Topos Theory (Dover Books on Mathematics)

One of the best books on a relatively new branch of mathematics, this text is the work of a leading authority in the field of topos theory. Suitable for advanced undergraduates and graduate students of mathematics, the treatment focuses on how topos theory integrates geometric and logical ideas into the foundations of mathematics and theoretical computer science.After a brief overview, the approach begins with elementary toposes and advances to internal category theory, topologies and sheaves, geometric morphisms, and logical aspects of topos theory. Additional topics include natural number objects, theorems of Deligne and Barr, cohomology, and set theory. Each chapter concludes with a series of exercises, and an appendix and indexes supplement the text. Preface Introduction Notes for the Reader Chapter 0: Preliminaries 0.1 Category Theory 0.2 Sheaf Theory 0.3 Grothendieck Topologies 0.4 Giraud's Theorem Exercises 0 Chapter 1: Elementary Toposes 1.1 Definition and Examples 1.2 Equivalence Relations and Partial Maps 1.3 The Category Eop 1.4 Pullback Functors 1.5 Image Factorizations Exercises 1 Chapter 2: Internal Category Theory 2.1 Internal Categories and Diagrams 2.2 Internal Limits and Colimits 2.3 Diagrams in a Topos 2.4 Internal Profunctors 2.5 Filtered Categories Exercises 2 Chapter 3: Topologies and Sheaves 3.1 Topologies 3.2 Sheaves 3.3 The Associated Sheaf Functor 3.4 shj(E) as a Category of Fractions 3.5 Examples of Topologies Exercises 3 Chapter 4: Geometric Morphisms 4.1 The Factorization Theorem 4.2 The Glueing Construction 4.3 Diaconescu's Theorem 4.4 Bounded Morphisms Exercises 4 Chapter 5: Logical Aspects of Topos Theory 5.1 Boolean Toposes 5.2 The Axiom of Choice 5.3 The Axiom (SG) 5.4 The Mitchell-Bénabou Language Exercises 5 Chapter 6: Natural Number Objects 6.1 Definition and Basic Properties 6.2 Finite Cardinals 6.3 The Object Classifier 6.4 Algebraic Theories 6.5 Geometric Theories 6.6 Real Number Objects Exercises 6 Chapter 7: Theorems of Deligne and Barr 7.1 Points 7.2 Spatial Toposes 7.3 Coherent Toposes 7.4 Deligne's Theorem 7.5 Barr's Theorem Exercises 7 Chapter 8: Cohomology 8.1 Basic Definitions 8.2 Čech Cohomology 8.3 Torsors 8.4 Profinite Fundamental Groups Exercises 8 Chapter 9: Topos Theory and Set Theory 9.1 Kuratowski-Finiteness 9.2 Transitive Objects 9.3 The Equiconsistency Theorem 9.4 The Filterpower Construction 9.5 Independence of the Continuum Hypothesis Exercises 9 Appendix: Locally Internal Categories Bibliography Index of Definitions Index of Notation Index of Names P. T. Johnstone. Includes Indexes. Bibliography: P. 347-356.