Charles Wells

Toposes, Triples

and Theories

Version 1.1

10 September 2000

Copyright 2000 by Michael Barr and Charles Frederick Wells.

This version may be downloaded and printed in unmodi¬ed form for private use

only. It is available at http://www.cwru.edu/artsci/math/wells/pub/ttt.html

and ftp.math.mcgill.ca/pub/barr as any of the ¬les ttt.dvi, ttt.ps, ttt.ps.zip,

ttt.pdf, ttt.pdf.zip.

Michael Barr

Peter Redpath Professor Emeritus of Mathematics, McGill University

barr@barrs.org

Charles Wells

Professor Emeritus of Mathematics, Case Western Reserve University

A¬liate Scholar, Oberlin College

charles@freude.com

To Marcia and Jane

Contents

Preface vi

1. Categories 1

1.1 De¬nition of category 1

1.2 Functors 11

1.3 Natural transformations 16

1.4 Elements and Subobjects 20

1.5 The Yoneda Lemma 26

1.6 Pullbacks 29

1.7 Limits 35

1.8 Colimits 48

1.9 Adjoint functors 54

1.10 Filtered colimits 67

1.11 Notes to Chapter I 71

2. Toposes 74

2.1 Basic Ideas about Toposes 74

2.2 Sheaves on a Space 78

2.3 Properties of Toposes 86

2.4 The Beck Conditions 92

2.5 Notes to Chapter 2 95

3. Triples 97

3.1 De¬nition and Examples 97

3.2 The Kleisli and Eilenberg-Moore Categories 103

3.3 Tripleability 109

3.4 Properties of Tripleable Functors 122

3.5 Su¬cient Conditions for Tripleability 128

3.6 Morphisms of Triples 130

3.7 Adjoint Triples 135

3.8 Historical Notes on Triples 142

4. Theories 144

4.1 Sketches 145

4.2 The Ehresmann-Kennison Theorem 149

4.3 Finite-Product Theories 152

4.4 Left Exact Theories 158

4.5 Notes on Theories 170

iv

5. Properties of Toposes 173

5.1 Tripleability of P 173

5.2 Slices of Toposes 175

5.3 Logical Functors 178

5.4 Toposes are Cartesian Closed 183

5.5 Exactness Properties of Toposes 186

5.6 The Heyting Algebra Structure on „¦ 193

6. Permanence Properties of Toposes 198

6.1 Topologies 198

6.2 Sheaves for a Topology 203

6.3 Sheaves form a topos 209

6.4 Left exact cotriples 211

6.5 Left exact triples 215

6.6 Categories in a Topos 220

6.7 Grothendieck Topologies 226

6.8 Giraud™s Theorem 231

7. Representation Theorems 240

7.1 Freyd™s Representation Theorems 240

7.2 The Axiom of Choice 245

7.3 Morphisms of Sites 249

7.4 Deligne™s Theorem 256

7.5 Natural Number Objects 257

7.6 Countable Toposes and Separable Toposes 265

7.7 Barr™s Theorem 272

7.8 Notes to Chapter 7 274

8. Cocone Theories 277

8.1 Regular Theories 277

8.2 Finite Sum Theories 280

8.3 Geometric Theories 282

8.4 Properties of Model Categories 284

9. More on Triples 291

9.1 Duskin™s Tripleability Theorem 291

9.2 Distributive Laws 299

9.3 Colimits of Triple Algebras 304

9.4 Free Triples 309

Bibliography 317

Index 323

Preface

Preface to Version 1.1

This is a corrected version of the ¬rst (and only) edition of the text, published by

in 1984 by Springer-Verlag as Grundlehren der mathematischen Wissenschaften

278. It is available only on the internet, at the locations given on the title page.

All known errors have been corrected. The ¬rst chapter has been partially

revised and supplemented with additional material. The later chapters are es-

sentially as they were in the ¬rst edition. Some additional references have been

added as well (discussed below).

