entwining structure

In addition to the usual distributive laws between 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. The distributive laws between a monad and a comonad are called “mixed”. In the bicategory of rings, bimodules and homomorphisms of bimodules, the *mixed distributive laws* are called entwinings. In that context they were rediscovered by T. Brzeziński and S. Majid in the context of the study of noncommutative principal bundles. Entwinings organize in a bicategory. To every entwining structure one associates the corresponding category of entwined modules.

- 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 - 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

Revised on November 30, 2012 20:47:57
by Zoran Škoda
(193.51.104.65)