The purpose of this post is to list some useful background material from category theory. Except from basic notions from category theory such as adjoint functors, Kan extensions, etc, which is covered in MacLane or any other introductory book, there are some important types of categories which pop up in the study of cohomology theories.
Additive and abelian categories. These are covered in any book on homological algebra. Additive categories are essentially just categories enriched over abelian groups (i.e. where each Hom group carry the structure of an abelian group). The theory of abelian categories is slightly more involved, and there is a series of “axioms” for abelian categories called AB3, AB4, and so on, which may or may not be satisfied for a given abelian category. Glossary entries: Additive category, Abelian category. The nLab page on additive and abelian categories is quite useful.
Triangulated categories.
Homotopical categories.
Tannakian categories.
nLab page on B20 Category theory background