The 2-category of Weak Entwining Structures

A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law. A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K , respectively. This observation is used to define a 2-category Entww(K ) of weak entwining structures in K . If the 2-category K admits EilenbergMoore constructions for both monads and comonads and idempotent 2-cells in K split, then there are 2-functors from Entww(K ) to the 2-category of monads and to the 2-category of comonads in K , taking a weak entwining structure (t,c) to a ‘weak lifting’ of t for c and a ‘weak lifting’ of c for t, respectively. The Eilenberg-Moore objects of the lifted monad and the lifted comonad are shown to be isomorphic. If K is the 2-category of functors induced by bimodules, then these isomorphic Eilenberg-Moore objects are isomorphic to the usual category of weak entwined modules.