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.


[1] Borceaux, F., “Handbook of Categorical Algebra II”, Encyclopedia Math. Appl. 51., Cambridge Univ. Press, 1994.
[2] Brzezinski, T., Vazquez-Marquez, A. and Vercryusse, J., The Eilenberg-Moore category and a Beck-type theorem for a Morita context, Appl. Categ. Structures 19(5) (2011), 821-858.
[3] Climent Vidal, J. and Soliveres Tur, J., Kleisli and Eilenberg-Moore constructions as part of a biadjoint situation, Extracta Math. 25(1) (2010), 1-61.
[4] Dubuc, E.J.,“Kan extensions in enriched category theory”, Lecture Notes in Math. 145, Springer, 1970.
[5] Mac Lane, S., “Categories for the working mathematician”, Grad. Texts in Math. 5, Springer, 1998.
[6] Mesablishvili, B. and Wisbauer, R., Notes on bimonads and Hopf monads, Theory Appl. Categ. 26(10) (2012), 281-303.
[7] Moerdijk, I., Monads on tensor categories, J. Pure Appl. Algebra 168(2-3) (2002), 189-208.
[8] Street, R., The formal theory of monads, J. Pure Appl. Algebra 2(2) (1972), 149-168.
[9] Tanaka, M., “Pseudo-Distributive laws and a unified framework for variable binding”, PhD Thesis, The University of Edinburg, 2005.
[10] Zawadoski, M., The formal theory of monoidal monads, J. Pure Appl. Algebra 216(8) (2012), 1932-1942.