Applications of the Kleisli and Eilenberg-Moore 2-adjunctions

Document Type: Research Paper


1 Research Coordination, CINVCAT, P.O. Box 36620, Irapuato, Gto. M'exico.

2 Instituto de Matemáticas, UNAM

3 Research Coordination, Universidad Incarnate Word Campus Bajío, P.O. Box 36821, Irapuato, Gto. M'exico.


In 2010, J. Climent Vidal and J. Soliveres Tur developed, among other things, a pair of 2-adjunctions between the 2-category of adjunctions and the 2-category of monads. One is related to the Kleisli adjunction and the other to the Eilenberg-Moore adjunction for a given monad.
Since any 2-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 2-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 2-category of monads, through the respective 2-adjunction. The same can be said for Kleisli extensions.
Several authors have been discovered this type of bijections and isomorphisms but these pair of 2-adjunctions can collect them all at once with an extra property, that of naturality.


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