJ(K)

'

where (as we warned) we are using restrictions. [A BJ(K)

, *

[K,1]

and

(5. 44)

*

LanK

loosely for their

By Proposition 4.23,

= [A

'

BJ(K)

and

[C BJ'~.K

,

for fully-faithful

[C,BJtK

*

K; (5.45)

and in this special case, (5.44) becomes the equivalence

La~

--1

of Theorem 4.99:

[K,1]: [C,BJJ!.K ~ [A,BJ(K)

(5.46)

B

(and in particular for

B

V)

=

K is not fully faithful, so that (5.46) then fails too.

whenever

Henceforth we suppose once again that proof of Theorem 5.1, a cocontinuous

* K;

K

but by the same Proposition 4.23, the first equality

in (5.45) fails for a general

KC

for fully faithful

K is dense.

As in the

S: C--+ B preserves the colimit

so that by Theorem 5.l(iv) we have

s

E

[C BJJ!.K

'

*

(from which,

using (5.44), were-find the conclusion of Theorem 5.l(i), that the restriction of

[K,l]

to

Cocts[C,BJ

is fully faithful).

We therefore

have inclusions Ladj[C,BJ

c

Cocts[C,] B

c

[c , B] *l!.K c [C,B ] tK c

CB I.,].

The first of these inclusions is by (5.43) an equality when cocomplete, but is proper in general - the inclusion into

(5. 4 7)

C is

Set of

the small full subcategory of finite sets is continuous but has no left adjoint.

The third inclusion is by (5.45) an equality when

K is

fully faithful, and in fact Theorem 5.29 then gives the common value explicity as [C BJtK

'

*

[C,BJtK

K

Cocts [C,BJ

for fully-faithful

K.

( 5. 48)

5.11

196 However the second and fourth inclusions above are in general proper

V ; Set,

even when

B are complete and cocomplete, and

C and

K is fully faithful.

For an example of this, let 1

E

= (O--+ 1);

2

maps

in

A----+ B

A; 0,

it is clearly dense.

B;

0 --+ B

A--+ B to

which

0

B;

and

When

B

= Ladj[2,B]

Cocts[2,B]

B.

BE

B.

[2,BJ~K.

is not so on

{0,1},

object of

[2,6]

this is sent by (D,C)

and let

[2,BJ: of

B

for

V = Set,

0

again

(B,A)

to

to the projection

E

--+

A in

[2,Set] ; Set x Set;

D x C

--+

C.

For an

is the disarete

2

B

B

and

K is clearly dense.

may be identified with a map [K,l]

C

K is surjective on objects.

B = Set P;

consists of those. B --+A

Clearly

An

Set;

and

LanK

[2,B]tK

isomorphic to such projections, while

consists of such projections for which the other projection

D )( C --+ D is an isomorphism - that is; the maps maps

[K,1]

given by those

K: 2 --+ 2 be the inclusion, where

category

*

[K,l],

K is not fully faithful, the second, third, and fourth

example, let

'

LanK

is epimorphic.

are complete and cocomplete, and

[2 BJtK

but

The replete image under

B

consists

Here of course

inclusions in (5.47) may all be proper, even when

sends

consists of all

[A,B]~K) = [A,BJ(K) = [A,BJ = B;

is the proper subcategory of --+

[2,B]

consists of those with

although necessarily fully faithful on

Ladj[2,BJ

Then

that are epimorphisms in

is fully faithful on

it sends

correspond to

while

the initial object of

of those

K: 1--+ 2

0 --+C.

of the maps

Finally (as in the last example) --+ 1

and

0

--+

1.

D --+ 1 Ladj[2,BJ

and the consists

Observe how badly (5.46) fails:

19 7

although

[K,l]

is conservative on

essentially surjective and

LanK

K),

and

[K,l]

but even a left adjoint SK;

of

(D,l)

Ladj[2,8]

K,

S: C--+ B may be

Since the unique map extension of

LanKG

(5.47),

2 --+ 1

K along

V =Set

With

--+

sends

[K,l]

B;

2--+ 1

A--+ B

for some

G must lie outside

again,, let

K =I'll: 2--+ 2.

is

I'll: 1--+ 2

to

LanK

sends

l:B--+B.

by Theorem 4.67; K by Proposition 5.7. C --+ D

lie in

B,

while

[A 8] (K)

• -*

consists of the maps Ladj[C,BJ

1: D - D.

consists of those

0 --+ D,

As in

0--+ B which LanK

which need not in general - for instance, if [A,B

to

Here

are epimorphic; and such an object is the image under --+

G other

K is not fully faithful - each consisting of the maps

the penultimate example,

C

while

is obviously initial, the left Kan

B cocomplete, we see that

even though 0

(O,C),

consists of the pairs

since the latter functor is clearly dense, so is

while

and

The example above does not illustrate this possibility, but

the following one does so.

Taking

Clearly

not only does Proposition 4.23 fail,

in which case, by (5.44) and

[A,BJ;K).

[2,8J~K;

(O,l).

For a non-fully-faithful

than

[2,8] = [2,B](K).

consists of the pairs of the forms

the replete image under (l,l)

(as it must be for an

it is not fully faithful on

is not fully faithful on

[2,BJ:K)

[2,8]

of any 0

8 = Set P -

J (K).

*

5.12 We now answer the question raised in 5.11 for a dense K: A--+ C by identifying the replete image of certain sub category

K-Com[A,BJ(K)

[K,l]: Ladj [C,8J

of

--+ [A,B]

[A,8J;K) c [A,8J;

the result is independent of the smallness of

A.

as a

in this form,

5.12

198

G: A--+ B a· K-aomodel if

We call a functor

K:

factorizes through

0

C--+ lA P,VJ

-

G for some

T: B --+ C.

as

(5.49)

KT

0!

K is fully faithful by Theorem 5.1,

Since

(5.49) more explicitly as a

We can write

V-natural isomorphism

B(GA,B)

=

(5.50)

C(KA,TB) .

K trivially factorizes through itself, we have:

The dense We write

K: A --+ C is always itself a

K-Com[A,BJ

K-comodels, and

for the subcategory of

K-Com[A,BJ(K)

exists.

More generally, for any

K-Com[A,BJ(Z)

for the intersection

The name

"K-comodel" for a

[A,BJ

A and

given by the

K-comodels Z: A--+

G for which

V,

we write

K-Com[A,BJ n [A,BJ(Z).

G: A--+ B satisfying (5.49) is

inspired by the fact (see 5.13 below) that a dense least for small

( 5. 51)

K-comodel.

for the intersection of this with

[A,BJ(K) - that is, the category of those LanKG

B--+ [A P,V]

T is unique to within isomorphism if it exists.

such a

Since

0

G:

cocomplete

C,

K: A--+ C,

at

may be thought of as a very

general kind of essentially-algebraic theory, a model of which in is a functor

0

H: A P --+ P

in the sense above. K-Mod[A0 P,PJ

of

such that

G

= H0 p:

A

0

P p

--+

P

is a comodel

These models constitute a full subcategory 0

[A P,PJ,

isomorphic to

0

(K-Com[A,P P]) 0 P,

If we talk

here of comodels rather than models, it is because we have elected to emphasize left Kan extensions and colimits, rather than their duals. Particular examples of such theories are given in 6.2

and 6.3 below.

