Gutsche, Sebastian: Constructive category theory and applications to algebraic geometryKonstruktive Kategorientheorie und Anwendungen in algebraischer Geometrie. 2017

Inhalt

Preface
Summary
Zusammenfassung
Contents
Chapter I. Introduction
Chapter II. Computability of categories

1. Computable functions and decidable sets
2. Categories with Hom-setoids
3. Categories with Hom-setoids vs. classical categories
4. Computable categories
5. Decidable properties
6. Preadditive categories
7. Additive categories
8. Preabelian categories
9. Abelian categories
10. Categorical notions

Chapter III. Implementation of graded modules

1. The category of graded module presentations
2. Computability of graded module presentations

Chapter IV. Generalized morphisms and Serre quotients

1. The category of generalized morphisms
2. Structure of the category of generalized morphisms
3. Generalized and pseudo-inverse
4. Serre quotients
5. Computability of Serre quotients

Chapter V. The category of coherent sheaves over a toric variety

1. Preliminaries from toric geometry
2. Equivalence of Serre quotient and coherent sheaves
3. Deciding membership of the kernel of the sheafification functor

Chapter VI. Application

1. Preliminaries
2. Bicomplexes
3. Internal Hom and Ext
4. Grade filtration
5. Spectral sequences
6. Filtered presentation
7. Coherent sheaves

Chapter VII. Implementation of computable categories

1. The concept of categorical programming
2. Main design goal and feature
3. Error messages for categorical operations
4. Undecidable realizations
5. Ensuring compatibility: WithGiven operations
6. Caching
7. Primitive and derived categorical operations
8. Logic Propagation: ToDoLists

Bibliography
Appendix A. Programming in Cap

1. An overview of installing categories
2. The category object
3. Functors and natural transformations: The category of categories
4. Special categories implemented in Cap

Appendix B. Logical theorems in Cap

1. Logic for all categories
2. Logic for preadditive categories
3. Logic for additive categories
4. Logic for abelian categories

Appendix C. All method names
Appendix D. Derivations
Appendix E. Final Derivations
Appendix F. Installed basic operations

1. Primitive operation index
2. Primitive operations for left module presentations
3. Primitive operations for right module presentations
4. Primitive operations for graded left module presentations
5. Primitive operations for graded right module presentations
6. Primitive operations for generalized morphisms by cospans
7. Primitive operations for generalized morphisms by spans
8. Primitive operations for generalized morphisms by three arrows
9. Primitive operations for Serre quotient by cospans
10. Primitive operations for Serre quotient by spans
11. Primitive operations for Serre quotient by three arrows

Appendix G. Application code

1. Function ResolutionFunctor
2. Function ResolutionFunctorToComplex
3. Function ResolutionFunctorToCocomplex
4. Function FreeResolutionComplex
5. Function FreeResolutionCocomplex
6. Function ResolutionTo
7. Function CAP INTERNAL HORSE SHOE HELPER
8. Function HorseShoeLemma
9. Function CartanEilenbergResolution
10. Function DualOnComplex
11. Function DualOnCocomplex
12. Function DualOnCochainMap
13. Function DualOnCocomplexCocomplex
14. Function TransposeComplexOfComplex
15. Function ResolutionLength
16. Function TotalComplexOfBicomplex
17. Function EmbeddingInObjectOfTotalComplex
18. Function ConnectingMorphismFromCocomplexToCartanEilenbergResolution
19. Function GeneralizedEmbeddingOfHomology
20. Function GeneralizedMorphismBetweenHomologies
21. Function GeneralizedEmbeddingOfSpectralSequenceEntry
22. Function PurityFiltrationBySpectralSequence

Index

Detailsuche

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