nLab
homological category

A category $C$ is homological if it is pointed, regular and protomodular.

Many of the standard results of classical homological algebra in abelian categories extend to homological categories: the five lemma, the nine lemma, the snake lemma, long exact sequence in homology, the Noether isomorphism theorem. There is also a version of the Jordan-Hölder theorem.

A homological category which is Barr-exact and has finite coproducts is semiabelian.

• Dominique Bourn, Francis Borceux, Mal'cev, protomodular, homological and semi-abelian categories, Kluwer 2004.