CONTENTS
========
Preface
1. Categories
1 Definition of category
2 Functors
3 Natural transformations
4 Elements and Subobjects
5 The Yoneda Lemma
6 Pullbacks
7 Limits
8 Colimits
9 Adjoint functors
10 Filtered colimits
11 Notes to Chapter I
2. Toposes 63
1 Basic Ideas about Toposes
2 Sheaves on a Space
3 Properties of Toposes
4 The Beck Conditions
5 Notes to Chapter 2
3. Triples 83
1 Definition and Examples
2 The Kleisli and Eilenberg-Moore Categories
3 Tripleability
4 Properties of Tripleable Functors
5 Suficient Conditions for Tripleability
6 Morphisms of Triples
7 Adjoint Triples
8 Historical Notes on Triples
4. Theories 123
1 Sketches
2 The Ehresmann-Kennison Theorem
3 Finite-Product Theories
4 Left Exact Theories
5 Notes on Theories
5. Properties of Toposes 148
1 Tripleability of P
2 Slices of Toposes
3 Logical Functors
4 Toposes are Cartesian Closed
5 Exactness Properties of Toposes
6 The Heyting Algebra Structure on
6. Permanence Properties of Toposes 170
1 Topologies
2 Sheaves for a Topology
3 Sheaves form a topos
4 Left exact cotriples
5 Left exact triples
6 Categories in a Topos
7 Grothendieck Topologies
8 Giraud's Theorem
7. Representation Theorems 207
1 Freyd's Representation Theorems
2 The Axiom of Choice
3 Morphisms of Sites
4 Deligne's Theorem
5 Natural Number Objects
6 Countable Toposes and Separable Toposes
7 Barr's Theorem
8 Notes to Chapter 7
8. Cocone Theories 240
1 Regular Theories
2 Finite Sum Theories
3 Geometric Theories
4 Properties of Model Categories
9. More on Triples 252
1 Duskin's Tripleability Theorem
2 Distributive Laws
3 Colimits of Triple Algebras
4 Free Triples
Bibliography 275
Index of exercises
Index
Author(s): Michael Barr, Charles Wells
Series: Grundlehren der mathematischen Wissenschaften
Edition: 1
Publisher: Springer
Year: 1984
Language: English
Commentary: Vector PDF, book cover, 2-level bookmarks, pagination.
Pages: 308
Front Cover......Page 1
Contents......Page 8
Preface......Page 10
1 Definition of category......Page 17
2 Functors......Page 26
3 Natural transformations......Page 30
4 Elements and Subobjects......Page 33
5 The Yoneda Lemma......Page 38
6 Pullbacks......Page 41
7 Limits......Page 47
8 Colimits......Page 57
9 Adjoint functors......Page 63
10 Filtered colimits......Page 74
11 Notes to Chapter I......Page 77
1 Basic Ideas about Toposes......Page 79
2 Sheaves on a Space......Page 83
3 Properties of Toposes......Page 89
4 The Beck Conditions......Page 94
5 Notes to Chapter 2......Page 97
1 Definition and Examples......Page 99
2 The Kleisli and Eilenberg-Moore Categories......Page 104
3 Tripleability......Page 109
4 Properties of Tripleable Functors......Page 120
5 Suficient Conditions for Tripleability......Page 125
6 Morphisms of Triples......Page 127
7 Adjoint Triples......Page 131
8 Historical Notes on Triples......Page 137
4. Theories......Page 139
1 Sketches......Page 140
2 The Ehresmann-Kennison Theorem......Page 144
3 Finite-Product Theories......Page 146
4 Left Exact Theories......Page 152
5 Notes on Theories......Page 161
1 Tripleability of P......Page 164
2 Slices of Toposes......Page 166
3 Logical Functors......Page 168
4 Toposes are Cartesian Closed......Page 173
5 Exactness Properties of Toposes......Page 175
6 The Heyting Algebra Structure on......Page 182
1 Topologies......Page 186
2 Sheaves for a Topology......Page 191
3 Sheaves form a topos......Page 196
4 Left exact cotriples......Page 198
5 Left exact triples......Page 201
6 Categories in a Topos......Page 205
7 Grothendieck Topologies......Page 211
8 Giraud's Theorem......Page 215
1 Freyd's Representation Theorems......Page 223
2 The Axiom of Choice......Page 227
3 Morphisms of Sites......Page 231
4 Deligne's Theorem......Page 237
5 Natural Number Objects......Page 238
6 Countable Toposes and Separable Toposes......Page 246
7 Barr's Theorem......Page 251
8 Notes to Chapter 7......Page 253
1 Regular Theories......Page 256
2 Finite Sum Theories......Page 259
3 Geometric Theories......Page 260
4 Properties of Model Categories......Page 262
1 Duskin's Tripleability Theorem......Page 268
2 Distributive Laws......Page 275
3 Colimits of Triple Algebras......Page 280
4 Free Triples......Page 284
References......Page 291
Index of exercises......Page 296
Index......Page 300
Back Cover......Page 308
