Wreaths, mixed wreaths and twisted coactions

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Abstract

Distributive laws between two monads in a 2-category K , as defined by Jon Beck in the case K = Cat, were pointed out by the author to be monads in a 2-category MndK of monads. Steve Lack and the author defined wreaths to be monads in a 2-category EMK 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 KlK of 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.

[1] Marcelo Aguiar, Walter Ferrer Santos and Walter Moreira, The Heisenberg product: From Hopf algebras and species to symmetric functions arXiv:1504.06315.

[2] Jon Beck, Distributive laws, Lecture Notes in Mathematics 80 (Springer, Berlin,1969) 119-140; www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html.

[3] Brian J. Day, Note on monoidal monads, J. Austral. Math. Soc. Ser. A 23(3) (1977) 292-311.

[4] Brian J. Day, Promonoidal functor categories, J. Austral. Math. Soc. Ser. A 23(3) (1977) 312-328.

[5] Beno Eckmann and Peter Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Mathematische Annalen 145(3) (1962) 227-255.

[6] 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.

[7] Daniela Hobst and Bodo Pareigis, Double quantum groups, J. Algebra 242 (2001) 460-494.

[8] André Joyal and Ross Street, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112.

[9] André Joyal and Ross Street, Tortile Yang-Baxter operators in tensor categories, Journal of Pure and Applied Algebra 71 (1991) 43-51.

[10] André Joyal and Ross Street, Braided tensor categories, Advances in Mathematics 102 (1993) 20-78.

[11] G. Max Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64 (Cambridge University Press, Cambridge, 1982).

[12] G. Max Kelly and Ross Street, Review of the elements of 2-categories, Lecture Notes in Mathematics 420 (Springer-Verlag, 1974) 75-103.

[13] Anders Kock, Monads for which structures are adjoint to units, Journal of Pure and Applied Algebra 104 (1995) 41-59.

[14] Stephen Lack and Ross Street, The formal theory of monads II, J. Pure Appl. Algebra 175(1-3) (2002) 243-265.

[15] 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.

[16] Walter Moreira, Products of representations of the symmetric group and non-commutative versions (PhD Thesis, Texas A&M University, 2008).

[17] A. John Power and Hiroshi Watanabe, Combining a monad and a comonad, Theoretical Computer Science 280 (2002) 137-162.

[18] Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149-168.

[19] Ross Street, Limits indexed by category-valued 2-functors, Journal of Pure and Applied Algebra 8 (1976) 149-181.

[20] Ross Street, Fibrations in bicategories, Cahiers de topologie et g´eom´etrie diff´erentielle 21 (1980) 111-160.

[21] Ross Street, Categorical structures, Handbook of Algebra Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0 444 82212 7) 529-577.

[22] 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).