Wreaths, mixed wreaths and twisted coactions

TL;DR

This paper explores the concept of mixed wreaths and twisted coactions in 2-category theory, extending distributive laws between monads and comonads, with applications to algebraic structures like Hopf algebras.

Contribution

It introduces and formalizes mixed wreaths and opwreaths as new structures in 2-category theory, connecting them to twisted coactions and convolution products.

Findings

Defines mixed wreaths as comonads in EMCK

Connects wreath convolution to Kleisli-like constructions

Provides examples involving bimonoids and Hopf algebras

Abstract

Distributive laws between two monads in a 2-category $\CK$, as defined by Jon Beck in the case $\CK=\mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\CK$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\CK$ of monads with different 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, 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 $\mathrm{EM}\CK$. There are also mixed opwreaths: comonoids in the Kleisli construction completion $\mathrm{Kl}\CK$ of $\CK$. 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.