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 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. Front Matter....Pages N1-xiii Categories....Pages 1-63 Toposes....Pages 64-89 Triples....Pages 90-139 Theories....Pages 140-170 Properties of Toposes....Pages 171-198 Permanence Properties of Toposes....Pages 199-245 Representation Theorems....Pages 246-287 Cocone Theories....Pages 288-302 More on Triples....Pages 303-331 Back Matter....Pages 333-347 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.