Abelian categories

Abelian categories , and antecedent structures like semi-additive categories , additive categories , and pre-abelian categories , axiomatize features of the category of abelian groups and, and more generally, the category of modules over a ring. Abelian categories were among the earliest doctrines to be studied, with the aim of abstracting and generalizing constructions in homological algebra.

Literature

Early, and now somewhat dated, books on abelian categories:

  • Freyd, 1964: Abelian categories (pdf)
  • Popescu, 1973: Abelian categories with applications to rings and modules

Textbook accounts:

  • Herrlich & Strecker, 1973: Category theory, Ch. XI: Pointed categories
  • Mac Lane, 1978: Categories for the working mathematician, Ch. VIII: Abelian categories (doi)
  • Borceux, 1994: Handbook of categorical algebra, Vol 2, Ch. 1: Abelian categories
  • Berrick & Keating, 2000: Categories and modules with K-theory in view, Ch. 2: Categories and exact sequences (doi)
  • Yekutieli, 2019: Derived categories (arxiv), Ch. 2: Abelian categories and additive functors (doi)