Kashiwara Masaki, Schapira Pierre. Categories and Sheaves

Springer-Verlag Berlin Heidelberg, 2006. — 495 p.The language of Mathematics has changed drastically since the middle of the twentieth century, in particular after Grothendieck’s ideas spread from algebraic geometry to many other subjects. As an enrichment for the notions of sets and functions, categories and sheaves are new tools which appear almost everywhere nowadays, sometimes simply in the role of a useful language, but often as the natural approach to a deeper understanding of mathematics.In this book, we present categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch and continuing with full proofs to an exposition of the most recent results in the literature, and sometimes beyond. We also present the main features and key results of related topics that would deserve a whole book for themselves (e.g., tensor categories, triangulated categories, stacks).The Language of Categories
Preliminaries: Sets and Universes
Categories and Functors
Morphisms of Functors
The Yoneda Lemma
Adjoint Functors
Exercises
Limits
Examples
Kan Extension of Functors
Inductive Limits in the Category Set
Cofinal Functors
Ind-lim and Pro-lim
Yoneda Extension of Functors
Exercises
Filtrant Limits
Filtrant Inductive Limits in the Category Set
Filtrant Categories
Exact Functors
Categories Associated with Two Functors
Exercises
Tensor Categories
Projectors
Tensor Categories
Rings, Modules and Monads
Exercises
Generators and Representability
Strict Morphisms
Generators and Representability
Strictly Generating Subcategories
Exercises
Indization of Categories
Indization of Categories and Functors
Representable Ind-limits
Indization of Categories Admitting Inductive Limits
Finite Diagrams in Ind(C)
Exercises
Localization
Localization of Categories
Localization of Subcategories
Localization of Functors
Indization and Localization
Exercises
Additive and Abelian Categories
Group Objects
Additive Categories
Abelian Categories
Injective Objects
Ring Action
Indization of Abelian Categories
Extension of Exact Functors
Exercises
π-accessible Objects and F-injective Objects
Cardinals
π-filtrant Categories and π-accessible Objects
π-accessible Objects and Generators
Quasi-Terminal Objects
F-injective Objects
Applications to Abelian Categories
Exercises
Triangulated Categories
Triangulated Categories
Localization of Triangulated Categories
Localization of Triangulated Functors
Extension of Cohomological Functors
The Brown Representability Theorem
Exercises
Complexes in Additive Categories
Differential Objects and Mapping Cones
The Homotopy Category
Complexes in Additive Categories
Simplicial Constructions
Double Complexes
Bifunctors
The Complex Hom•
Exercises
Complexes in Abelian Categories
The Snake Lemma
Abelian Categories with Translation
Complexes in Abelian Categories
Example: Koszul Complexes
Double Complexes
Exercises
Derived Categories
Derived Categories
Resolutions
Derived Functors
Bifunctors
Exercises
Unbounded Derived Categories
Derived Categories of Abelian Categories with Translation
The Brown Representability Theorem
Unbounded Derived Category
Left Derived Functors
Exercises
Indization and Derivation
of Abelian Categories
Injective Objects in Ind(C)
Quasi-injective ObjectsXDerivation of Ind-categories
Indization and Derivation
Exercises
Grothendieck Topologies
Sieves and Local Epimorphisms
Local Isomorphisms
Localization by Local Isomorphisms
Exercises
Sheaves on Grothendieck Topologies
Presites and Presheaves
Sites
Sheaves
Sheaf Associated with a Presheaf
Direct and Inverse Images
Restriction and Extension of Sheaves
Internal Hom
Exercises
18 Abelian Sheaves
R-modules
Tensor Product and Internal Hom
Direct and Inverse Images
Derived Functors for Hom and Hom
Flatness
Ringed Sites
Cech Coverings
Exercises
Stacks and Twisted Sheaves
Prestacks
Simply Connected Categories
Simplicial Constructions
Stacks
Morita Equivalence
Twisted Sheaves
Exercises