nLab entwining structure

In addition to the usual distributive laws between monad and a monad (in a bicategory) there are many other combinations like between monad and comonad, comonad and endofunctor, action of a monoidal category and a monad and so on. The distributive laws between a monad and a comonad are called “mixed”. In the bicategory of rings, bimodules and homomorphisms of bimodules, the mixed distributive laws are called entwinings. In that context they were rediscovered by T. Brzeziński and S. Majid in the context of the study of noncommutative principal bundles. Entwinings organize in a bicategory. To every entwining structure one associates the corresponding category of entwined modules.

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