Sheaves in Geometry and Logic : A First Introduction to Topos Theory (Universitext)

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Sheaves in Geometry and Logic : A First Introduction to Topos Theory (Universitext)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 629 p.
  • 言語 ENG
  • 商品コード 9780387977102
  • DDC分類 512.55

Full Description


Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.

Table of Contents

Preface                                            vii
Prologue 1 (9)
Categorical Preliminaries 10 (14)
Categories of Functors 24 (40)
The Categories at Issue 24 (5)
Pullbacks 29 (2)
Characteristic Functions of Subobjects 31 (4)
Typical Subobject Classifiers 35 (4)
Colimits 39 (5)
Exponentials 44 (4)
Propositional Calculus 48 (2)
Heyting Algebras 50 (7)
Quantifiers as Adjoints 57 (7)
Exercises 62 (2)
Sheaves of Sets 64 (42)
Sheaves 65 (4)
Sieves and Sheaves 69 (4)
Sheaves and Manifolds 73 (6)
Bundles 79 (4)
Sheaves and Cross-Sections 83 (5)
Sheaves as Etale Spaces 88 (7)
Sheaves with Algebraic Structure 95 (2)
Sheaves are Typical 97 (2)
Inverse Image Sheaf 99 (7)
Exercises 103(3)
Grothendieck Topologies and Sheaves 106(55)
Generalized Neighborhoods 106(3)
Grothendieck Topologies 109(7)
The Zariski Site 116(5)
Sheaves on a Site 121(7)
The Associated Sheaf Functor 128(6)
First Properties of the Category of Sheaves 134(6)
Subobject Classifiers for Sites 140(5)
Subsheaves 145(5)
Continuous Group Actions 150(11)
Exercises 155(6)
First Properties of Elementary Topoi 161(57)
Definition of a Topos 161(6)
The Construction of Exponentials 167(4)
Direct Image 171(5)
Monads and Beck's Theorem 176(4)
The Construction of Colimits 180(4)
Factorization and Images 184(6)
The Slice Category as a Topos 190(8)
Lattice and Heyting Algebra Objects in a 198(6)
Topos
The Beck-Chevalley Condition 204(6)
Injective Objects 210(8)
Exercises 213(5)
Basic Constructions of Topoi 218(49)
Lawvere-Tierney Topologies 219(4)
Sheaves 223(4)
The Associated Sheaf Functor 227(6)
Lawvere-Tierney Subsumes Grothendieck 233(2)
Internal Versus External 235(2)
Group Actions 237(3)
Category Actions 240(7)
The Topos of Coalgebras 247(9)
The Filter-Quotient Construction 256(11)
Exercises 263(4)
Topoi and Logic 267(80)
The Topos of Sets 268(9)
The Cohen Topos 277(7)
The Preservation of Cardinal Inequalities 284(7)
The Axiom of Choice 291(5)
The Mitchell-Benabou Language 296(6)
Kripke-Joyal Semantics 302(13)
Sheaf Semantics 315(3)
Real Numbers in a Topos 318(6)
Brouwer's Theorem: All Functions are 324(7)
Continuous
Topos-Theoretic and Set-Theoretic 331(16)
Foundations
Exercises 343(4)
Geometric Morphisms 347(72)
Geometric Morphisms and Basic Examples 348(5)
Tensor Products 353(8)
Group Actions 361(5)
Embeddings and Surjections 366(12)
Points 378(6)
Filtering Functors 384(6)
Morphisms into Grothendieck Topoi 390(4)
Filtering Functors into a Topos 394(5)
Geometric Morphisms as Filtering Functors 399(8)
Morphisms Between Sites 407(12)
Exercises 414(5)
Classifying Topoi 419(51)
Classifying Spaces in Topology 420(3)
Torsors 423(9)
Classifying Topoi 432(2)
The Object Classifier 434(3)
The Classifying Topos for Rings 437(8)
The Zariski Topos Classifies Local Rings 445(5)
Simplicial Sets 450(5)
Simplicial Sets Classify Linear Orders 455(15)
Exercises 466(4)
Localic Topoi 470(56)
Locales 471(2)
Points and Sober Spaces 473(2)
Spaces from Locales 475(5)
Embeddings and Surjections of Locales 480(7)
Localic Topoi 487(4)
Open Geometric Morphisms 491(9)
Open Maps of Locales 500(6)
Open Maps and Sites 506(5)
The Diaconescu Cover and Barr's Theorem 511(3)
The Stone Space of a Complete Boolean 514(5)
Algebra
Deligne's Theorem 519(7)
Exercises 521(5)
Geometric Logic and Classifying Topoi 526(46)
First-Order Theories 527(3)
Models in Topoi 530(3)
Geometric Theories 533(6)
Categories of Definable Objects 539(14)
Syntactic Sites 553(6)
The Classifying Topos of a Geometric Theory 559(7)
Universal Models 566(6)
Exercises 569(3)
Appendix: Sites for Topoi 572(24)
1. Exactness Conditions 572(3)
2. Construction of Coequalizers 575(3)
3. The Construction of Sites 578(9)
4. Some Consequences of Giraud's Theorem 587(9)
Epilogue 596(7)
Bibliography 603(10)
Index of Notation 613(4)
Index 617