Flatness in finitely accessible additive categories

TL;DR

This paper investigates flat objects in finitely accessible additive categories, characterizing their structure and conditions for the existence of enough flat and projective objects, extending concepts from Grothendieck categories.

Contribution

It generalizes flatness concepts to finitely accessible additive categories and provides characterizations of when these categories are preabelian or abelian.

Findings

Flat objects form direct unions of small flat subobjects under certain conditions

Characterization of when the category has enough flat and projective objects

Closure of flat objects under pure subobjects in specific cases

Abstract

Motivated by some problems proposed by Cuadra and Simson related to flat objects in finitely accessible Grothendieck categories, we study flatness in the more general setting of finitely accessible additive categories. For such category $\mathcal{A}$, we characterize when $\mathcal{A}$ is preabelian and abelian. We prove that if the class of flat objects in $\mathcal A$ is closed under pure subobjects, then every flat object is a direct union of \textit{small} flat subobjects. Finally, we characterize when $\mathcal{A}$ has enough flat and projective objects and we prove that, in this case, the class of flat objects is closed under pure subobjects.