A criterion for reflectiveness of normal extensions

TL;DR

This paper introduces a new criterion ensuring the reflectiveness of normal extensions within certain categorical structures, with applications to Barr-exact categories and their subcategories.

Contribution

It provides a novel sufficient condition for reflectiveness of normal extensions in admissible Galois structures, applicable to Barr-exact categories and their subcategories.

Findings

The criterion is satisfied in the context of S-special objects in Barr-exact S-protomodular categories.

The adjunction between a Barr-exact unital category and its abelian core is shown to be admissible.

Concrete examples demonstrate the applicability of the criterion.

Abstract

We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when X is the (protomodular) reflective subcategory of S-special objects of a Barr-exact S-protomodular category C, where S is the class of split epimorphic trivial extensions in C. Next to some concrete examples where the criterion may be applied, we also study the adjunction between a Barr-exact unital category and its abelian core, which we prove to be admissible.