Monoids and pointed $S$-protomodular categories

Abstract

We investigate the notion of pointed S-protomodular category, with respect to a suitable class S of split epimorphisms, and we prove that these categories satisfy, relatively to the class S, many partial aspects of the properties of Mal’tsev and protomodular categories, like the split short five lemma for the S-split exact sequences, or the fact that a reflexive S-relation is transitive. The main examples of S-protomodular categories are the category of monoids and, more generally, any category of monoids with operations, where the class S is the class of Schreier split epimorphisms.