Our text is intended primarily as an exposition of the mathematics, not a

historical treatment of it. In particular, if we state a theorem without attribution

we do not in any way intend to claim that it is original with this book. We

note speci¬cally that most of the material in Chapters 4 and 8 is an extensive

reformulation of ideas and theorems due to C. Ehresmann, J. B´nabou, C. Lair

e

and their students, to Y. Diers, and to A. Grothendieck and his students. We

learned most of this material second hand or recreated it, and so generally do not

know who did it ¬rst. We will happily correct mistaken attributions when they

come to our attention.

The bibliography

We have added some papers that were referred to in the original text but

didn™t make it into the bibliography, and also some texts about the topics herein

that have been written since the ¬rst edition was published. We have made no

attempt to include research papers written since the ¬rst edition.

vi

vii

Acknowledgments

We are grateful to the following people who pointed out errors in the ¬rst

ˇ

edition: D. Cubri´, Samuel Eilenberg, Felipe Gago-Cuso, B. Howard, Peter John-

c

stone, Christian Lair, Francisco Marmolejo, Colin McLarty, Jim Otto, Vaughan

Pratt, Dwight Spencer, and Fer-Jan de Vries.

When (not if) other errors are discovered, we will update the text and increase

the version number. Because of this, we ask that if you want a copy of the text,

you download it from one of our sites rather than copying the version belonging

to someone else.

Preface to the First Edition

A few comments have been added, in italics, to the original preface. As its title

suggests, this book is an introduction to three ideas and the connections between

them. Before describing the content of the book in detail, we describe each

concept brie¬‚y. More extensive introductory descriptions of each concept are in

the introductions and notes to Chapters 2, 3 and 4.

A topos is a special kind of category de¬ned by axioms saying roughly that

certain constructions one can make with sets can be done in the category. In that

sense, a topos is a generalized set theory. However, it originated with Grothen-

dieck and Giraud as an abstraction of the properties of the category of sheaves

of sets on a topological space. Later, Lawvere and Tierney introduced a more

general idea which they called “elementary topos” (because their axioms were

¬rst order and involved no quanti¬cation over sets), and they and other math-

ematicians developed the idea that a theory in the sense of mathematical logic

can be regarded as a topos, perhaps after a process of completion.

The concept of triple originated (under the name “standard constructions”)

in Godement™s book on sheaf theory for the purpose of computing sheaf cohomol-

ogy. Then Peter Huber discovered that triples capture much of the information

of adjoint pairs. Later Linton discovered that triples gave an equivalent approach

to Lawvere™s theory of equational theories (or rather the in¬nite generalizations

of that theory). Finally, triples have turned out to be a very important tool for

deriving various properties of toposes.

Theories, which could be called categorical theories, have been around in one

incarnation or another at least since Lawvere™s Ph.D. thesis. Lawvere™s original

insight was that a mathematical theory”corresponding roughly to the de¬nition

of a class of mathematical objects”could be usefully regarded as a category with

viii

structure of a certain kind, and a model of that theory”one of those objects”

as a set-valued functor from that category which preserves the structure. The

structures involved are more or less elaborate, depending on the kind of objects

involved. The most elaborate of these use categories which have all the structure

of a topos.

Chapter 1 is an introduction to category theory which develops the basic

constructions in categories needed for the rest of the book. All the category

theory the reader needs to understand the book is in it, but the reader should

be warned that if he has had no prior exposure to categorical reasoning the book

might be tough going. More discursive treatments of category theory in general

may be found in Borceux [1994], Mac Lane [1998], and Barr and Wells [1999];

the last-mentioned could be suitably called a prequel to this book.

Chapters 2, 3 and 4 introduce each of the three topics of the title and develop

them independently up to a certain point. Each of them can be read immediately

after Chapter 1. Chapter 5 develops the theory of toposes further, making heavy

use of the theory of triples from Chapter 3. Chapter 6 covers various fundamen-

tal constructions which give toposes, with emphasis on the idea of “topology”,

a concept due to Grothendieck which enables us through Giraud™s theorem to

partially recapture the original idea that toposes are abstract sheaf categories.

