Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 005, 20 pages      arXiv:1503.02783

Weighted Tensor Products of Joyal Species, Graphs, and Charades

Ross Street
Centre of Australian Category Theory, Macquarie University, Australia

Received August 18, 2015, in final form January 14, 2016; Published online January 17, 2016

Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota-Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graphs in a monoidal additive category. Weighted derivations are lifted to the categorical level.

Key words: weighted derivation; Hurwitz series; monoidal category; Joyal species; convolution; Rota-Baxter operator.

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  1. Aguiar M., Ferrer Santos W., Moreira W., The Heisenberg product: from Hopf algebras and species to symmetric functions, arXiv:1504.06315.
  2. Aguiar M., Mahajan S., Monoidal functors, species and Hopf algebras, CRM Monograph Series, Vol. 29, Amer. Math. Soc., Providence, RI, 2010.
  3. Aguiar M., Moreira W., Combinatorics of the free Baxter algebra, Electron. J. Combin. 13 (2006), R17, 38 pages, math.CO/0510169.
  4. Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742.
  5. Cartier P., On the structure of free Baxter algebras, Adv. Math. 9 (1972), 253-265.
  6. Chikhladze D., Lack S., Street R., Hopf monoidal comonads, Theory Appl. Categ. 24 (2010), 554-563, arXiv:1002.1122.
  7. Day B., Construction of biclosed categories, Ph.D. Thesis, University of New South Wales, 1970, available at
  8. Day B., On closed categories of functors, in Reports of the Midwest Category Seminar, IV, Lecture Notes in Math., Vol. 137, Springer, Berlin, 1970, 1-38.
  9. Day B., Street R., Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), 99-157.
  10. Ebrahimi-Fard K., Guo L., Free Rota-Baxter algebras and rooted trees, J. Algebra Appl. 7 (2008), 167-194, math.RA/0510266.
  11. Garner R., Street R., Coalgebras governing both weighted Hurwitz products and their pointwise transforms, arXiv:1510.05323.
  12. Guo L., Keigher W., On differential Rota-Baxter algebras, J. Pure Appl. Algebra 212 (2008), 522-540, math.RA/0703780.
  13. Joyal A., Une théorie combinatoire des séries formelles, Adv. Math. 42 (1981), 1-82.
  14. Joyal A., Foncteurs analytiques et espèces de structures, in Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985), Lecture Notes in Math., Vol. 1234, Springer, Berlin, 1986, 126-159.
  15. Joyal A., Street R., Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), 43-51.
  16. Joyal A., Street R., Braided tensor categories, Adv. Math. 102 (1993), 20-78.
  17. Joyal A., Street R., The category of representations of the general linear groups over a finite field, J. Algebra 176 (1995), 908-946.
  18. Kapranov M.M., Analogies between the Langlands correspondence and topological quantum field theory, in Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993), Progr. Math., Vol. 131, Birkhäuser Boston, Boston, MA, 1995, 119-151.
  19. Kelly G.M., Street R., Review of the elements of $2$-categories, in Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math., Vol. 420, Springer, Berlin, 1974, 75-103.
  20. Moreira W., Products of representations of the symmetric group and non-commutative versions, Ph.D. Thesis, Texas A&M University, 2008.
  21. Rota G.-C., Baxter algebras and combinatorial identities. I, Bull. Amer. Math. Soc. 75 (1969), 325-329.
  22. Street R., Monoidal categories in, and linking, geometry and algebra, Bull. Belg. Math. Soc. Simon Stevin 19 (2012), 769-821, arXiv:1201.2991.
  23. Zhang S., Guo L., Keigher W., Monads and distributive laws for Rota-Baxter and differential algebras, Adv. in Appl. Math. 72 (2016), 139-165, arXiv:1412.8058.

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