(5 . 49)- (5. 54)

199

Rather than establish directly the promised equivalence Ladj[A,BJ

~

K-Com[A,BJ(K),

we first prove a more general result,

itself of value in practice:

A~

Theorem 5.52 Let the dense K: of

V and a fully-faithful

Z: A-+

z

Theorem 5.13, both

and

C be (isomorphic to) the composite V-+ C (so that, by

J:

are also dense).

J

Then

~e

have an

equivalence Lanz ~hose

[Z,l]: J-Com[V,BJ ~ K-Com[A,BJ(Z),

--1

unit and counit are the canonical maps.

Proof Let HE J-Com[V,B],

so that, corresponding to (5.50), we have

B(HD,B) for some

T:

C(JZA,TB)

~

(HZ) (K

(5. 53)

B--+ C.

explicitly as

D = ZA

Putting

C(KA,TB);

C(JD,TB)

showing that KT.

(5.54)

in this gives

HZ

is a

Since (4.49) gives

B(HZA,B)

~

K-comodel, and giving Z

~

KJ,

we have

being fully faithful)

which by (4.20) shows that

H

~

Lanz(HZ);

moreover the unit of this

Kan extension is easily calculated by Theorem 4.6 to be 1: HZ--+ HZ. Since we have seen that (Z)

K-Corn[A,BJ

that

Z~

H

for

KJ,

is a

K-cornodel, it does indeed lie in

. G E K-Corn[A,BJ

Now let Writing

HZ

LanZG, we have

(Z)

,

so that

G;;;

KT

for some

T: B--+ C.

using (4.20) for the first step, and recalling

5.12

200

C(JD, TB),

B(HD,B) H = Lan 2G belongs to

which is (5.54), showing that Putting

D

=

ZA

G ~HZ,

B(HZA,B) ~ C(KA,TB); the

in (5.54) gives as before

latter now being

B(GA,B)

J-Com[V,BJ

by (5.50), Yoneda gives an isomorphism

which is easily verified to be the unit

G --+ (Lan 2 G)Z

of

0

the Kan extension.

Since the unit and counit of the equivalence (5.53) are the canonical maps, it follows that J-Com[V,BJ

and

[V BJJ/.Z

c

'

*

K-Com[A,BJ(Z)

[A,BJ(Z) ,

c

(5.55)

""

and that the equivalence (5.53) is just the restriction of the

[V BJJ/.Z ~ [A BJ(Z)

equivalence

'

*

, *

of (5.44).

Now consider the special case of Theorem 5.52 in which J

=

1c,

and

for which

Z

K.

=

B(SC,B)

-adjoint functor.

~

A

1c-comodel is by (5.50) a functor

C(C,TB)

for some

T;

V

=

C ,

S: C--+

B

that is, it is a left-

Thus Theorem 5.52 (together with (5.43)) gives the

following answer to the problem proposed at the beginning of 5.11:

Theorem 5.56 For a dense LanK

-I

[K, l]: [C ,BJ:K

K: A--+ C and any

o::::

K-Com[A,B](K)

Here, if C is cocomplete and A is smaZ!, Cocts[C,BJ;

aZong

K,

and if every

as when

may be replaced by

the equivalence

~ [A,BJ;K) restricts t(l an ef?uivale>tce

La~ --j [K,l]: .Ladj [C,BJ

by

B,

Ladj[C,B]

may be replaced

G: A--+ B admits a left Kan extension

A is smaU and B is cocompZete, K-Com[A,BJ.

(':>.57)

0

K-Com[A,BJ(K)

(5.55)-(5.58)

201

A is small and

Note that, when Y:

A--+ [A

0

by (4.30). (4.55),

P,VJ,

K

is the Yoneda embedding

A~ B,

Y-comodel is any functor

a

since

Y

Thus (5.57) is another generalization of the equivalence

different from

the generalization given by Theorem 5.31.

In the situation of Theorem 5.52, combining that theorem with Theorem 5.56 gives a diagram of multiple equivalences

~

Ladj[C,BJ

~

J-Com[V,BJ(J)

K-Com[A,BJ(K)

n

n

~

J-Com[V,BJ

K-Com[A,SJ(Z),

(5.58)

in which the horizontal maps from left to right are restrictions, and those from right to left are left Kan extensions. inclusions here, although equalities when cocomplete, are in general proper, even if

V

and

are small.

A= V

For let

and

Z

1,

J-comodel, since

1

Y

take

need not exist.

J:

V --+

and take

2;

To see this, let

since

{1}

by Theorem 5,17,

whence

a retract of

Clearly

[{2}

0

P,SetJ;

object of

2.

so that

{2}.

fails to exist.

Since

B

A is small and C

V;

then

{2}

A

1:

V--+ V JJ;

and let

is a Lan 1 3 0

yet J

be the

determined by the single two-

Set

by (5.17), so is

{1,2}

is dense by Proposition 5.20, as

JO

*

= V = Set,

{2}

is

be fully faithful and dense,

is dense in

{2}

JO

C

B

is cocomplete and

is isomorphic by (4.22) to

inclusion of the full subcategory -element set

C

The vertical

is

lv

bO,

the initial object of

would, if it existed, be the initial

has no initial object,

LanJlV

is

5.13

202 5.13

We now settle on the precise definition of an essentially

algebraic theory:

We have:

Theorem 5.59 For any dense equivalence

is

K: A--+ C with

K-Mod[A 0 P,VJ ~ C', UJhere

given by those C

C

A small and

V itself are usually called its

The models of the theory in

cocomplete.

algebras.

K: A--+ C with

namely, a dense

such that

C

X rh C

C'

small, there is an

is the subcategory of C

exists for aU

K-Mod[ Aop, VJ ~ C.

cotensored,

A

X

E

V.

Thus if

This is so in partiau lar if

C is cocomplete. Proof Since

V0 p

Ladj[C,V 0 P],

or equivalently

is cocomplete, Theorem 5.56 gives

-adjoint functors

C p--+

0

0

K-Mod[A P,V] ~ (Ladj[C,V p]) p; 0

this last category is equivalent to 0

0

K-Com[A,V P] ~

0

Radj[C P,V].

and

But all right-

V lie in the subcategory C c [C 0 P,V]

given by the representables, and constitute in fact the category of the theorem.

C'

The last assertion follows from Proposition 5.15. ~

This leads to the following point of view.

If

C is a cocomplete

category admitting a small dense subcategory, any choice of a dense K: A--+ C with small which

A gives an essentially algebraic theory, for

C is the category of algebras.

As long as we consider models

of the theory only in complete categories

.

K·

is independent of the choice of called a

C-object in

P,

the concept of a model

a model of the theory in

P,

also

is by Theorem 5.56 just a continuous (and

hence right-adjoint) functor P .__,. P'

P,

0

C P

P.