Chapter 7 provides the basic representation theorems for toposes. Theories are

then carried further in Chapter 8, making use of the representation theorems and

the concepts of topology and sheaf. Chapter 9 develops further topics in triple

theory, and may be read immediately after Chapter 3. Thus in a sense the book,

except for for Chapter 9, converges on the exposition of theories in Chapters 4

and 8. We hope that the way the ideas are applied to each other will give a

coherence to the many topics discussed which will make them easier to grasp.

We should say a word about the selection of topics. We have developed the in-

troductory material to each of the three main subjects, along with selected topics

for each. The connections between theories as developed here and mathematical

logic have not been elaborated; in fact, the point of categorical theories is that

it provides a way of making the intuitive concept of theory precise without using

concepts from logic and the theory of formal systems. The connection between

topos theory and logic via the concept of the language of a topos has also not

been described here. Categorical logic is the subject of the book by J. Lambek

and P. Scott [1986] which is nicely complementary to our book.

Another omission, more from lack of knowledge on our part than from any

philosophical position, is the intimate connection between toposes and algebraic

geometry. In order to prevent the book from growing even more, we have also

omitted the connection between triples and cohomology, an omission we particu-

larly regret. This, unlike many advanced topics in the theory of triples, has been

ix

well covered in the literature. See also the forthcoming book, Acyclic Models, by

M. Barr.

Chapters 2, 3, 5, 6 and 7 thus form a fairly thorough introduction to the theory

of toposes, covering topologies and the representation theorems but omitting the

connections with algebraic geometry and logic. Adding chapters 4 and 8 provides

an introduction to the concept of categorical theory, again without the connection

to logic. On the other hand, Chapters 3 and 9 provide an introduction to the

basic ideas of triple theory, not including the connections with cohomology.

It is clear that among the three topics, topos theory is “more equal” than

the others in this book. That re¬‚ects the current state of development and, we

believe, importance of topos theory as compared to the other two.

Foundational questions

It seems that no book on category theory is considered complete without some

remarks on its set-theoretic foundations. The well-known set theorist Andreas

Blass gave a talk (published in Gray [1984]) on the interaction between category

theory and set theory in which, among other things, he o¬ered three set-theoretic

foundations for category theory. One was the universes of Grothendieck (of which

he said that one advantage was that it made measurable cardinals respectable in

France) and another was systematic use of the re¬‚ection principle, which probably

does provide a complete solution to the problem; but his ¬rst suggestion, and one

that he clearly thought at least reasonable, was: None. This is the point of view

we shall adopt.

For example, we regard a topos as being de¬ned by its elementary axioms,

saying nothing about the set theory in which its models live. One reason for our

attitude is that many people regard topos theory as a possible new foundation

for mathematics. When we refer to “the category of sets” the reader may choose

between thinking of a standard model of set theory like ZFC and a topos satisfying

certain additional requirements, including at least two-valuedness and choice.

We will occasionally use procedures which are set-theoretically doubtful, such

as the formation of functor categories with large exponent. However, our conclu-

sions can always be justi¬ed by replacing the large exponent by a suitable small

subcategory.

Terminology and notation

With a few exceptions, we usually use established terminology and standard

notation; deviations from customary usage add greatly to the di¬culties of the

x

reader, particularly the reader already somewhat familiar with the subject, and

should be made only when the gain in clarity and e¬ciency are great enough to

overcome the very real inconvenience they cause. In particular, in spite of our

recognition that there are considerable advantages to writing functions on the

right of the argument instead of the left and composing left to right, we have

conformed reluctantly to tradition in this respect: in this book, functions are

written on the left and composition is read right to left.

We often say “arrow” or “map” for “morphism”, “source” for “domain” and

“target” for “codomain”. We generally write “±X” instead of “±X ” for the

component at X of the natural transformation ±, which avoids double subscripts

and is generally easier to read. It also suppresses the distinction between the