Exactness of direct limits via an injective cogenerator

We show that a complete abelian category with an injective cogenerator has exact direct limits (i.e. satisfies Grothendieck's AB5 condition) if and only if one (or any) injective cogenerator is pure-injective in the sense of Jensen and Lenzing (i.e. the summing maps from coproducts extend to products). The result is motivated by tilting theory and the proof uses a representation theorem for the dual setting of cocomplete abelian categories with a projective generator via additive monads on the category of sets. This is an account on joint work with Leonid Positselski (arXiv:1805.05156).