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Michael Barr
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


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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.




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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
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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
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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

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