Open Access
June 2017 Wreaths, mixed wreaths and twisted coactions
Ross Street
Tbilisi Math. J. 10(3): 1-22 (June 2017). DOI: 10.1515/tmj-2017-0100


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.


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Ross Street. "Wreaths, mixed wreaths and twisted coactions." Tbilisi Math. J. 10 (3) 1 - 22, June 2017.


Received: 6 May 2017; Revised: 11 May 2017; Published: June 2017
First available in Project Euclid: 20 April 2018

zbMATH: 06786072
MathSciNet: MR3663440
Digital Object Identifier: 10.1515/tmj-2017-0100

Primary: 18D10
Secondary: 05A15 , 16T30 , 18A32 , 18D05 , 20H30

Keywords: bialgebra , comonad , convolution , Heisenberg product , mixed distributive law , monad , twisted action , wreath

Rights: Copyright © 2017 Tbilisi Centre for Mathematical Sciences

Vol.10 • No. 3 • June 2017
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