--+

At this level, any continuous

carries models to models; and any model in

by applying the continuous functor

0

C p --+ p

P is so obtained

to the generic model

(5. 59) in

203

0

C p

1.· C0 p - C0 P.

given by

generic model in

0

C P

corresponds to

0

c,

K: A --+

In terms of 0

0

K P: A P--+ C P,

this

which is a

model by (5.51). The choice of

P when P is not complete.

a model in

dense functor with in

C of

C affects to some extent the notion of

K: A->-

A'

K and of

Theorem 5.13.

small, let

.

K'.

V

be the union of the full images

then the inclusion

J:

V->-

C is dense by

Thus Theorem 5.52 gives, in an evident notation,

K-Mod[A P,P](Z) ~ K'-Mod[A' 0 P,PJ(Z'). 0

0

and

K': A'--+ C is another

If

K'-Nod[A' P,P]

coincide when

all right Kan extensions along

It follows that

K-Mod[A 0 P,PJ

P is "complete enough" to admit

Z: A--+

V

and

V.

Z': A'--+

A

concrete example of such "sufficient completeness" will occur in 6.4 below. We end with three remarks.

First, it is easily seen that, for

P and P' • a right-adjoint functor P --+ P'

any to

K-models.

Secondly, the dense

theory of an object"; for, for any

P.

Thirdly, if

I

K-models

takes

I: I-->- V of (5.17) is "the

being

1,

we have

0

I-Mod[I p ,P] ar P

P too is cocomplete with a small dense

subcategory (and hence complete by Proposition 5.15), the notion of a C-object in right-adjoint

P coincides with that of a P-object in C: C0 p --+ p has as its left adjoint

in effect a right adjoint

5.14

S:

C ~B.

0

p --+ C p

what is

0

P p->- C.

We now determine the image, under the equivalence

K-Com[A,BJ(K)

for a

Ladj[C,BJ ~

of (5.57), of the subcategory of equivaZenaes

We do not suppose

A small nor C cocomplete; but we do

restrict ourselves to the simpler case of a fuZZy-faithful dense K: A --+ C.

5.14

204

B to be equivalent

We thus get a criterion for a given category to

C,

in terms of there being a

G: A--+ B

K-comodel

satisfying

our characterization of the image in question; this contains as a

B to be

special case the criterion of Theorem 5.26 above for 0

[A P,VJ.

equivalent to

We give only a few applications below;

numerous others can be found in the article the results (in the case

V

= Set)

If there is an equivalence

under (5.57) is the B(GA,B) = B(SKA,B) (5.50), exhibits -inverse of

S.

~

this

T

T: B

'="

C,

the image of

SK: A--+

B.

Since we have

K-comodel

G

C(KA,TB),

the unique

G~

=

KT

S

T which, as in (5.49) or

K-comodel, is precisely the equivalenceK-comodel

G: A--+ B,

as in (5.49), and seek conditions for

to be an equivalence.

This isomorphism B(GA,B) :;;; C(KA,TB) the form

---1

We therefore start with any

exhibited as such by

of Diers, to whom

are due.

S

G as being a

[21]

a=

a:

G-

KT,

written as a

V-natural isomorphism

as in (5.50), is (by Yoneda in the form (1.48)) of

C(p,l)T

(1.39) expressing the

for some

V-natural

V-naturality of y

p

---------------~

p: K--+ TG.

The diagram

may be written as

KK

GG - - ---::G:-----+ KTG 0

where

K

is the map of (4.22), according ·to which

isomorphism exactly when

G is fully faithful.

K

G

Since

is an K is fully

faithful, both the horizontal maps here are isomorphisms; whence,

205

since

K is fully faithful,

is an isomorphism if and only if G

p

is fuZZy faithful. By Theorem 5.56,

admits a left adjoint

T

admits a left Kan extension

S

n

an equivalence precisely when

TSK.

Let

Y

*

F

y

of

py

c; ;,

so.

Since the left-adjoint

finally that n

and the coZimits Once

n

KT,

preserves these colimits if and only i f

TS

S

with K

n:

1 --+ TS

is an isomorphism and TS

p

preserves and reflects these colimits, and because conclude that

is

A under the

Because the fully-faithful

.

T

It is

p: K--+ TG

It now follows that

~.

is an isomorphism if and only if preserves the coUmits of

are isomorphisms.

is the closure of

C

Then

be a density presentation of

Y YE

as in Theorem 5.19, so that

E

G

Suppose for the moment

identifies

= (F ,P ) r

t

K-absolute colimits

and

G ~ SK

clear that the isomorphism ~

K.

precisely when

n,E: S ~ T be the adjunction.

that this is so, and let

nK: K

along

S

K both we GS

does

preserves all colirnits, we have

is an isomorphism if and onZy if G is fully faithful F

y

*

SP

y

are

G-ahsoZute.

is known to be an isomorphism,

by the triangular equation

Tc.nT

=

T£

is an isomorphism

1 of the adjunction.

So

£

too

is an isomorphism if T is conservative; which of course T must be if it is an equivalence.

Since

G - KT

T is conservative if and onZy if

G

where

K

is fully faithful,

is conservative; that is, if

and only if G: A--+ B is strongly generating.

(In contrast to 3.6,

we are using "strongly generating" here without thereby implying that

A

is small.)

5.14

206

s.

It remains to ensure the existence of the left adjoint

y ~ r,

colimits for all

F -indexed

B admits all

Propositions 3.36 and 3.37 give this if

y

but this is more than we need.

The proof of

Proposition 3.36 shows that we only need the (inductive) existence of

Fy * SPy

the particular colimits

in

B.

Since we can run

simultaneously the inductive proof of the existence of

n: 1

inductive proof that

TS

--+

S

and the

is an isomorphism, we have finally:

Theorem 5.60 Let K: A --+ C be fully faithful and dense, with (F ,P )

density presentation K-aomodd with

G :o:

y

y y

Then

KT.

~

r T

>

G: A --+ B be a

and let

is an equivalenae if and only if the

following aonditions are satisfied (whereupon

K ~ TG):

is fuZly faithful.

(i)

G

(ii)

G is strongly generating (or, equivalently,

(iii)

Whenever exists in is

Py

;;;

T'- [A P,VHYPY-,Y'\) 0

y

(6.9)

corresponds by (3.5) to a map

9 : F y

y

*

YP

y

--...,. YA

y

(6.10)

(6. 7)-(6.12)

219

in the cocomplete maps; because

8

Theorem 6.11

is small, the set of codomains in (6.10) is small, so

A

If the regulus

~-Com[A,BJ

then ~-Alg

for any category

is the reflective subcategory

when the cylinders Proof

8 corresponds as above to the sketch

= 0-Com[A,BJ

8-Alg

K: A --->- ~Alg

corresponding dense functor

B(G-,B)

P,V](F y

*

In particular

of [A0 P,V].

The

is fuZZy faithful precisely

G: A --+

is orthogonal in

0

that is, when each

B.

