(also nonabelian homological algebra)
This entry provides a hyperlinked index for the book
Masaki Kashiwara, Pierre Schapira,
Categories and Sheaves
Grundlehren der Mathematischen Wissenschaften 332
Springer (2006)
on basics of category theory and the foundations of homological algebra and abelian sheaf cohomology.
See also the related lecture notes
Summary
The book discusses the theory of presheaves and sheaves with an eye towards their application in homological algebra and with an outlook on stacks.
A self-contained introduction of the basics of presheaf-categories with detailed discussion of representable functors and the corresponding notions of limits, colimits, adjoint functors and ind-objects forms the first third of the book.
The second part describes central concepts and tools of modern category-theoretic homological algebra in terms of derived triangulated categories.
The last part merges these two threads in a discussion of sheaves in general and abelian sheaves in particular. This provides the machinery for the consideration of abelian sheaf cohomology conceptually embedded into the general notion of cohomology and higher stacks, on which the last section provides an outlook.
The organization and emphasis of the book (for instance of the category of sheaves as a localization of the category of presheaves) makes it a suitable 1-categorical preparation for the infinity-categorical discussion of sheaves in
and of triangulated categories, i.e. stable infinity-categories, in
On the other hand, topos-theoretic aspects of the category of sheaves are not emphasized, here
is the natural complementary reading. In particular sections V and VII there are directly useful for supplementing the concept of geometric morphism and its relation to localization.
The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.
For a pedagogical motivation of the general topic under consideration here see
general examples
concrete examples
classical examples: sets with extra structure
functor and presheaf categories
sheafification with respect to a Lawvere-Tierney topology
operations
hindsight motivation
outloook and further reading