Homological Lemmas in a Non-pointed Context

TL;DR

This paper extends classical homological lemmas to non-pointed contexts within regular protomodular categories, broadening their applicability to various algebraic and topological structures.

Contribution

It introduces non-pointed versions of homological lemmas applicable in regular protomodular categories with specific subcategories, expanding the theoretical framework.

Findings

Homological lemmas hold in non-pointed regular protomodular categories.

Examples include categories of rings, Boolean, Heyting, and MV-algebras.

Applicability to topological models and elementary toposes.

Abstract

We show that non-pointed versions of the classical homological lemmas hold in regular protomodular categories equipped with a suitable posetal monocoreflective subcategory. Examples of such categories are all protomodular varieties of universal algebras having more than one constant, like the ones of unitary rings, Boolean algebras, Heyting algebras and MV-algebras, their topological models, and the dual category of every elementary topos.