Eilenberg-Moore and Kleisli Type Categories for Bimonads on Arbitrary Categories

We discuss categorical aspects of the theory of bimonads on arbitrary categories as introduced in Mesablishvili and Wisbauer (J K Theory 7:349-388, 2011). These includ explicit descriptions of limits and colimits as well as characterizations of monomorphisms and epimorphisms in the Eilenberg-Moore category of a bimonad. Two Kleisli type categories for bimonads are considered and it is shown that the classical Kleisli (co)monad adjunctions can be extended to this setting. Furthermore, the Kleisli categories are shown to be equivalent with the categories of free and respectively cofree bimodules.