~.

are already colimit-cylinders in A.

~Y

By the remarks at the end of 6.2,

precisely when each

0

for the set of these

{e } yyEf

e is indeed a regulus in the sense of 6.2.

that

[A

=

B is a

0

[A P,VJ

8-comodel

to each

e ; y

0

[A P,V](8 ,1): [A P,V](YA ,B(G-,B))--->Y y YP ,B(G-,B)) is an isomorphism. By (3.5) and Yoneda, y

this map is isomorphic to a map

B(GA ,B) y

----+

[L 0 P,V](F ,B(GP -,B)) , y

y

(6.12)

y

whose unit is easily verified to be (6.8); to say that it is an isomorphism is therefore by (3.5) to say that (6.8) is a colimit-cylinder, or that G is a

~-comodel.

Theorem 5.59 the equality By Proposition 5.16, contains the representab1es

~-Alg

gives by

K is fully faithful if and only if A(-,A).

To ask each

A(-,A)

¢-Alg

to be an

0

A(A y ,A)--+ [L yP,VJ(F y ,A(Py -,A))

be an isomorphism, and hence by (3.5) that each

0

Vp

= 8-Alg.

algebra is by (6.12) to ask that each

-cylinder.

0

Applying this to the comodels in

~

y

be a colimit-

6.3

In particular,

· · is re fl ect1ve 1n

~Alg

of all small colimit-cylinders in

[ V op , S e t'-.I

v0 '

r,

A0 p, V -]

when

"'~

consists

V0 " in

A; the reflectivity of

of 3.12 above is an example of this.

0

A very important special case of Theorem 6.11 is that where

F

F-cocomplete for some set

=

(A 0 P,~)

F-Cocts[A,BJ

~-Alg

and

~

of the sketch

F-colimits in

0

small

V

For instance, with

([52], [ 7 ]) Ap

In this case

0

F-Cts[A P,Vj.

The full name 0

might here be replaced by

also suggestive to refer to a small

F-theory.

=

A.

0

is a small category

A p

0

A p

It is as an

a Lawvere-Benabou theory with finite products; while a

sometimes called a finitar;~

with finite limits is

essentially-aZgebraic theory.

(A P,F).

F-complete category

Set,

=

is

~

of small indexing-types, and where

consists of (the units of) all the ¢-Com[A,BJ

A

(Joyal suggests calling the latter a

cartesian theory; in that finite limits may be said to go back to Descartes, {(x,y)

I

0

A p-Alg

who introduced both the product

f(x,y) = g(x,y) }.) (F

R x R

and the equalizer

This terminology moreover allows

being understood) as a useful abbreviation for

0

F-Cts[A P, VJ. 0

CA P,~)

Consider now a sketch

where

~

is given by (6.7); let

F be a set of small indexing-types containing each O·

'

and let

and

B

be a small

L: B--+ 8°p~Alg

=

F-cocomplete category. 0

F-Cts[B P,V]

observing by Theorem 6.11 since

L

is also an

that

0

a

-y

Write

occurring in K: A--+

~-Alg

for the respective dense functors, is fully faithful.

Note that,

L-comodel by (5.51), we have:

Proposition 6.13 For a small L: B--+ Bop_Alg

L

F

F-Cts[B P,V]

F-cocomplete

B, the dense inclusion

preserves and reflects

F-colimits.

0

(6.13)-(6.16)

221

Consider a

~-comodel

G: A

~B.

F-cocontinuous functor, the composite Taking

0

=Vp

V

here,

If QG

B - - 7 V is any

Q:

is clearly another

0

we see that

$-comodel.

0

[G P,l]: [8°P,VJ--+ [A P,VJ

restricts to a functor (6.14) Such a functor between the categories of algebras, induced by a 0

realization (that is, a model) often said to be algebraic: F-cocomplete and

Gp

of one theory in the other, is

especially in the case where

~

K: $-Alg --+ [A 0 p, VJ

and 0

fully faithful, since the left adjoint adjoint

-

* L,

A too is

0

0

[c P,l]: [B P,V]--+ [A P,VJ

LanGop•

and since

L

are

has by Theorem 4.50

has by Theorem 4.51 the left

it follows that the algebraic functor

G

has the

left adjoint (-* L).

The value of this at is also

(KP)

*

P

(LG);

E

K

LanG 0 P. 0

B p-Alg

is

(Lan

so that by (4.17)

G0 P

(KP))

* L,

which by (4.65)

we have (6.15)

Lan (LG). K

In fact

G is recoverable from

G*.

First observe that

Proposition 6.13 gives:

Proposition 6.16

In the circumstances above,

-comodel. 0

of '!>-ComlA,B P-Alg]

Thus

given by_ the

6.3 222 Using this and (6.15), we have:

Theorem 6.17

Let the small

B be

F-cocompLete, wher•e the set

(A 0 P,~);

respective dense functors.

y

0

K: A~ $-Alg

and let

occurring in the sketch

F

small indexing-types contains aLl the

F of

and L: B ~ B P-Alg be the

Then the composite of the full inclusion sending

~-Com[A,BJ

G to

LG,

and the

equivalence

of Theorem 5.56• is isomorphic to a functor ( ) : ~-Com[A,B] __,. Ladj [)-Alg,B P-Alg] 0

(6 .18)

* sending

..

G to the left adjoint

G K

*

and the functor

( )*

of (6.18)

L as

According~y

(6.19)

is fuZZy faithful, ita image consisting

SK ~

LG

for which

SK

factors

for some (tssentiaZZy

D

We may consider this theorem in the case where F-cocomplete, and

=

0

(A P, F).

functors between these, and the the latter, form a form more; for

we have

LG

0

through the fully-faithful G.

=

S: -Alg ~ B P -Alg

of those left-adjoint

unique)

of c*.

G

F-theories

V-Cat

the

F-continuous

V-natural transformations between

2-category - that is,. a

F-CoctsrA,B]

We saw in 2.3 that

The

A too is

Cat-category.

is not just a category but a

In fact they V-category,

is symmetric monoidal closed; and what the

F-theories actually constitute is a

(V-Cat)-category.

There is a

(6.17)-(6.20)

223

( V-Cat) -functor into

V-CAT

sending the

and defined on hom-objects by (6.18).

A0 p

F-theory

A0 P-Alg

to

It is locally fully faithful,

in the sense that (6.18) is fully faithful.

Theorem 6.17 above

determines, in a certain sense, its local image; not as absolutely as 0

S: A p-Alg ~ B0 P-Alg

we should like, since the criterlon for

in this local image refers back to the inclusions of

to be

B.

A and

The

rem.aining step in determining globally the image of this

(V-Cat)-functor

is that of characterizing the

0

V-categories of the form

A p-Alg.

