Wreaths, mixed wreaths and twisted coactions.

R. Street

Distributive laws between two monads in a 2-category Ҝ, as defined by Jon Beck in the case Ҝ=Cat, were pointed out by the author to be monads in a 2-category MndҜ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category EMҜ of monads with different 2-cells from MndҜ.

Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in MndҜ. 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 EMҜ. There are also mixed opwreaths: comonads in the Kleisli construction completion KlҜ of Ҝ. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. A wreath determines a monad structure on 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 law. Monoidality of the Kleisli-like construction is also discussed. 

Tbilisi Mathematical Journal, Vol. 10(3) (2017), pp. 1-22