A new approach to S-protomodular categories

TL;DR

This paper introduces a novel approach to S-protomodular categories using generalized points, enhancing the understanding of split and regular Schreier epimorphisms in monoids.

Contribution

It redefines S-protomodular categories with generalized points, providing a more convenient framework for analyzing Schreier epimorphisms in monoids.

Findings

Generalized points are composable pairs of morphisms with pullback stable regular epimorphisms.

The new approach clarifies the connection between split and regular Schreier epimorphisms.

Enhanced categorical framework for monoid epimorphisms.

Abstract

We propose a new approach to S-protomodular categories in the sense of D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral. Instead of points (=split epimorphisms) it uses generalized points, which we define as composable pairs of morphisms whose composites are pullback stable regular epimorphisms. This approach is convenient in describing the connection between split and regular Schreier epimorphisms of monoids.