In addition to the usual distributive laws between a monad and a monad (in a bicategory) there are many other combinations like between monad and comonad, comonad and endofunctor, action of a monoidal category and a monad and so on. Suppose is any monoidal category with coequalizers and where the tensor product preserves coequalizers, 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 4 coherences so that is (after postcomposing by the associativity coherence) a mixed distributive law . This amounts to ensuring the lifting of to a comonad on -modules (which are the same as left -modules) or equivalently the lifting of monad to a monad on -comodules. Right entwining structure is a morphism such that defines similarly a distributive law ensuring liftings 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 October 2, 2023 at 08:15:20. See the history of this page for a list of all contributions to it.