Internal precategories relative to split epimorphisms

TL;DR

This paper introduces and studies internal precategories relative to split epimorphisms in categories, generalizing known equivalences and exploring additive and semi-additive cases.

Contribution

It defines reflexive graphs and precategories relative to split epimorphisms and extends the equivalence between precategories and 2-chain complexes to new categorical contexts.

Findings

Generalization of precategory equivalence to additive and semi-additive cases

Introduction of relative reflexive graphs and precategories

Application to strongly unital categories

Abstract

For a given category B we are interested in studying internal categorical structures in B. This work is the starting point, where we consider reflexive graphs and precategories (i.e., for the purpose of this note, a simplicial object truncated at level 2). We introduce the notions of reflexive graph and precategory relative to split epimorphisms. We study the additive case, where the split epimorphisms are "coproduct projections", and the semi-additive case where split epimorphisms are "semi-direct product projections". The result is a generalization of the well known equivalence between precategories and 2-chain complexes. We also consider an abstract setting, containing, for example, strongly unital categories.