TITLE: A generalization of abelian categories including Grp
QUESTION [7 upvotes]: The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp such that the 5- and 9- lemmas are still provable by arrow chasing?

REPLY [8 votes]: Semi-abelian categories are good contexts for doing homological algebra, and $\mathbf{Grp}$ is semi-abelian. One of the axioms is the split short five lemma, and the five lemma, nine lemma, and snake lemma are theorems. A category is abelian if and only if it is semi-abelian and its opposite is semi-abelian.