Wreaths, mixed wreaths and twisted coactions

Abstract

Distributive laws between two monads in a 2-category $\mathscr K$, as defined by Jon Beck in the case $\mathscr{K} = \mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\mathscr{K}$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\mathscr{K}$ of monads with different 2-cells from $\mathrm{Mnd}\mathscr{K}$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in $\mathrm{Mnd}\mathscr{K}$. 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 $\mathrm{EM}\mathscr{K}$. There are also mixed opwreaths: comonads in the Kleisli construction completion $\mathrm{Kl}\mathscr{K}$ of $\mathscr{K}$. 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.