Applied Categorical Structures

, Volume 6, Issue 2, pp 177–191 | Cite as

Fusion Operators and Cocycloids in Monoidal Categories

  • Ross Street


The Yang–Baxter equation has been studied extensively in the context of monoidal categories. The fusion equation, which appears to be the Yang–Baxter equation with a term missing, has been studied mainly in the context of Hilbert spaces. This paper endeavours to place the fusion equation in an appropriate categorical setting. Tricocycloids are defined; they are new mathematical structures closely related to Hopf algebras.

bialgebra Hopf algebra fusion equation 3-cocycle monoidal category tensor category braiding string diagram Tannaka duality 


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  1. 1.
    Aitchison, I.: String diagrams for non-Abelian cocycle conditions, Handwritten notes, talk presented at Louvain-la-Neuve, Belgium, 1987.Google Scholar
  2. 2.
    Par Saad Baaj and Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres, Ann. Scient. Éc. Norm. Sup. 26 (1993), 425–488.Google Scholar
  3. 3.
    Day, B. J.: On closed categories of functors, in Midwest Category Seminar Reports IV, Lecture Notes in Math. 137, Springer, 1970, pp. 1–38.Google Scholar
  4. 4.
    Day, B. J.: Promonoidal functor categories, J. Austral. Math. Soc. Ser. A 23 (1977), 312–328.Google Scholar
  5. 5.
    Eilenberg, S. and Mac Lane, S.: On the groups H(П, n), I, II, Ann. of Math. 58 (1953), 55–106; 70 (1954), 49–137.Google Scholar
  6. 6.
    Gordon, R., Power, A. J. and Street, R.: Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), #558.Google Scholar
  7. 7.
    Joyal, A. and Street, R.: Tortile Yang–Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), 43–51.Google Scholar
  8. 8.
    Joyal, A. and Street, R.: The geometry of tensor calculus I, Adv. in Math. 88 (1991), 55–112.Google Scholar
  9. 9.
    Joyal, A. and Street, R.: Braided tensor categories, Adv. in Math. 102 (1993), 20–78.Google Scholar
  10. 10.
    Joyal, A. and Street, R.: An introduction to Tannaka duality and quantum groups, in Category Theory, Proceedings, Como 1990, Part II of Lecture Notes in Math. 1488, Springer-Verlag, Berlin, 1991, pp. 411–492.Google Scholar
  11. 11.
    Lyubashenko, V.: Tangles and Hopf algebras in braided categories, J. Pure Appl. Algebra 98 (1995), 245–278.Google Scholar
  12. 12.
    Lyubashenko, V.: Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995), 279–327.Google Scholar
  13. 13.
    Moore, G. and Seiberg, N.: Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177–254.Google Scholar
  14. 14.
    Skandalis, G.: Operator algebras and duality, in Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990, The Mathematical Society of Japan, 1991, pp. 997–1009.Google Scholar
  15. 15.
    Street, R.: The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), 283–335.Google Scholar
  16. 16.
    Street, R.: Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995), 29–77.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Ross Street
    • 1
  1. 1.School of Mathematics, Physics, Computing & ElectronicsMacquarie UniversityAustralia

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