Topoi The Categorial Analysis of Logic
by Goldblatt, RobertEdition: Revised
Format: Paperback
Pub. Date: 2006-04-28
Publisher(s): Dover Publications
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Summary
A classic exposition of the branch of mathematical logic known as category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Robert Goldblatt is Professor of Pure Mathematics at New Zealand's Victoria University.
Table of Contents
| 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. |
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