Wreaths, mixed wreaths and twisted coactions

Ross Street*

*Corresponding author for this work

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    Abstract

    Distributive laws between two monads in a 2-category K, as defined by Jon Reek in the case K = Cat, were pointed out by the author to be monads in a 2-category Mnd K of monads. Steve Lack and the author defined wreaths to he monads in a 2-category EM K of monads with different 2-cells from MndK.

    Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in MndK. 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 EMK. There are also mixed opwreaths: comonads in the Kleisli construction completion KIK of K. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. A wreath determines a monad structure 011 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 Monoidality of the Kleisli-like construction is also discussed.

    Original languageEnglish
    Pages (from-to)1-22
    Number of pages22
    JournalTbilisi mathematical journal
    Volume10
    Issue number3
    DOIs
    Publication statusPublished - 2017

    Bibliographical note

    Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

    Keywords

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

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