A criterion for reflectiveness of normal extensions

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 $\mathbb{X}$ is the (protomodular) reflective subcategory of $\mathcal{S}$-special objects of a Barr-exact $\mathcal{S}$-protomodular category $\mathbb{C}$, where $\mathcal{S}$ is the class of split epimorphic trivial extensions in $\mathbb{C}$. 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.