Wreaths, mixed wreaths and twisted coactions

Distributive laws between two monads in a 2-category K, as defined by Jon Reek in the case K = Cat, were pointed out by the author to be monads in a 2-category Mnd K of monads. Steve Lack and the author defined wreaths to he monads in a 2-category EM K of monads with different 2-cells from MndK.

Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in MndK. Actually, as pointed out by John Power and Hiroshi Watanabe, there are a number of dual possibilities for mixed distributive laws.

It is natural then to consider mixed wreaths as we do in this article: they are comonads in EMK. There are also mixed opwreaths: comonads in the Kleisli construction completion KIK of K. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. A wreath determines a monad structure 011 the composite of the two endomorphisms involved; this monad is called the wreath product. For mixed wreaths, corresponding to this wreath product, is a convolution operation analogous to the convolution monoid structure on the set of morphisms from a comonoid to a monoid. In fact, wreath convolution is composition in a Kleisli-like construction. Walter Moreira's Heisenberg product of linear endomorphisms on a Hopf algebra, is an example of such convolution, actually involving merely a mixed distributive Monoidality of the Kleisli-like construction is also discussed.