| Preface | p. ix |
| Preface to Second Edition | p. xiv |
| Preface to Dover Edition | p. xv |
| Prospectus | p. 1 |
| Mathematics = Set Theory? | p. 6 |
| Set theory | p. 6 |
| Foundations of mathematics | p. 13 |
| Mathematics as set theory | p. 14 |
| What Categories Are | p. 17 |
| Functions are sets? | p. 17 |
| Composition of functions | p. 20 |
| Categories: first examples | p. 23 |
| The pathology of abstraction | p. 25 |
| Basic examples | p. 26 |
| Arrows Instead of Epsilon | p. 37 |
| Monic arrows | p. 37 |
| Epic arrows | p. 39 |
| Iso arrows | p. 39 |
| Isomorphic objects | p. 41 |
| Initial objects | p. 43 |
| Terminal objects | p. 44 |
| Duality | p. 45 |
| Products | p. 46 |
| Co-products | p. 54 |
| Equalisers | p. 56 |
| Limits and co-limits | p. 58 |
| Co-equalisers | p. 60 |
| The pullback | p. 63 |
| Pushouts | p. 68 |
| Completeness | p. 69 |
| Exponentiation | p. 70 |
| Introducing Topoi | p. 75 |
| Subobjects | p. 75 |
| Classifying subobjects | p. 79 |
| Definition of topos | p. 84 |
| First examples | p. 85 |
| Bundles and sheaves | p. 88 |
| Monoid actions | p. 100 |
| Power objects | p. 103 |
| [Omega] and comprehension | p. 107 |
| Topos Structure: First Steps | p. 109 |
| Monics equalise | p. 109 |
| Images of arrows | p. 110 |
| Fundamental facts | p. 114 |
| Extensionality and bivalence | p. 115 |
| Monics and epics by elements | p. 123 |
| Logic Classically Conceived | p. 125 |
| Motivating topos logic | p. 125 |
| Propositions and truth-values | p. 126 |
| The propositional calculus | p. 129 |
| Boolean algebra | p. 133 |
| Algebraic semantics | p. 135 |
| Truth-functions as arrows | p. 136 |
| [epsilon]-semantics | p. 140 |
| Algebra of Subobjects | p. 146 |
| Complement, intersection, union | p. 146 |
| Sub(d) as a lattice | p. 151 |
| Boolean topoi | p. 156 |
| Internal vs. external | p. 159 |
| Implication and its implications | p. 162 |
| Filling two gaps | p. 166 |
| Extensionality revisited | p. 168 |
| Intuitionism and its Logic | p. 173 |
| Constructivist philosophy | p. 173 |
| Heyting's calculus | p. 177 |
| Heyting algebras | p. 178 |
| Kripke semantics | p. 187 |
| Functors | p. 194 |
| The concept of functor | p. 194 |
| Natural transformations | p. 198 |
| Functor categories | p. 202 |
| Set Concepts and Validity | p. 211 |
| Set concepts | p. 211 |
| Heyting algebras in P | p. 213 |
| The subobject classifier in Set[superscript p] | p. 215 |
| The truth arrows | p. 221 |
| Validity | p. 223 |
| Applications | p. 227 |
| Elementary Truth | p. 230 |
| The idea of a first-order language | p. 230 |
| Formal language and semantics | p. 234 |
| Axiomatics | p. 237 |
| Models in a topos | p. 238 |
| Substitution and soundness | p. 249 |
| Kripke models | p. 256 |
| Completeness | p. 264 |
| Existence and free logic | p. 266 |
| Heyting-valued sets | p. 274 |
| High-order logic | p. 286 |
| Categorial Set Theory | p. 289 |
| Axioms of choice | p. 290 |
| Natural numbers objects | p. 301 |
| Formal set theory | p. 305 |
| Transitive sets | p. 313 |
| Set-objects | p. 320 |
| Equivalence of models | p. 328 |
| Arithmetic | p. 332 |
| Topoi as foundations | p. 332 |
| Primitive recursion | p. 335 |
| Peano postulates | p. 347 |
| Local Truth | p. 359 |
| Stacks and sheaves | p. 359 |
| Classifying stacks and sheaves | p. 368 |
| Grothendieck topoi | p. 374 |
| Elementary sites | p. 378 |
| Geometric modality | p. 381 |
| Kripke-Joyal semantics | p. 386 |
| Sheaves as complete [Omega]-sets | p. 388 |
| Number systems as sheaves | p. 413 |
| Adjointness and Quantifiers | p. 438 |
| Adjunctions | p. 438 |
| Some adjoint situations | p. 442 |
| The fundamental theorem | p. 449 |
| Quantifiers | p. 453 |
| Logical Geometry | p. 458 |
| Preservation and reflection | p. 459 |
| Geometric morphisms | p. 463 |
| Internal logic | p. 483 |
| Geometric logic | p. 493 |
| Theories as sites | p. 504 |
| References | p. 521 |
| Catalogue of Notation | p. 531 |
| Index of Definitions | p. 541 |
| Table of Contents provided by Ingram. All Rights Reserved. |
