Relative category theory and geometric morphisms: a logic approach
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1992
|
| Schriftenreihe: | Oxford logic guides
16 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 263 S. |
| ISBN: | 0198534345 9780198534341 |
Internformat
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| 100 | 1 | |a Chapman, Jonathan |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Relative category theory and geometric morphisms |b a logic approach |c Jonathan Chapman and Frederick Rowbottom |
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| 300 | |a XI, 263 S. | ||
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Datensatz im Suchindex
| _version_ | 1804119799349903360 |
|---|---|
| adam_text | CONTENTS
1. Introduction l
1.1 Introduction 1
1.2 The logical approach to topos theory 1
1.3 Frames for category theory 2
1.4 L , a language for the S frame 3
1.5 Categories in S 3
1.6 Discussion of categories in S 4
1.7 A representation theorem for topoi over S 5
1.8 An analogous result for cartesian closed categories 6
1.9 A language for topoi in S and the relative Giraud theorem 6
1.10 A note for logicians 6
1.11 Prerequisites 7
1.12 Some historical comments 7
1.13 Notation 7
2. Local set theories 9
2.1 Introduction 9
2.2 Preliminaries 9
2.3 A language, L 12
2.4 Topoi 14
2.5 Interpretations of a language in a topos 16
2.6 The proper substitutions of L 17
2.7 The interpretation of substitution 21
2.8 Validity and set theories 23
2.9 The theory of an interpretation (soundness) 26
2.10 Logic 27
2.11 Further constructions in a set theory 29
2.12 The theory of a topos 29
3. Partial function theory, L 31
3.1 Introduction 31
3.2 The motivation for L 31
3.3 The language of L 32
3.4 The definitional operator, D 32
viii Contents
3.5 Substitution 33
3.6 Axioms for the definitional operator 33
3.7 Equality 34
3.8 Logical equivalence 35
3.9 The // axioms and rules 36
3.10 Some theorems of L 36
3.11 Equality 37
3.12 Extensionality for sets 37
3.13 Domains of definition for the logical operators 37
3.14 Quantification 38
3.15 Interchangeability, = 38
4. The interpretation of L 40
4.1 Introduction 40
4.2 The interpretation of L in L 40
4.3 Validity 41
4.4 L is a conservative extension of L 41
4.5 Soundness for L 41
4.6 L partial functions 42
4.7 Internal partial functions 42
4.8 Description operators 43
4.9 Internal functions defined by terms (functional abstraction) 43
4.10 Illustrative proof in L 44
4.11 Internal functions defined by description 45
4.12 Morphisms and subobjects in S, defined using L 45
4.13 Alternative notions of set in L 47
4.14 Relative definition and interchangeability 48
5. Equationals 50
5.1 Overview 50
5.2 Some categorical structure for partial functions 50
5.3 Equationals 54
5.4 The left exact interpretation of equationals 55
5.5 The categorical notion of validity coincides with validity in L 56
5.6 Left exact functors preserve equationals 56
5.7 Reflection of equationals 57
Contents ix
6. Categories in a topos 59
6.1 Introduction 59
6.2 Classes and operations in S 59
6.3 A note on foundations 60
6.4 Examples of classes in S 61
6.5 Isos, monos, and epis in CLASS(S) 62
6.6 Categories, functors, and natural transformations in a topos 62
6.7 Examples 65
6.8 Isos in a category € in S 71
6.9 Adjunctions in S 72
7. Topoi in a topos 76
7.1 Introduction 76
7.2 Terminals 76
7.3 Pullbacks 76
7.4 Finite completeness and products of pairs 78
7.5 Left exact functors 79
7.6 Monos 80
7.7 Subobject classifier 81
7.8 Cartesian closed categories 82
7.9 Topoi and geometric morphisms 83
7.10 Preservation of topoi 83
8. A representation theorem for geometric morphisms 84
8.1 Introduction 84
8.2 A representation of geometric morphisms 84
8.3 A characterization of topoi over if 85
8.4 y y* nas copowers of 1 89
8.5 S = GK^ y,) over S: proof 94
8.6 Extending the representation theorem to geometric morphisms over S 95
8.7 A representation theorem for cartesian morphisms to S 97
9. Local set theories in S 100
9.1 Introduction 100
9.2 Preliminaries 100
9.3 A typed, set theoretic language i£ in S 106
9.4 The global part of a language in S 109
9.5 Interpretations of a language in a topos in S 111
x Contents
9.6 The proper substitutions of jSf 115
9.7 The interpretation of substitution 118
9.8 Validity and set theories 120
9.9 The theory of an interpretation (soundness) 125
9.10 Logic 130
