Barr M., Wells W. - Toposes, Triples and Theories.pdf

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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 unmodied 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
Aliate Scholar, Oberlin College
charles@freude.com
To Marcia and Jane
Contents
Preface
vi
1. Categories
1
1.1 Denition 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 Denition 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 Sucient 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

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