For example, the 5-lemma, 9-lemma?, the snake lemma are some of the “elements” which can be generalized to a wide class of categories. Typical classes of categories suitable for some elements of homological algebra are protomodular categories, homological categories, and semi-abelian categories.
Though they may be shown to be related the subject of nonabelian cohomology is a bit different from nonabelian homological algebra.
The nonabelian groups were historical motivation for much of the subject. While abelian groups are the template of an object in an abelian category, large part of work was directed toward understanding of what is the right generalization of concept of a group, what lead to a definition of a semi-abelian category.
A recent independent development is the work of Alexander Rosenberg listed below in references. It features also the nonadditive triangulated categories and a new version of algebraic K-theory as byproducts. In unfinished part of the work further generalization of the homological algebra in the fibered categories is considered.
Francis Borceux, Dominique Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004