nLab Categories and Sheaves



Category theory

Topos Theory

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories


This entry provides links related to the book

on basics of category theory and the foundations of homological algebra and abelian sheaf cohomology.

See also the related lecture notes:


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

1 The language of categories

2 Limits

3 Filtrant Limits

4 Tensor Categories

5 Generators and Representability

6 Indization of Categories

7 Localization

8 Additive and Abelian Categories

9 π\pi-accesible Objects and FF-injective Objects

10 Triangulated Categories

11 Complexes in Additive Categories

12 Complexes in Abelian Categories

13 Derived Categories

14 Unbounded Derived Categories

15 Indization and Derivation of Abelian Categories

16 Grothendieck Topologies

17 Sheaves on Grothendieck Topologies

18 Abelian Sheaves

19 Stacks and Twisted Sheaves

category: reference

Last revised on June 19, 2024 at 08:48:27. See the history of this page for a list of all contributions to it.