Kleisli and Eilenberg-Moore constructions as parts of biadjoint situations

Abstract

We consider monads over varying categories, and by defining the morphisms of Kleisli and of Eilenberg-Moore from a monad to another and the appropriate transformations (2-cells) between morphisms of Kleisli and between morphisms of Eilenberg-Moore, we obtain two 2-categories Mndₖₗ and MndEM. Then we prove that MndKl and MndEM are, respectively, 2-isomorphic to the conjugate of Kl and to the transpose of EM, for two suitably defined 2-categories Kl and EM, related, respectively, to the constructions of Kleisli and of Eilenberg-Moore. Next, by considering those morphisms and transformations of monads that are simultaneously of Kleisli and of Eilenberg-Moore, we obtain a 2-category Mndalg, of monads, algebraic morphisms, and algebraic transformations, and, to con¯rm its naturalness, we, on the one hand, prove that its underlying category can be obtained by applying the Ehresmann-Grothendieck construction to a suitable contravariant functor, and, on the other, we provide an explicit 2-embedding of a certain 2-category, Sigpd, of many- sorted signatures (hence also of another 2-category Spfpd, of many-sorted specifications), arising from the field of many-sorted universal algebra, into a 2-category of the type Mndalg. Moreover, we investigate for the adjunctions between varying categories the counterparts of the concepts previously defined for the monads, obtaining several 2-categories of adjunc- tions, as well as several 2-functors from them to the corresponding 2-categories of monads, and all in such a way that the classical Kleisli and Eilenberg-Moore constructions are left and right biadjoints, respectively, for these 2-functors. Finally, we define a 2-category Adalg, of adjunctions, algebraic squares, and algebraic transformations, and prove that there exists a canonical 2-functor Mdalg from Adalg to Mndalg.