For this we can get something from Theorem 5.60; in the absence of any particularly simple density presentation of

* K

can just use the canonical one

KC

Proposition 6.20

A is

0

A P-Alg

be an

If the small

~

0

K: A--+ A p-Alg,

C from Theorem 5.19.

we

We have:

F-cocomplete and K: A--+

is the fully-faithfUl dense functor as above, let

G: A--+ V

F-cocontinuous functor, and hence a K-comodel; and define

T: V--+ A0 P-Alg

by

G~

KT.

In order that

T be an equivalence,

the following conditions are necessary and sufficient: (i)

V is cocomplete.

(ii)

G is fully faithful and strongly generating.

(iii)

The canonical map

0

0

A P-Alg(KA,C) --+ V(GA,A P-Alg(K-,C)

*

G)

is an isomorphism. Proof

The conditions are clearly necessary.

refer not to Theorem 5.60 but to its proof. G admits a left Kan extension a left adjoint to

T.

For

n

S and

along ~

K,

For their sufficiency we Since

V

is cocomplete,

which by Theorem 5.56 is

to be isomorphisms, we need

6.3

224 that

G

is fully faithful and strongly generating (which is (ii)) KC * SK

and that the colimit

last requirement is that and as

G

givin!-'

(iii).

is

G-absolute.

*

V(GA,KC

G)

SK ~ G,

Since

be isomorphic to

is fully faithful the latter is

KC * A(A,-)

this

*

KC or

V(GA,G-);

(KC)A,

II

Again this recognition theorem is not as absolute as we should

A0 p-Alg.

like, in that condition (iii) refers in detail to 0

when both

V and

0

A p ~ A p-Alg

better description of the image of

A far

is often available

F are suitably restricted - see for example

[45].

The fact is that we are coming to the end of what can usefully be said in the present generality.

6.4 One last series of importaPt observations can however be made. Call a sketch If all the

FY

(A P,~) 0

small if the set

~

-types, the smallness of

of cylinders is small.

lie in a small set

~

occurring in

¢

F of indexing-

is automatic; for the small

A admits

but a small set of cylinders with any given indexing-type.

Let

Proposition 6.21

0

(A P,~)

indexing-tyFes cnZy fPom the

be a small sketch b'hich 1"nvC'!.ves

smaZ~

set

be the corresponding dense functor. F-colimits in

J:

V--> C;

dense, so that in particular

--+

C = 4>-Alg

V be

it is small by 3.5.

Z: A--+ V

by TheoPem 5.13 both

J: C--+ [V0 P,VJ is

assert that the r~pZete image of

K: A

Let V be the closure under

C of the fuZZ image of K;

the factorization of K through fuZZ inclusion

and Zet

J is

fol~owed

z and

J

Let

by the are

fuZ~y faithful.

F-Cts[V0 P,V].

We

(6. 21)

225

Proof

One direction is immediate.

V,

constructon of and hence

JC

so that

is

C(J-,C)

pop: vop -+V

-comodel by (5.51),

is

F-continuous.

and since the fully-faithful

p

-comodel.

Since

too is a

-comodel.

By Theorem 5. 56, we have

Vop

Theorem 4.47, ~

Lan Pz 2

SJ,

1: PZ--+ PZ

gives a map SJ

preserve

~:

SJ

since

K

=

JZ

with

PZ =

s

-

JZ

c

is

=

SK

t.;here the

LanKPZ,

By

fully faithful, we have

J

A an isomorphism.

with unit

Applying Theorem 4.43 to

as in

~:

SJ

--~

P

with

~Z

an isomorphism.

F-colimits, and since

--+

P

we have

Z

V

Since both

is the closure under

(which is the full image of

K),

P

and

F-colimits

we conclude

is an isomorphism.

The right-adjoint C(-,C),

v

of this left Kan extension being an isomorphism.

of the full image of that

J:

K

PZ: A__,. vop

F-cocontinuous,

is a left-adjoint functor given by

PZ __,. SK

unit

is

Since

Z: A-+V

F-colimits,

itself a

c __,.

V,

sends them to limits in

not only preserves but also reflects

S:

F-colimits by the

F-continuous.

Suppose now that is a

J preserves

S0 p: C0 p--+ V being necessarily a representable

p 0 P ~ S0 pJop ~ C(J-,C)

Now Theorem 6.11 gives:

= JC,

as required.

0

6.4

226 Corollary 6.22

In thene cirewnstanccs the functor

to the inclusion V ~ V0 P-Alg th.at for any

B

?..'e

of the sketch

¢,

F

V,

so

0

here is bounded only below by the knowledge 0

we cannot expect the comodels of the sketches

to "coincide" in eve:ry

and

F-cocomplete smaZZ

J-Com[V,B] = F-CoctsiV,BJ.

haVI?

Since the size of

of the

ts isomm-phic

J

B·

B

that they do so when

'

(A P,¢)

is

F-cocomplete is an example of the "sufficient completeness" referred to in the penultimate paragraph of 5.13.

The precise result is:

e~ery

Theorem 6.23

In the cir>cwnstances of Proposition 6.21,

~-comodel

A --+ B admits a left Kan extension alonJ Z: A --+ V

whenever

G:

B is Lan

F-cocomplete; and we have an equivalence

2

--1

[Z,l]: F-Cocts[V,B] ~ ¢-Com[A,BJ

whose unit and counit are the canonical maps.

Proof By Corollary 6.22 and TI1eorem 5.52, we have only to show that every

-comodel G: A-+ B has the form

HZ

for some

F-cocontinuous

H: V--+ B. We suppose first that

L: B--+ 6°p-Alg

B is small, and consider the inclusion

of Proposition 6.13.

LG,

Then

being itself a

-comodel as in Proposition 6.16, is by Theorem 5.56 isomorphic to for some left-adjoint of (6.19).

B c:

8°P-Alg

The set

S: -Alg--+ 8°p-Alg;

H of those objects

left-adjoint

.c (

c

certainly contains all the objects

and it is closed under S

s

in fact

is the

G*

sc

lies in

for which KA,

since

SK

-

F-colimits by Proposition 6.13, since the

preserves colimits; hence

H

contains

SK

V

by the

LG;

(6.22)-(6.23)

227

definition of this latter, is clearly and

(L

SJ ~ LH

Thus

F-cocontinuous since SJ

H: V--+ B,

for some

is so.

LG ~ SK

Now

being fully faithful) we have the desired result

Passing to the case where the let

B'

be the closure under

the

~-comodel

A--+ B;

G:

F-colimits,

G'

Now

G-;: HZ, 0

We may say that

(A P,~), 0

where

is the

V p

(A P,~) 0

or that

G'

= H'Z H

B.

Since

~-comodel;

is itself a

small by 3.5, we have as above H': V--+ B',

N: B' --+

=

is

G':

N

A--+

B'

preserves and

and since

for some NH'

G-;: HZ.

of the full image of

this latter factorizing as

followed by the full inclusion reflects

B

SJZ -;: LHZ;

=

B need not be small,

F-cocomplete

F-colimits in

which

B'

is

F-cocontinuous

0

F-cocontinuous.

F-theory generated by the sketch

is a sketch of the

0

F-theory

V P.

