A functorial approach to $0$-abelian categories

We use functorial methods to define and study $0$-abelian categories, which we propose to be the case $n = 0$ of Jasso's $n$-abelian categories. In particular, we define a bifunctor for $0$-abelian categories with enough injectives or projectives, which is analogous to the extension bifunctor for an abelian category. We prove a few results concerning this bifunctor, including $0$-abelian versions of the long exact sequence involving the extension bifunctors, of a conjecture due to Auslander on the direct summands of the extension functors, and of the Hilton-Rees theorem. These results are then applied to the study of the stable categories of a $0$-abelian category, and a similar discussion is carried out for the stable categories of an abelian category. Moreover, by specializing our results to modules over rings, we show that $0$-abelian categories with additive generators are in correspondence with semi-hereditary rings. We present applications to these rings.