A note on strong protomodularity, actions and quotients

TL;DR

This paper explores properties of the fibration of points to characterize protomodular and strong protomodular categories, providing insights into actions and quotients within categorical algebra.

Contribution

It offers new characterizations of protomodular and strong protomodular categories among quasi-pointed regular and semi-abelian categories, respectively.

Findings

Characterization of protomodular categories among quasi-pointed regular ones

Characterization of strong protomodular categories in the semi-abelian case

Results expressed in terms of internal actions

Abstract

In order to study the problems of extending an action along a quotient of the acted object and along a quotient of the acting object, we investigate some properties of the fibration of points. In fact, we obtain a characterization of protomodular categories among quasi-pointed regular ones, and, in the semi-abelian case, a characterization of strong protomodular categories. Eventually, we return to the initial questions by stating the results in terms of internal actions.