nLab entwining structure

Idea

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 $V$ is any symmetric monoidal category with coequalizers preserved by the tensor product, for example $V$ is the category of $k$-modules where $k$ is a commutative ring. Then for any $k$-algebra $A$, endofunctors $A\otimes -$ and $-\otimes A$ are monads on $V$; similarly if $C$ is a $k$-coalgebra, then endofunctors $C\otimes -$ and $-\otimes C$ are comonads on $V$.

Definition

A left entwining structure is a morphism $\phi:A\otimes C\to C\otimes A$ in $V$ satisfying four coherence axioms so that $\phi\otimes -$ is (after postcomposing by the associator) a mixed distributive law $A\otimes(C\otimes -)\to C\otimes(A\otimes -)$.

This amounts to ensuring $C\otimes -$ lifts to a comonad on $A\otimes -$-modules (which are the same as left $A$-modules) or, equivalently, that $A\otimes -$ lifts to a monad on $C\otimes -$-comodules.

A right entwining structure is a morphism $\psi:C\otimes A\to A\otimes C$ such that $-\otimes\psi$ defines a distributive law ensuring a lift of the comonad $-\otimes C$ to $-\otimes A$-modules, that is right $A$-modules. Entwining structures in the monoidal category of endofunctors are the usual distributive laws.

In category of vector spaces (and $k$-modules for $k$ 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.

Literature

• T. Brzeziński, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 arXiv version
• Gabriella Böhm, Internal bialgebroids, entwining structures and corings, Algebraic structures and their representations, 207-226, Contemp. Math., 376, Amer. Math. Soc., Providence, RI, 2005, arXiv:math/0311244
• Donovan van Osdol, Bicohomology Theory, Trans. Amer. Math. Soc. 183 (1973) 449–476 jstor:1996479]
• Zoran Škoda, Bicategory of entwinings, arXiv:0805.4611
• B. Mesablishvili, Entwining structures in monoidal categories, J. Algebra 319:6 (2008) 2496–2517 doi
• Bachuki Mesablishvili, Robert Wisbauer, Galois functors and entwining structures, J. Algebra 324:3 (2010) 464–506 doi