In addition to the usual distributive laws between a monad and a monad (in a bicategory) there are many other combinations between monad and comonad, comonad and endofunctor, action of a monoidal category and a monad and so on.
Suppose is any symmetric monoidal category with coequalizers preserved by the tensor product, for example is the category of -modules where is a commutative ring. Then for any -algebra , endofunctors and are monads on ; similarly if is a -coalgebra, then endofunctors and are comonads on .
A left entwining structure is a morphism in satisfying four coherence axioms so that is (after postcomposing by the associator) a mixed distributive law .
This amounts to ensuring lifts to a comonad on -modules (which are the same as left -modules) or, equivalently, that lifts to a monad on -comodules.
A right entwining structure is a morphism such that defines a distributive law ensuring a lift of the comonad to -modules, that is right -modules. Entwining structures in the monoidal category of endofunctors are the usual distributive laws.
In category of vector spaces (and -modules for being a commutative ring) (right) entwining structure was introduced by Brzeziński and Majid 1998 in the context of the study of noncommutative principal bundles, not being aware at the time of the mixed distributive laws in category theory (Van Osdol, 1973). Namely, they noticed that some examples of noncommutative principal bundles do not have a bialgebra as a structure “group”, but a coalgebra together with an entwining. Entwinings organize into a bicategory. Every entwining induces a coring (as observed by Takeuchi; in Škoda 2008 this correspondence is extended to a 2-functor). To every entwining structure one associates the corresponding category of entwined modules.
Last revised on January 17, 2024 at 12:33:42. See the history of this page for a list of all contributions to it.