Intrinsic Schreier split extensions

TL;DR

This paper introduces an intrinsic categorical concept of Schreier split epimorphisms in regular unital categories, extending their homological properties and providing new insights into the structure of monoids.

Contribution

It defines an intrinsic version of Schreier split epimorphisms in a categorical setting, generalizing their properties and applications beyond monoids.

Findings

Schreier split epimorphisms satisfy homological properties similar to those in monoids.

New examples of S-protomodular categories are identified.

Enhanced understanding of monoids' homological behavior categorically.

Abstract

In the context of regular unital categories we introduce an intrinsic version of the notion of a Schreier split epimorphism, originally considered for monoids. We show that such split epimorphisms satisfy the same homological properties as Schreier split epimorphisms of monoids do. This gives rise to new examples of S-protomodular categories, and allows us to better understand the homological behaviour of monoids from a categorical perspective.