The aim of this talk is to introduce di-exact categories [3], which are defined by a simple axiom system capturing self-dual aspects of the context of Janelidze-Márki-Tholen semi-abelian categories [2,1]. A Borceux-Bourn homological category [1] is Barr-exact if and only if it is di-exact. We explain that classical diagram lemmas such as the Snake Lemma hold in di-exact categories, and give an overview of examples and related contexts.

[1] F. Borceux and D. Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Math. Appl., vol. 566, Kluwer Acad. Publ., 2004
[2] G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, Journal of Pure and Applied Algebra 168 (2002), no. 2, 367-386
[3] G. Peschke and T. Van der Linden, A Homological View of Categorical Algebra, arXiv:2404.15896