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Categories and Sheaves

Image from Gyazo

Authors: Masaki Kashiwara, Pierre Schapira

  • Introduction

    1. The Language of Categories
    • 1.1 Preliminaries: Sets and Universes
    • 1.2 Categories and Functors
    • 1.3 Morphisms of Functors
    • 1.4 The Yoneda Lemma
    • 1.5 Adjoint Functors
    • Exercises
    1. Limits
    • 2.1 Limits
    • 2.2 Examples
    • 2.3 Kan Extension of Functors
    • 2.4 Inductive Limits in the Category Set
    • 2.5 Cofinal Functors
    • 2.6 Ind-lim and Pro-lim
    • 2.7 Yoneda Extension of Functors
    • Exercises
    1. Filtrant Limits
    • 3.1 Filtrant Inductive Limits in the Category Set
    • 3.2 Filtrant Categories
    • 3.3 Exact Functors
    • 3.4 Categories Associated with Two Functors
    • Exercises
    1. Tensor Categories
    • 4.1 Projectors
    • 4.2 Tensor Categories
    • 4.3 Rings, Modules and Monads
    • Exercises
    1. Generators and Representability
    • 5.1 Strict Morphisms
    • 5.2 Generators and Representability
    • 5.3 Strictly Generating Subcategories
    • Exercises
    1. Indization of Categories
    • 6.1 Indization of Categories and Functors
    • 6.2 Representable Ind-limits
    • 6.3 Indization of Categories Admitting Inductive Limits
    • 6.4 Finite Diagrams in Ind(C)
    • Exercises
    1. Localization
    • 7.1 Localization of Categories
    • 7.2 Localization of Subcategories
    • 7.3 Localization of Functors
    • 7.4 Indization and Localization
    • Exercises
    1. Additive and Abelian Categories
    • 8.1 Group Objects
    • 8.2 Additive Categories
    • 8.3 Abelian Categories
    • 8.4 Injective Objects
    • 8.5 Ring Action
    • 8.6 Indization of Abelian Categories
    • 8.7 Extension of Exact Functors
    • Exercises
    1. π-accessible Objects and F-injective Objects
    • 9.1 Cardinals
    • 9.2 π-filtrant Categories and π-accessible Objects
    • 9.3 π-accessible Objects and Generators
    • 9.4 Quasi-Terminal Objects
    • 9.5 F-injective Objects
    • 9.6 Applications to Abelian Categories
    • Exercises
    1. Triangulated Categories
    • 10.1 Triangulated Categories
    • 10.2 Localization of Triangulated Categories
    • 10.3 Localization of Triangulated Functors
    • 10.4 Extension of Cohomological Functors
    • 10.5 The Brown Representability Theorem
    • Exercises
    1. Complexes in Additive Categories
    • 11.1 Differential Objects and Mapping Cones
    • 11.2 The Homotopy Category
    • 11.3 Complexes in Additive Categories
    • 11.4 Simplicial Constructions
    • 11.5 Double Complexes
    • 11.6 Bifunctors
    • 11.7 The Complex Hom•
    • Exercises
    1. Complexes in Abelian Categories
    • 12.1 The Snake Lemma
    • 12.2 Abelian Categories with Translation
    • 12.3 Complexes in Abelian Categories
    • 12.4 Example: Koszul Complexes
    • 12.5 Double Complexes
    • Exercises
    1. Derived Categories
    • 13.1 Derived Categories
    • 13.2 Resolutions
    • 13.3 Derived Functors
    • 13.4 Bifunctors
    • Exercises
    1. Unbounded Derived Categories
    • 14.1 Derived Categories of Abelian Categories with Translation
    • 14.2 The Brown Representability Theorem
    • 14.3 Unbounded Derived Category
    • 14.4 Left Derived Functors
    • Exercises
    1. Indization and Derivation of Abelian Categories
    • 15.1 Injective Objects in Ind(C)
    • 15.2 Quasi-injective Objects
    • 15.3 Derivation of Ind-categories
    • 15.4 Indization and Derivation
    • Exercises
    1. Grothendieck Topologies
    • 16.1 Sieves and Local Epimorphisms
    • 16.2 Local Isomorphisms
    • 16.3 Localization by Local Isomorphisms
    • Exercises
    1. Sheaves on Grothendieck Topologies
    • 17.1 Presites and Presheaves
    • 17.2 Sites
    • 17.3 Sheaves
    • 17.4 Sheaf Associated with a Presheaf
    • 17.5 Direct and Inverse Images
    • 17.6 Restriction and Extension of Sheaves
    • 17.7 Internal Hom
    • Exercises
    1. Abelian Sheaves
    • 18.1 R-modules
    • 18.2 Tensor Product and Internal Hom
    • 18.3 Direct and Inverse Images
    • 18.4 Derived Functors for Hom and Hom
    • 18.5 Flatness
    • 18.6 Ringed Sites
    • 18.7 Čech Coverings
    • Exercises
    1. Stacks and Twisted Sheaves
    • 19.1 Prestacks
    • 19.2 Simply Connected Categories
    • 19.3 Simplicial Constructions
    • 19.4 Stacks
    • 19.5 Morita Equivalence
    • 19.6 Twisted Sheaves
    • Exercises

  • References

  • List of Notations

  • Index

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