Admissibility of Localizations of Crossed Modules

TL;DR

This paper explores the relationship between conditional flatness and Galois admissibility in the context of localizations of crossed modules, establishing new equivalences and properties for regular-epi localization functors.

Contribution

It establishes an equivalence between conditional flatness and Galois admissibility for regular-epi localizations in crossed modules, and shows nullification functors are admissible without kernel acyclicity.

Findings

Conditional flatness and Galois admissibility are equivalent for regular-epi localizations.

Nullification functors are admissible even when kernels are not acyclic.

The results extend the understanding of localization in the category of crossed modules.

Abstract

The correspondence between the concept of conditional flatness and admissibility in the sense of Galois appears in the context of localization functors in any semi-abelian category admitting a fiberwise localization. It is then natural to wonder what happens in the category of crossed modules where fiberwise localization is not always available. In this article, we establish an equivalence between conditional flatness and admissibility in the sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors. We use this equivalence to prove that nullification functors are admissible for the class of regular epimorphisms, even if the kernels of their localization morphisms are not acyclic.