Topoi The Categorial Analysis of Logic

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ISBN: 9780486450261 | 0486450260
Cover: Paperback
Copyright: 4/28/2006

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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.

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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Topoi The Categorial Analysis of Logic

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