nLab
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