Wreaths, mixed wreaths and twisted coactions

Distributive laws between two monads in a 2-category $\text{CK}$, as defined by Jon Beck in the case $\text{CK}=\mathrm{C}\mathrm{a}\mathrm{t}$, were pointed out by the author to be monads in a 2-category $\mathrm{M}\mathrm{n}\mathrm{d}\text{CK}$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{E}\mathrm{M}\text{CK}$ of monads with different 2-cells from $\mathrm{M}\mathrm{n}\mathrm{d}\text{CK}$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others, they are comonads in $\mathrm{M}\mathrm{n}\mathrm{d}\text{CK}$. 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 $\mathrm{E}\mathrm{M}\text{CK}$. There are also mixed opwreaths: comonoids in the Kleisli construction completion $\mathrm{K}\mathrm{l}\text{CK}$ of $\text{CK}$. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. Corresponding to the wreath product on the mixed side is wreath convolution, which 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.