9.11 Further constructions in a set theory 138
10. The theory of a topos in S 140
10.1 Introduction 140
10.2 The theory of S 140
10.3 Basic results in the theory of i 141
10.4 3!, descriptions and cartesian closure 145
10.5 Characterization of finite limits in £ 148
10.6 Finite coproducts 149
10.7 Kernels, coequalizers, and equivalence relations 153
10.8 Set indexed disjunctions and coproducts 157
10.9 Colimits 163
10.10 Topoi over S correspond to cocomplete topoi in S 167
11. Topologies and sheaves 172
11.1 Introduction 172
11.2 Topologies 172
11.3 Sheaves 177
11.4 The Associated Sheaf Functor 184
12. The relative Giraud theorem 188
12.1 Introduction 188
12.2 The cocompiete topos yc™ 188
12.3 Generators 195
12.4 Topologies given by a class of monos 200
12.5 The relative Giraud theorem 208
Contents xi
Appendix 215
Al. Introduction 215
A2. /) Sets, functions, and partial functions 216
A3. The topos structure of S^, 220
A4. The categories S^, are essentially the comma categories (SJ./) 222
A5. ^ Classes and categories 224
A6. Isos, 0 adjunctions, and equivalences 228
A7. Linking the various categories of internal sets 231
A8. Finite 0 limits 233
A9. Further examples of 0 categories and functors 235
A10. Small (^ limits 237
All. Preservation and creation of finite and small ^ limits 240
A12. The general adjoint functor theorem 243
A13. Two special adjoint functor theorems 247
References 255
Index 259
|
| any_adam_object | 1 |
| author | Chapman, Jonathan Rowbottom, Frederick |
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| ctrlnum | (OCoLC)644055881 (DE-599)BVBBV005586953 |
| discipline | Mathematik |
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| id | DE-604.BV005586953 |
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| indexdate | 2024-07-09T16:31:56Z |
| institution | BVB |
| isbn | 0198534345 9780198534341 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003498918 |
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| physical | XI, 263 S. |
| publishDate | 1992 |
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| publisher | Clarendon Press |
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| series | Oxford logic guides |
| series2 | Oxford logic guides |
| spelling | Chapman, Jonathan Verfasser aut Relative category theory and geometric morphisms a logic approach Jonathan Chapman and Frederick Rowbottom Oxford Clarendon Press 1992 XI, 263 S. txt rdacontent n rdamedia nc rdacarrier Oxford logic guides 16 Morphismus (DE-588)4149340-0 gnd rswk-swf Kategorientheorie (DE-588)4120552-2 gnd rswk-swf Topos Mathematik (DE-588)4185717-3 gnd rswk-swf Topos Mathematik (DE-588)4185717-3 s DE-604 Kategorientheorie (DE-588)4120552-2 s Morphismus (DE-588)4149340-0 s Rowbottom, Frederick Verfasser aut Oxford logic guides 16 (DE-604)BV000013997 16 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003498918&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chapman, Jonathan Rowbottom, Frederick Relative category theory and geometric morphisms a logic approach Oxford logic guides Morphismus (DE-588)4149340-0 gnd Kategorientheorie (DE-588)4120552-2 gnd Topos Mathematik (DE-588)4185717-3 gnd |
| subject_GND | (DE-588)4149340-0 (DE-588)4120552-2 (DE-588)4185717-3 |
| title | Relative category theory and geometric morphisms a logic approach |
| title_auth | Relative category theory and geometric morphisms a logic approach |
| title_exact_search | Relative category theory and geometric morphisms a logic approach |
| title_full | Relative category theory and geometric morphisms a logic approach Jonathan Chapman and Frederick Rowbottom |
| title_fullStr | Relative category theory and geometric morphisms a logic approach Jonathan Chapman and Frederick Rowbottom |
| title_full_unstemmed | Relative category theory and geometric morphisms a logic approach Jonathan Chapman and Frederick Rowbottom |
| title_short | Relative category theory and geometric morphisms |
| title_sort | relative category theory and geometric morphisms a logic approach |
| title_sub | a logic approach |
| topic | Morphismus (DE-588)4149340-0 gnd Kategorientheorie (DE-588)4120552-2 gnd Topos Mathematik (DE-588)4185717-3 gnd |
| topic_facet | Morphismus Kategorientheorie Topos Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003498918&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000013997 |
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