The

latter form of words comes closer to explicating Ehresmann's choice of the word "sketch". small category

V =Set),

As he used the word (in the case

A

the

was not in general explicitly given, but presented

by generators and relations; often this presentation was finite, and ~

was a finite set of finite cones (the colimits involved in

classically being conical ones), (A p .~) 0

(the presentation of)

In such cases, one can literally see as a finitary way of ''sketching" the or the still more

generally infinite and complicated structure of complicated structure of

When

V =Set

ar.d

for finite

L,

the sketch

(A P,~)

the 0

of finite cones. that are the

F

0

V p-Alg.

F

consists of the functors

61:

L ~Set

F-theories are the cartesian theories of 6.3 above; gives rise to such a theory if

It is exactly the categories

~-Alg

locally finitely presentable categories;

¢

is a small set for such a

V c ¢-Alg

Q

is

6.4

228 a small dense subcategory composed of objects which (since finite limits commute with filtered colimits in

Set)

V

all the examples of categories

0

are finitely presentable.

that we declared locally finitely

presentable in 6.1, it is easy to see that ~.

sketch

V

0

= ~-Alg

than

for such a

For further information on the relation of locally presentable

categories to reguli and sketches, see the classical newer

In

[45]

[31J,

which extends the ideas to certain important

or the

V

other

Set.

V include

These

Cat

and

Gpd,

which are the

appropria~e

base

categories for the study of categories with essentially-algebraic structure.

At this level, as is well known, equivalence becomes more

central than isomorphism, and the naturally-occurring "functors" and "transformations" are more general than the

V-functors and the

V-natural transformations; which leads to various problems of coherence. The study of essentially-algebraic theories in this context must take account of these more general notions; this is not touched on in but a beginning is made in

6.5

[45],

[46].

Before ending, we give a final application of Theorem 6.23.

consider the where

F

(V-Cat)-category of small

We

F-cocomplete categories

is a small set of indexing-types.

Since

A,

F-CoctsfA,BJ

denotes the

(V-Cat)-valued hom here, an appropriate name for this

category is

F-Cocts.

If

B and

C are

F-cocomplete, so of course is

F-colimits formed pointwise. this is again

The full subcategory

[B,CJ,

with

F-Cocts[B,C]

F-cocomple te, since colimi ts commute with colimi ts.

of

(6.24)

229

Q: A--+ F-Cocts[B,C]

The functors P: A~

B--+

correspond to those functors

C for which each partial functor

F-cocontinuous.

If

A

too is

B--+

P(A,-):

Q is itself

F-cocomplete, the functor

F-cocontinuous precisely when the other partial functors are also

F-cocontinuous.

P: A ~ B --+ C which is Such a

P

T: L --+A,

F-cocontinuous in each variable separately. 0

((A~ B) P,~).

is a cylinder in

we can consider for

A, B

E

F: L0 p

where B

L

to

(TL,B),

and called

If

--+

V and

the functor

--,T,---~-=-B--+ A ~

sending

P(-,B): A--+ C

We are thus led to the notion of a

is a comodel for a sketch

A: F--+ A(T-,M)

C is

T ®B.

B

Then we can consider the

cylinder F --+ A(T- ,M) ;;;-ACT- ,M) ~I ~1 . A(T-,M) ~ B(B ,B)= (A® B) ( (T-,B) '(M,B))'

A

~J

and call this if

A is one in

colimits. A ®

A ®B.

~.

A;

Note that this is a colimit-cylinder in essentially because

If we take

where

~

A is an

F-colimit-cylinder in precisely those

P

- ® Y: V

to consist of all the F-colimit-cylinder in

B,

~-comodels

the

that are

--+

A® B

V preserves

A ® B and all the A and

P: A® E

--+

~

is an C are

F-cocontinuous in each variable.

In fact

more than this is true; it is easy to see that ~-Com[A ~

B,CJ

-

for the

Write ((A® B) 0 P,~).

F-Cocts[A,F-cocts[B,C]]. F-theory generated by the sketch

In view of Theorem 6.23, we get

F-Cocts[A atFB,CJ

::::

F-cocts[A,F-Cocts[B,CJJ •

(6.24)

6.5

230 The symmetry in

B,

A,

C

of

F-Cocts[A,F-Cocts[B,F-Cocts[C,VJJJ

now

easily gives (6.25)

and (6.26)

If

A

is the free

F-cocompletion of

clear that (for any small

A

and

A

as in Theorem 5.35, it is

B) (6.27)

it is also clear that, for any small

A

F-cocontinuous functors

correspond to the functors

P:

A ® B --~

C

A®FB--+ C

for which each

P(A,-)

and for

is

F-cocomplete

F-cocontinuous.

B,

the

It follows

that

(6.28)

where

1

is the free

is the closure of

I

F-cocompletion of in

V under

I;

which by Theorem 5.35

F-colimits.

The coherence, in the appropriate sense, of the maps (6.25), (6.26), and (6.28), follows from the universal property (V-Cat)-category

(6.24).

Thus the

F-Cocts is endowed with something like a symmetric

monoidal closed structure - the difference being that we have equivalences in place of isomorphisms.

(b.~5)-(b.28)

2 31

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2 32

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E. Brown, Cohomology theories, Ann. Math. 75(1962), 467-484.

[ 12J

B.J. Day, On closed categories of functors,Lecture Notes in tJ,,LI ..

[13

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1-'Jtl.

B.J. Day, Construction of biclosed categories

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B.J. Day, A reflection theorem for closed categories, J. Pure

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B.J. Day, On adjoint-functor factorization, Lecture f,'otes i"l Math. 420(1974), 1-19.

[16]

B.J. Day, On closed categories of functors II, Lecture l.'otes

in [17]

Math. 420(1974), 20-54.

B.J. Day, An embedding theorem for closed categories, Lzcture Notes

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in Math. 420(1974), 55-64.

B.J. Day, An adjoint-functor theorem over topoi, P•~i.

15(1976), 381-394.

B.J. Day and G.M. Kelly, On topological quotient by pullbacks or products, PY'oc. ('rqr>broi?;;" p;.,n.

~aps E~n.

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67( 1970),

S S l- S'i R.

[20]

B.J. Day and G.M. Kelly, Enriched functor categories, L~~turr rJotes {n Math.

[21]

106(1969), 178-191..

Y. Diers, Type de densit~ d'une sous-cat~gorie pleine, A;;n. S(le.

Scirnt. Bruxelles

90(1976), 25-47.

Index [22]

233 E.J. Dubuc, Kan extensions in enriched category theory,

Lectu:t'e

Notes in Math. 145(1970). [23]

E.J. Dubuc and H. Porta, Convenient categories of topological algebras, and their duality theory, J. Pure Appl. Algebra 3(1971)' 281-316.

[24]

C. Ehresmann, Esquisses et types de structures algebriques,

Bul. Inst. Polit. [25:

Ia~i

S. Eilenberg and G.M. Kelly, A generalization of the functorial calculus, J. Algebra

[26]

14(1968), 1-14.

3(1966), 366-375.

