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.

• 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.