entwined module

Given an entwining structure between a kk-algebra AA and a kk-coalgebra CC one defines the corresponding analogue of Hopf modules: they are AA-modules with structure of CC-comodules with a compatibility dictated by the entwining structure. If an algebra and a coalgebra are generalized to a monad and a comonad, then entwining structure generalizes to a mixed distributive law. Every mixed distributive law defines the composed comonad; the entwined modules are then precisely the objects in the Eilenberg-Moore category of the composed comonad.

Historically, the entwined modules are first introduced under the name “bialgebras” by van Osdol in the more general case of monads and comonads instead of kk-algebras and kk-coalgebras.