S. Eilenberg and G.M. Kelly, Closed categories, Proc. Conf. on

Categorical Algebra

(La Jolla, 1965), Springer-Verlag, Berlin-

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J. Fisher-Palmquist and P.H. Palmquist, Morita contexts of enriched categories, Proc. Amer. Math. Soc. 50(1975), 55-60.

[28]

P.J. Freyd, Abelian categories:

an introduction to the theory

of functors, Harper and Row, New York, Evanston, and London, 1964. [29]

P.J. Freyd, Several new concepts:

lucid and concordant functors,

pre-limits, pre-completeness, the continuous and concordant completions of categories, Lectu:t'e Notes in Math. 99(1969), 196-241. [30]

P.J. Freyd and G.M. Kelly, Categories of continuous functors I,

J. Pure Appl. Algebra Algebra

4(1974), 121.

2(1972), 169-191; Erratum, J. Pu:t'e Appl.

Bibliography

234 [31J

P. Gabriel and F. Ulmer,

Lecture Notes in Math. [32J

Kategorien~

221(1971).

J.W. Gray, Closed categories, lax limits and homotopy limits,

J. Pure Appl. Algebra [33]

Lokal prasentierbare

19(1980), 127-158.

M.F. Gouzou and R. Grunig, U-distributeurs et

cat~gories

de

modules, Manuscript, Univ. de Paris 7, 1975. [34]

A. Grothendieck,

Cat~gories

geometrie algebrique de [35]

fibr~es

l'I.H.E.S.~

et descente, Seminaire de Paris, 1961.

J.R. Isbell, Adequate subcategories, Illinois J. !42th. 4(1960), 541-552.

[36]

J.R. Isbell, Structure of categories, Bull. Amer. !'2th. Soc. 72(1966), 619-655.

[37]

J.R. Isbell, General functorial semantics I, Amer. J. Uath. 94(1972), 535-596.

[38J

D.N. Kan, Adjoint functors, Trans. Amer. ~ath. Soc. 87(1958), 294-329.

[39]

G.M. Kelly, On Mac Lane's conditions for coherence of natural associativities, cornmutativities, etc., J. Algebra

1(1964),

397-402.

[40]

G.M. Kelly, Tensor products in categories, J. AlJ~bra

2(1965),

l 5- \7.

[41]

G.M. Kelly, Monornorphisms, epimorphisrns, and pull-backs,

J. Austral. Math. Soc. 9(1969), 124-142.

Index [42]

235 G.M. Kelly, Adjunction for enriched categories, Lecture Notes

in Math. 420(1969), 166-177. [43]

G.M. Kelly, Doctrinal adjunction, Lecture Notes in Math. 420(1974), 257-280.

[44]

G.M. Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on,

:45~

Bu~~.

AustraZ. Math. Soc. 22(1980), 1-83.

G.M. Kelly, Structures defined by finite limits in the enriched context, Cahiers de Top. et

[46]

G~om.

Diff. 1981, to appear.

G.M. Kelly, Categories with structure- biadjoints for algebraic functors, to appear.

[47~

G.M. Kelly and S. Mac Lane,

J. Pure AppZ. AZgebra A~gebra

[48]

Coherence in closed categories,

1(1971), 97-140; Erratum, J. Pure

App~.

1(1971), 219.

G.M. Kelly and S. Mac Lane, Closed coherence for a natural transformation, Lecture Notes in Math. 281(1972), 1-28.

[49J

G.M. Kelly and R.H. Street, Review of the elements of

2-categories,

Lecture Notes in Math. 420(1974), 75-103. I50J

J.F. Kennison, On limit-preserving functors, IZZinois J. Math. 12(1968), 616-619.

[51]

A. Kock, Closed categories generated by commutative monads,

J. AustraZ. Math. Soc. 12(1971), 405-424.

Bibliography i

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I

F .lv. Lawvere, Functorial semantics of algebraic theories,

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50(1963), 869-872.

F.W. Lawvere, Teoria delle categorie sopra un topos di base, Mimeographed notes, Perugia 1972/1973.

[54]

F. \v. Lawvere, Metric spaces, generalized logic, and closed

categories, Rend. del Sem. Mat. e Fis. di Milano

43(1973),

135-166. [55 I

H. Lindner, Morita equivalences of enriched categories,

Cahiers de Top. et [56]

Diff. 15(1974), 377-397.

F.E.J. Linton, Autonomous categories and duality of functors,

J. Algebra [57]

G~om.

2(1965), 315-349; Erratum, J. Algebra

4(1966), 172.

F.E.J. Linton, Relative functorial semantics: adjointness results,

Lecture Notes in Math. 99(1969), 384-418. [58]

S. Mac Lane, Natural associativity and commutativity, Rice

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49(1963), 28-46.

S. Mac Lane, Categorical algebra, Bull. Amer.

~2th.

Soc.

71(1965), 40-106. [60]

S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York-Heidelberg-Berlin, 1971.

[61]

J.P. May, The geometry of iterated loor sr~ces,

in Math. 271( 1972).

Lecture Notes

Index [62]

237 J.P. May, Pairings of categories and spectra, J. Pure Appl.

Algebra [63]

19(1980), 299-346.

B. Mitchell, Low dimensional group cohomology as monoidal structures, to appear.

[64]

Ju. A. Mitropol'skii and L.A. Nazorova, editors:

Matrix Problems,

Akad.Nauk Ukrain. SSR Inst. Mat., Kiev, 1977. (in Russian) [65]

K.

~~rita,

Duality for modules and its application to the

theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku

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27(1979), 479-494.

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[69J

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[70]

R.H. Street, The formal theory of monads, J. Pure Appl. Algebra 2(1972), 149-168.

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R.H. Street, Fibrations and Yoneda's lemma in a

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R.H. Street, extract from a letter to F. Foltz; to appear in

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R.H. Street and R.F.C. Walters, The comprehensive factorization of a functor, Bull. Amer. Math. Soc. 79(1973), 936-941.

