Applications of the Kleisli and Eilenberg-Moore ۲-adjunctions

In ۲۰۱۰, J. Climent Vidal and J. Soliveres Tur developed, among other things, a pair of ۲-adjunctions between the ۲-category of adjunctions and the ۲-category of monads. One is related to the Kleisli adjunction and the other to the Eilenberg-Moore adjunction for a given monad.Since any ۲-adjunction induces certain natural isomorphisms of categories, these can be used to classify bijections and isomorphisms for certain structures in monad theory. In particular, one important example of a structure, lying in the ۲-category of adjunctions, where this procedure can be applied to is that of a lifting. Therefore, a lifting can be characterized by the associated monad structure,lying in the ۲-category of monads, through the respective ۲-adjunction. The same can be said for Kleisli extensions.Several authors have been discovered this type of bijections and isomorphisms but these pair of ۲-adjunctions can collect them all at once with an extra property, that of naturality.

Keywords:

۲-categories , ۲-adjunctions , monad theory , liftings for algebras , monoidal monads