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Distributive laws between two monads in a 2-category $\CK$, as defined by JonBeck in the case $\CK=\mathrm{Cat}$, were pointed out by the author to bemonads in a 2-category $\mathrm{Mnd}\CK$ of monads. Steve Lack and the authordefined wreaths to be monads in a 2-category $\mathrm{EM}\CK$ of monads withdifferent 2-cells from $\mathrm{Mnd}\CK$.  Mixed distributive laws were also considered by Jon Beck, Mike Barr and,later, various others, they are comonads in $\mathrm{Mnd}\CK$. Actually, aspointed out by John Power and Hiroshi Watanabe, there are a number of dualpossibilities for mixed distributive laws.  It is natural then to consider mixed wreaths as we do in this article, theyare comonads in $\mathrm{EM}\CK$. There are also mixed opwreaths: comonoids inthe Kleisli construction completion $\mathrm{Kl}\CK$ of $\CK$. The main examplestudied 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'sHeisenberg product of linear endomorphisms on a Hopf algebra, is an example ofsuch convolution, actually involving merely a mixed distributive law.Monoidality of the Kleisli-like construction is also discussed.

Wreaths, mixed wreaths and twisted coactions