UID:
almahu_9947367866902882
Format:
1 online resource (569 p.)
Edition:
2nd ed.
ISBN:
1-4832-9921-X
Series Statement:
Studies in Logic and the Foundations of Mathematics ; Volume 98
Content:
The first of its kind, this book presents a widely accessible exposition of topos theory, aimed at the philosopher-logician as well as the mathematician. It is suitable for individual study or use in class at the graduate level (it includes 500 exercises). It begins with a fully motivated introduction to category theory itself, moving always from the particular example to the abstract concept. It then introduces the notion of elementary topos, with a wide range of examples and goes on to develop its theory in depth, and to elicit in detail its relationship to Kripke's intuitionistic semantics,
Note:
Description based upon print version of record.
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Front Cover; Topoi: The Categorial Analysis of Logic; Copyright Page; Dedication; PREFACE; PREFACE TO SECOND EDITION; Table of Contents; PROSPECTUS; CHAPTER 1. MATHEMATICS = SET THEORY?; 1. Set theory; 2. Foundations of mathematics; 3. Mathematics as set theory; CHAPTER 2. WHAT CATEGORIES ARE; 1. Functions are sets?; 2. Composition of functions; 3. Categories: first examples; 4. The pathology of abstraction; 5. Basic examples; CHAPTER 3. ARROWS INSTEAD OF EPSILON; 1. Monic arrows; 2. Epic arrows; 3. Iso arrows; 4. Isomorphic objects; 5. Initial objects; 6. Terminal objects; 7. Duality
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8. Products9. Co-products; 10. Equalisers; 11. Limits and co-limits; 12. Co-equalisers; 13. Thepullback; 14. Pushouts; 15. Completeness; 16. Exponentiation; CHAPTER 4. INTRODUCING TOPOI; 1. Subobjects; 2. Classifying subobjects; 3. Definition of topos; 4. First examples; 5. Bundles and sheaves; 6. Monoid actions; 7. Power objects; 8. Ωand comprehension; CHAPTER 5. TOPOS STRUCTURE: FIRST STEPS; 1. Monies equalise; 2. Images of arrows; 3. Fundamental facts; 4. Extensionality and bivalence; 5. Monies and epics by elements; CHAPTER 6. LOGIC CLASSICALLY CONCEIVED; 1. Motivating topos logic
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2. Propositions and truth-values3. The prepositional calculus; 4. Boolean algebra; 5. Algebraic semantics; 6. Truth-functions as arrows; 7.E-semantics; CHAPTER 7. ALGEBRA OF SUBOBJECTS; 1. Complement, intersection, union; 2. Sub(d) as a lattice; 3. Boolean topoi; 4. Internal vs. external; 5. Implication and its implications; 6. Filling two gaps; 7. Extensionality revisited; CHAPTER 8. INTUITIONISM AND ITS LOGIC; 1. Constructivist philosophy; 2. Heyting's calculus; 3. Heyting algebras; 4. Kripke semantics; CHAPTER 9. FUNCTORS; 1. The concept of functor; 2. Natural transformations
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3. Functor categoriesCHAPTER 10. SET CONCEPTS AND VALIDITY; 1. Set concepts; 2. Heyting algebras in P; 3. The subobject classifier inSetp; 4. The truth arrows; 5. Validity; 6. Applications; CHAPTER 11. ELEMENTARY TRUTH; 1. The idea of a first-orderlanguage; 2. Formal language andsemantics; 3. Axiomatics; 4. Models in a topos; 5. Substitution and soundness; 6. Kripke models; 7. Completeness; 8. Existence and free logic; 9. Heyting-valued sets; 10. High-order logic; CHAPTER 12. CATEGORIAL SET THEORY; 1. Axioms of choice; 2. Natural numbers objects; 3. Formal set theory; 4. Transitive sets
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5. Set-objects6. Equivalence of models; CHAPTER 13. ARITHMETIC; 1. Topoi as foundations; 2. Primitive recursion; 3. Peano postulates; CHAPTER 14. LOCAL TRUTH; 1. Stacks and sheaves; 2. Classifying stacks and sheaves; 3. Grothendiecktopoi; 4. Elementary sites; 5. Geometric modality; 6. Kripke-Joyalsemantics; 7. Sheaves as completeΩ-sets; 8. Number systems as sheaves; CHAPTER 15. ADJOINTNESS AND QUANTIFIERS; 1. Adjunctions; 2. Some adjoint situations; 3. The fundamental theorem; 4. Quantifiers; CHAPTER 16. LOGICAL GEOMETRY; 1. Preservation and reflection; 2. Geometric morphisms
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3. Internal logic
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English
Additional Edition:
ISBN 1-322-47986-0
Additional Edition:
ISBN 0-444-86711-2
Language:
English
