Conditional flatness, fiberwise localizations, and admissible reflections

TL;DR

This paper generalizes the concept of conditional flatness for localization functors from groups to pointed categories, exploring its implications in homological and semi-abelian contexts, and establishing criteria for specific localizations.

Contribution

It extends the notion of conditional flatness to broader categorical settings and provides existence results and criteria for localizations in semi-abelian categories.

Findings

Conditional flatness characterizes admissibility of adjunctions.

Existence theorems for localization functors in semi-abelian categories.

Criteria for torsion-free reflections in fiberwise localizations.

Abstract

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply in particular to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor $L$ with what is called right properness for the $L$-local model structure.