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 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 $k$-algebras and $k$-coalgebras. The same terminology is used by Power and Watanabe.
Donovan van Osdol, Bicohomology Theory, Transactions of the American Mathematical Society 183 (1973) 449-476 [jstor:1996479]
John Power, Hiroshi Watanabe, Combining a monad and a comonad, Theoretical Computer Science 280 1–2 (2002) 137–162 [doi:10.1016/S0304-3975(01)00024-X]
Last revised on October 2, 2023 at 08:54:49. See the history of this page for a list of all contributions to it.