Tbilisi Mathematical Journal

Wreaths, mixed wreaths and twisted coactions

Ross Street

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1524232072

Digital Object Identifier
doi:10.1515/tmj-2017-0100

Mathematical Reviews number (MathSciNet)
MR3663440

Zentralblatt MATH identifier
06786072

Subjects
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

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

Citation

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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References

  • Marcelo Aguiar, Walter Ferrer Santos and Walter Moreira, The Heisenberg product: From Hopf algebras and species to symmetric functions arXiv:1504.06315.
  • Jon Beck, Distributive laws, Lecture Notes in Mathematics 80 (Springer, Berlin,1969) 119–140; www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html.
  • Brian J. Day, Note on monoidal monads, J. Austral. Math. Soc. Ser. A 23(3) (1977) 292–311.
  • Brian J. Day, Promonoidal functor categories, J. Austral. Math. Soc. Ser. A 23(3) (1977) 312–328.
  • Beno Eckmann and Peter Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Mathematische Annalen 145(3) (1962) 227–255.
  • R. Harmer, M. Hyland and P.-A. Mellies, Categorical combinatorics for innocent strategies, Logic in Computer Science, Proc. 22nd Annual IEEE Symposium on Logic in Computer Science (LICS, IEEE, 2007) 379–388.
  • Daniela Hobst and Bodo Pareigis, Double quantum groups, J. Algebra 242 (2001) 460–494.
  • André Joyal and Ross Street, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55–112.
  • André Joyal and Ross Street, Tortile Yang-Baxter operators in tensor categories, Journal of Pure and Applied Algebra 71 (1991) 43–51.
  • André Joyal and Ross Street, Braided tensor categories, Advances in Mathematics 102 (1993) 20–78.
  • G. Max Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64 (Cambridge University Press, Cambridge, 1982).
  • G. Max Kelly and Ross Street, Review of the elements of 2-categories, Lecture Notes in Mathematics 420 (Springer-Verlag, 1974) 75–103.
  • Anders Kock, Monads for which structures are adjoint to units, Journal of Pure and Applied Algebra 104 (1995) 41–59.
  • Stephen Lack and Ross Street, The formal theory of monads II, J. Pure Appl. Algebra 175(1-3) (2002) 243–265.
  • F. William Lawvere, Ordinal sums and equational doctrines, in: Sem. on Triples and Categorical Homology Theory, ETH, Zürich, 1966/1967, (Springer, Berlin, 1969) 141–155.
  • Walter Moreira, Products of representations of the symmetric group and non-commutative versions (PhD Thesis, Texas A&M University, 2008).
  • A. John Power and Hiroshi Watanabe, Combining a monad and a comonad, Theoretical Computer Science 280 (2002) 137–162.
  • Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149–168.
  • Ross Street, Limits indexed by category-valued 2-functors, Journal of Pure and Applied Algebra 8 (1976) 149–181.
  • Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle 21 (1980) 111–160.
  • Ross Street, Categorical structures, Handbook of Algebra Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0 444 82212 7) 529–577.
  • Ross Street, Quantum Groups: a path to current algebra, Australian Math. Society Lecture Series 19 (Cambridge University Press; 18 January 2007; ISBN-978-0-521-69524-4).