Distributive laws between two monads in a 2-category K , as defined by Jon Beck in the case K = Cat, were pointed out by the author to be monads in a 2-category MndK of monads. Steve Lack and the author defined wreaths to be monads in a 2-category EMK 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 KlK of K . 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.
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