Bourn-normal monomorphisms in regular Mal'tsev categories

TL;DR

This paper explores Bourn-normal monomorphisms in regular Mal'tsev categories, clarifying their relationship with internal equivalence relations and providing detailed descriptions in specific categorical settings.

Contribution

It establishes connections between internal equivalence relations and Bourn-normal monomorphisms in regular Mal'tsev categories, extending understanding in quasi-pointed cases.

Findings

Characterizes Bourn-normal monomorphisms in regular Mal'tsev categories.

Provides a full description in quasi-pointed regular Mal'tsev categories.

Clarifies the role of pushouts of split monomorphisms.

Abstract

Normal monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this subject often focus on protomodular settings, where normality becomes a property. This paper clarifies the connections between internal equivalence relations and Bourn-normal monomorphisms in regular Mal'tesv categories with pushouts of split monomorphisms along arbitrary morphisms, whereas a full description is achieved for quasi-pointed regular Mal'tsev categories with pushouts of split monomorphisms along arbitrary morphisms.