Exact Sequences in Non-Exact Categories (An Application to Semimodules)

TL;DR

This paper introduces a new concept of exact sequences in non-exact categories, specifically applied to semimodules over semirings, enabling homological analysis in broader algebraic contexts.

Contribution

It defines and studies a novel notion of exact sequences relative to a factorization structure, extending homological tools to semimodules and non-exact categories.

Findings

Established a new notion of exact sequences in non-exact categories.

Proved versions of the Short Five Lemma and Snake Lemma in this context.

Extended homological results to semimodules over semirings and commutative monoids.

Abstract

We consider a notion of exact sequences in any -not necessarily exact- pointed category relative to a given (E;M)-factorization structure. We apply this notion to introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties to relative homological varieties.