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R.F.C. Walters, Sheaves and Cauchy-complete categories,

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239

Index Absolute colirnit, 187 K-absolute colirnit, 170 Accessible functor, 144 weakly accessible functor, 147 Adequate, 158 Adjoint, left, 49 partial, 50 right, 49 Adjoint functor theorem, general, 147 special, 150 Adjunction, 49 Algebra, 202 for a regulus, 218 for a sketch, 218 Algebraic functor, 221 Atom, 149 Bicategory, 34 Biclosed bicategory, 34 monoidal category, 34 Boolean algebra, 35 Cartesian closed category, 33 Cartesian rnonoidal category, 21 Cartesian theory, 220 Category, cartesian closed 33 cartesian monoidal, 21 enriched over V, 23 functor -, 59 monoidal, 21 monoidal biclosed, 34 monoidal closed, 33 of elements of F, 48 symmetric monoidal, 30 symmetric monoidal closed, 35 underlying, 26 2-category, 24 Cauchy complete, 176 Cauchy completion, 176 Closed monoidal category, 33 Closure under a family of colimits, 86 under a family of cylinders, 154 Co complete, 74 F-cocomplete, 183 Cocompletion, free, 126 Cocontinuous, 75 F-cocontinuous, 183 Codense, 158 Coend, 101

240 Cogenerator, strong, 88 Coherence axioms, 21, 28, 29 Cointersection, 209 Co limit, 7l absolute, 187 classical, 81 conical, 81, 96, 99 F-colimit, 183 filtered, 134 K-absolute, 170 iterated, 98 pointwise, 77 Comodel, 198 for a regulus, 218 for a sketch, 218 Complete, 74 finitely, 132 Completion, Cauchy, 176 idempotent-splitting, 186 under F-colimi ts, 185 under finite colimits, 188 Component, connected, 84 of a V-natural transformation, 25 Composition law, 23 Comprehensive factorization system, 136 Cone, 81 canonical inductive, 82 projective, 81 Conical limit or colimit, classical, 81 in a V-category, 94, 96, 99 Connected component, 84 Conservative, 23, 87 Continuous, 75 Continuity of a limit in its index, 80 Contravariant functor, 31 respresentable, 37 Copower, 92 Cotensor product, 91 Cotensored, 92 Counit, of a general end, 100 of a limit, 72 of an adjunction, 49 of an end in V, 54 of a representation, 47 of a right Kan extension, 113 Cowellpoweredness, weak, 91 Cylinder, 73 K-cylinder, 152

Index

241

Index Dense, 158 Density presentation, 172 Diagonal fill-in, 208 Diagram, 71 Directed, 133 Discrete Grothendieck construction, Discrete op-fibration, 136 Distributor, 176 Dual V-category, 30

139

(E,~I)-generator, 209 Element, of X E V, 23 of a set-valued functor, 48 End, in V, 54 in any V-category, 100 Epimorphism, extremal, 84 Equivalence, 51 Essentially algebraic theory, 198, 202 finitary, 220 Essentially surjective, 51 Evaluation, 33, 59 Extremal epimorphism, 84 Extraordinary V-naturality, 40

F-colirni t, 183 F-cocomplete, 183 F-cocontinuous, 183 F-theory, 220 Factorization system, 138 comprehensive, 136 proper, 208 Filtered category, 133 a-filtered, 136 Filtered colimit, 134 Final diagram, 130 Final functor, 130 Finitary, 220 Finite-colimit completion, 188 Finitely complete, 132 Finitely presentable, 155 Flat, 190 Free cocompletion, 126 Free V-category, 68 Fubini theorem, for ends in V, general, 78 Full image, 53 Full subcategory, 25

Fully faithful, 25 locally, 221

57

Index

242 Functor, algebraic, 221 conservative, 23 contravariant, 31 contravariant representable, enriched over V, 24 flat, 190 fully faithful, 25 generating, 88 of two variables, 31 partial, 31 representable, 36, 47 underlying, 2 7 2-functor, 25 Functor category, 59

37

General adjoint functor theorem, 147 Generating functor, 88 Generator, 88 (E,M)-generator, 209 regular, 152 strong, 88 Generic model, 202 Grothendieck construction, discrete, 139 Heyting algebra, 35 Hom, internal, 33 Idempotent, split, 186 Idempotent monad, 53 Idempotent-splitting completion, Identity element, 23 Image, full, 53 replete, 53 Indexed colimit, 73 Indexed limit, 72 Indexing type, 71 Inductive cone, canonical, 82 Initial diagram, 129 Initial functor, 130 Interchange, of ends in V, 57 of limits in general, 80 Internal hom, 33 Intersection of subobjects, 149 of M-subobjects, 208 Iterated colimit, 98 K-absolute colimit, 170 K-comodel, 198 K-cylinder, 152 K-model, 198 Kan adjoint, 125 Kan extension, 113' 115 pointwise, 123 weak, 122

186

Index

243

Left adequate, 158 Left adjoint, 49 Left exact, 132 Left Kan extension, 115 weak, 122 Limit, conical, 81, 94, 99 indexed, 72 pain twise, 77 Limit cone, 81, 94 Locally bounded, 209, 210 Locally finitely presentable, Locally fully faithful, 223 Locally presentable, 211 Locally small, 23

211

~~subobject,

208 52 M-wellpowered, 209 Model, 198 for a regulus, 218 for a sketch, 218 generic, 202 Monad, idempotent, 53 Monoidal category, 21 biclosed, 34 cartesian, 21 strict, 22 symmetric, 30 Monoidal closed category, Morita equivalence, 176 M~tes,

Opposite V-category, Orthogonal, 215

33

30

Partial adjoint, SO Partial functor, 31 Pointwise limit or colimit, 77 Pointwise Kan extension, 123 Power, 92 Presentation of density, 172 Preservation, of a limit, 75 of a right-Kan extension, 114 Profunctor, 176 Projective cone, 81 Proper factorization system, 208 Realization, 221 Reflect limits, 85 Reflective full subcategory, 53 Regular generator, 152 Regulus, 218 Replete full subcategory, 53

Index

244 Replete image, 53 Repletion, 53 Representable V-functor, Representation, 47 Right adequate, 158 Right adjoint, 49 Right exact, 132 Right Kan extension, 113

36' 4 7

Sketch, 218 of an F-theory, 2Z 7 small, 224 Small limit, 74 Cl-small, 136 Small-projective, 174 Small sketch, 224 Small V-ca tegory, 24, 55 Solution-set condition, 148 Special adjoint functor theorem, 150 Split coequalizer diagram, 151 Split idempotent, 186 Strict monoidal category, 22 Strong cogenerator, strongly cogenerating, Strong generator, strongly generating, 58 Subobject, 149 M-subobject, 208 Surjective, essentially, (on objects), 51 Symmetric monoidal category, 30 Symmetric monoidal closed category, 35 Symmetry, 28 Tensor product, 23, 92 of F-theories, 229 of V-categories, 30 Tensored, 92 Theory, essentially algebraic, 198, 202 F-theory, 220 finitary, 220 Transformation, V-natural, 25 Triangular equations, 49 Underlying category, 26 Underlying functor, 27 Union, of M-subo~jects, 208 Unit, of a colimit, 73 of a left Kan extension, I IS of an adjunction, 49 of a representation, 47 Unitobject, 23 Unit V-category, 26 Universal V-natural family, 54

88

Index

245

V-ca tegory,

23 cocomplete, 74 complete, 74 dual, 30 free, 68 opposite, 30 small, 24, 55 unit,

2b

V-functor, 24 accessible, 144 cocontinuous, 75 conservative, 87 continuous, 75 dense, 158 representable, 36, 47 strongly generating, 88 V-natural family, extraordinary, universal, 54 V-natural transformation, 25 Weak Kan extension, 122 Weakly accessible, 147 Weakly cowellpowered, 91 Wellpowered: M-wellpowered, Yoneda embedding, 67 Yoneda lemma, weak, 45 strong, 65 Yoneda isomorphism, 65, 74

209

40