Ebook: Toposes, Triples and Theories

CONTENTS
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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