nLab 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 entwining structure defines the composed coring (as observed by Takeuchi); the entwined modules are then precisely the objects in the category of comodules (Eilenberg-Moore category) of the composed coring.


Historically, the entwined modules are first introduced under the name “bialgebras” by van Osdol for the mixed distributive laws between monads and comonads instead of kk-algebras and kk-coalgebras. The same terminology is used by Power and Watanabe.

Last revised on October 2, 2023 at 08:54:49. See the history of this page for a list of all contributions to it.