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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 Entw^w(K) of weak entwining structures in K. If the 2-category K admits Eilenberg-Moore constructions for both monads and comonads and idempotent 2-cells in K split, then there are pseudo-functors from Entw^w(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 equivalent. If K is the 2-category of functors induced by bimodules, then these Eilenberg-Moore objects are isomorphic to the usual category of weak entwined modules.