Topos Theory

by P.T. Johnstone

Overview

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.

Product Details

ISBN-13:	9780486783093
Publisher:	Dover Publications
Publication date:	12/08/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	400
File size:	34 MB
Note:	This product may take a few minutes to download.

About the Author

P. T. Johnstone is Professor of the Foundations of Mathematics at the University of Cambridge and a Fellow of St. John's College, Cambridge.

Table of Contents

Preface v

Introduction xi

Notes for the Reader xxi

Chapter 0 Preliminaries 1

0.1 Category Theory 1

0.2 Sheaf Theory 8

0.3 Grothendieck Topologies 12

0.4 Giraud's Theorem 15

Exercises 0 18

Chapter 1 Elementary Toposes 23

1.1 Definition and Examples 23

1.2 Equivalence Relations and Partial Maps 27

1.3 The Category ε^op 31

1.4 Pullback Functors 35

1.5 Image Factorizations 40

Exercises 1 43

Chapter 2 Internal Category Theory 47

2.1 Internal Categories and Diagrams 47

2.2 Internal Limits and Colimits 50

2.3 Diagrams in a Topos 53

2.4 Internal Profunctors 59

2.5 Filtered Categories 65

Exercises 2 72

Chapter 3 Topologies and Sheaves 76

3.1 Topologies 76

3.2 Sheaves 81

3.3 The Associated Sheaf Functor 84

3.4 sh_j(ε) as a Category of Fractions 90

3.5 Examples of Topologies 93

Exercises 3 99

Chapter 4 Geometric Morphisms 103

4.1 The Factorization Theorem 103

4.2 The Glueing Construction 107

4.3 Diaconescu's Theorem 112

4.4 Bounded Morphisms 119

Exercises 4 132

Chapter 5 Logical Aspects of Topos Theory 136

5.1 Boolean Toposes 136

5.2 The Axiom of Choice 140

5.3 The Axiom (SG) 145

5.4 The Mitchell-Benabou Language 152

Exercises 5 161

Chapter 6 Natural Number Objects 165

6.1 Definition and Basic Properties 165

6.2 Finite Cardinals 173

6.3 The Object Classifier 180

6.4 Algebraic Theories 190

6.5 Geometric Theories 198

6.6 Real Number Objects 210

Exercises 6 220

Chapter 7 Theorems of Deligne and Barr 224

7.1 Points 224

7.2 Spatial Toposes 229

7.3 Coherent Toposes 232

7.4 Deligne's Theorem 240

7.5 Barr's Theorem 249

Exercises 7 254

Chapter 8 Cohomology 259

8.1 Basic Definitions 259

8.2 Cech Cohomology 266

8.3 Torsors 272

8.4 Profinite Fundamental Groups 283

Exercises 8 290

Chapter 9 Topos Theory and Set Theory 296

9.1 Kuratowski-Finiteness 296

9.2 Transitive Objects 303

9.3 The Equiconsistency Theorem 312

9.4 The Filterpower Construction 319

9.5 Independence of the Continuum Hypothesis 323

Exercises 9 330

Appendix: Locally Internal Categories 334

Bibliography 347

Index of Definitions 357

Index of Notation 361

Index of Names 366