Exactness of direct limits for abelian categories with an injective cogenerator

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. (C) 2018 Elsevier B.V.

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Keywords

• Exactness
• direct
• limits
• for
• abelian
• categories
• with
• injective
• cogenerator