Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585310120190101Applications of the Kleisli and Eilenberg-Moore 2-adjunctions11715676725ENJuan Luis L'opez HernándezResearch Coordination, CINVCAT, P.O. Box 36620, Irapuato, Gto. M\'exico.Luis JesúsTurcioInstituto de Matemáticas, UNAMAdrian Vazquez-MarquezResearch Coordination, Universidad Incarnate Word Campus Bajío, P.O. Box 36821, Irapuato, Gto. M\'exico.Journal Article20180131In 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.<br />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.<br />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.http://cgasa.sbu.ac.ir/article_76725_4c74fe2ffb149c7099e49e4c27eeb355.pdf