Generalized lax epimorphisms in the additive case

TL;DR

This paper introduces generalized lax epimorphisms for functors between rings with several objects and abelian categories, providing characterizations and conditions for localizations and equivalences in module categories.

Contribution

It defines and characterizes generalized lax epimorphisms in the additive setting, linking them to abelian localizations and equivalences of module categories.

Findings

Characterization of generalized lax epimorphisms via right cancellation.

Conditions for functors to induce abelian localizations.

Necessary and sufficient conditions for ring morphisms to induce equivalences.

Abstract

In this paper we call generalized lax epimorphism a functor defined on a ring with several objects, with values in an abelian AB5 category, for which the associated restriction functor is fully faithful. We characterize such a functor with the help of a conditioned right cancellation of another, constructed in a canonical way from the initial one. As consequences we deduce a characterization of functors inducing an abelian localization and also a necessary and sufficient condition for a morphism of rings with several objects to induce an equivalence at the level of two localizations of the respective module categories.