# nLab entwined module

Given an entwining structure between a $k$-algebra $A$ and a $k$-coalgebra $C$ one defines the corresponding analogue of Hopf modules: they are $A$-modules with structure of $C$-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 $k$-algebras and $k$-coalgebras.

• D. H. van Osdol, Bicohomology theory, Trans. Amer. Math. Soc. 183 (1973), 449–476.

Created on November 29, 2012 at 22:36:33. See the history of this page for a list of all contributions to it.