Torsion Theories in a non-pointed Context II: iterated torsion theories

Abstract

Generalizing a known result in the context of homological categories, we show that, in a regular protomodular category $\mathcal{C}$ equipped with a posetal, monocoreflective subcategory whose coreflector inverts monomorphisms, every torsion theory, in a non-pointed sense, satisfying suitable properties gives rise to a torsion theory in the category $\mathrm{Arr}(\mathcal{C})$ of arrows in $\mathcal{C}$. Moreover, we show that the induced torsion theory in $\mathrm{Arr}(\mathcal{C})$ satisfies the same good properties of the original one in $\mathcal{C}$, allowing to iterate the process. Examples of this situation can be found in protomodular varieties of universal algebras with more than one constant or in duals of elementary toposes.