Exactness of direct limits for abelian categories with an injective cogenerator

TL;DR

This paper establishes a criterion linking the exactness of direct limits in certain abelian categories to pure-injectivity, with applications to tilting theory and inverse limits in monad categories.

Contribution

It provides a new characterization of when direct limits are exact in abelian categories with an injective cogenerator, connecting to pure-injectivity and monad theory.

Findings

Exactness of direct limits is equivalent to a pure-injectivity condition on the cogenerator J.

The result applies to tilting theory, offering new insights.

Inverse limits in Eilenberg-Moore categories preserve regular epimorphisms under certain conditions.

Abstract

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg-Moore category of a monad on the category of sets preserve regular epimorphisms.