nLab homological category

Definition

A category $C$ is called homological if it is

1. pointed

2. regular

3. protomodular.

Properties

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

• the long exact sequence in homology

• the Noether isomorphism theorem

• a version of the Jordan-Hölder theorem.

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

Examples

Example

The category Grp of all groups (including non-abelian groups) is homological. Namely it is

1. regular, by this example,

2. pointed protomodular by this example.

Related entries

• abelian category, semiabelian category

• Barr-exact category

• homological algebra

• pointed category

• protomodular category

• regular category

References

• Francis Borceux, Dominique Bourn, Chapter IV in: Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer, 2004 (doi:10.1007/978-1-4020-1962-3)

• Francis Borceux, Marco Grandis, Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra 208 (2007), no. 2, 665–689. (doi, MR2007k:18021)

• Tamar Janelidze, Foundations of relative non-abelian homological algebra, 2009 (pdf, pdf, hdl:11427/4891)