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