Flatness in finitely accessible additive categories

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.