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.

- T. Brzeziński, S. Majid,
*Coalgebra bundles*, Comm. Math. Phys. 191 (1998), no. 2, 467–492 (arXiv version). - T. Brzeziński, R. Wisbauer,
*Corings and comodules*, London Math. Soc. Lec. Note Series**309**, Cambridge 2003. - Z. Škoda,
*Some equivariant constructions in noncommutative algebraic geometry*, Georgian Mathematical Journal**16**(2009), No. 1, 183–202, arXiv:0811.4770 MR2011b:14004

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.