On monads of exact reflective localizations of Abelian categories

TL;DR

This paper introduces Gabriel monads as idempotent monads linked to exact reflective localizations in Abelian categories, characterizing their properties and their role in relating Serre quotient categories to localizing subcategories.

Contribution

It defines Gabriel monads in the context of Abelian categories and characterizes their properties, connecting localizations with Serre quotients and local objects.

Findings

Gabriel monads are characterized by simple properties.

The coimage of a Gabriel monad is a Serre quotient category.

An equivalence exists between the coimage and the image of a Gabriel monad.

Abstract

In this paper we define Gabriel monads as the idempotent monads associated to exact reflective localizations in Abelian categories and characterize them by a simple set of properties. The coimage of a Gabriel monad is a Serre quotient category. The Gabriel monad induces an equivalence between its coimage and its image, the localizing subcategory of local objects.