Front cover image for Topoi : the Categorial Analysis of Logic

Topoi : the Categorial Analysis of Logic

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
eBook, English, 2014
Elsevier Science, Amsterdam, 2014
1 online resource (569 pages)
9781483299211, 148329921X
1041741227
Front Cover; Topoi: The Categorial Analysis of Logic; Copyright Page; Table of Contents; Dedication; PREFACE; PREFACE TO SECOND EDITION; 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. 8. Products9. Co-products; 10. Equalisers; 11. Limits and co-limits; 12. Co-equalisers; 13. The pullback; 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. ] and 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. 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. 3. Functor categoriesCHAPTER 10. SET CONCEPTS AND VALIDITY; 1. Set concepts; 2. Heyting algebras in P; 3. The subobject classifier in Setp; 4. The truth arrows; 5. Validity; 6. Applications; CHAPTER 11. ELEMENTARY TRUTH; 1. The idea of a first-order language; 2. Formal language and semantics; 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. 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. Grothendieck topoi; 4. Elementary sites; 5. Geometric modality; 6. Kripke-Joyal semantics; 7. Sheaves as complete ]-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
3. Internal logic