Prenormal categories

TL;DR

This paper introduces prenormal categories, a new categorical framework inspired by regular categories, replacing certain notions with cokernels and kernels, and explores their properties, examples, and extensions to non-pointed contexts.

Contribution

It defines prenormal categories, studies their properties, and extends the concept to non-pointed contexts, providing a new framework for algebraic structures.

Findings

Characterisation via factorisation system with normal epimorphisms

Categorical version of Noether's third isomorphism theorem

Examples including the category of commutative monoids

Abstract

In this paper we introduce the notion of (pointed) prenormal category, modelled after regular categories, but with the key notions of coequaliser and kernel pair replaced by those of cokernel and kernel. This framework provides a natural setting for extending certain classical results in algebra. We study the fundamental properties of prenormal categories, including a characterisation in terms of a factorisation system involving normal epimorphisms, and a categorical version of Noether's so-called `third isomorphism theorem'. We also present a range of examples, with the category of commutative monoids constituting a central one. In the second part of the paper we extend prenormality and its related properties to the non-pointed context, using kernels and cokernels defined relative to a distinguished class of trivial objects.