Toposes, Triples and Theories (Grundlehren der mathematischen Wissenschaften)

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 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 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 briefly. 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 defined 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 Grothendieck and Giraud as an abstraction of the of the category of sheaves of sets on a topological space. Later, properties Lawvere and Tierney introduced a more general id~a which they called "elementary topos" (because their axioms did not quantify over sets), and they and other mathematicians 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 construc­ in Godement's book on sheaf theory for the purpose of computing tions") sheaf cohomology. Then Peter Huber discovered that triples capture much of the information of adjoint pairs. Later Linton discovered that triples gave an equivalent approach to Lawverc's theory of equational theories (or rather the infinite generalizations of that theory). Finally, triples have turned out to be a very important tool for deriving various properties of toposes. 1. Categories 2. Toposes 3. Triples 4. Theories 5. Properties of Toposes 6. Permanence Properties of Toposes 7. Representation Theorems 8. Cocone Theories 9. More on Triples Index to Exercises. Michael Barr, Charles Wells. Includes Indexes. Bibliography: P. 333-338.