Tbilisi Mathematical Journal

Wreaths, mixed wreaths and twisted coactions

Ross Street

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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.

Article information

Tbilisi Math. J., Volume 10, Issue 3 (2017), 1-22.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 18A32: Factorization of morphisms, substructures, quotient structures, congruences, amalgams 18D05: Double categories, 2-categories, bicategories and generalizations 20H30: Other matrix groups over finite fields 16T30: Connections with combinatorics

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


Street, Ross. Wreaths, mixed wreaths and twisted coactions. Tbilisi Math. J. 10 (2017), no. 3, 1--22. doi:10.1515/tmj-2017-0100. https://projecteuclid.org/euclid.tbilisi/1524232072

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