Letters in Mathematical Physics

, Volume 107, Issue 3, pp 553–590 | Cite as

Tensor functors between Morita duals of fusion categories

  • César Galindo
  • Julia Yael Plavnik


Given a fusion category \({\mathcal C}\) and an indecomposable \({\mathcal C}\)-module category \({\mathcal M}\), the fusion category \({\mathcal C}^*_{_{{\mathcal M}}}\) of \({\mathcal C}\)-module endofunctors of \({\mathcal M}\) is called the (Morita) dual fusion category of \({\mathcal C}\) with respect to \({\mathcal M}\). We describe tensor functors between two arbitrary duals \({\mathcal C}^*_{_{{\mathcal M}}}\) and \({\mathcal D}^*_{\mathcal N}\) in terms of data associated to \({\mathcal C}\) and \({\mathcal D}\). We apply the results to G-equivariantizations of fusion categories and group-theoretical fusion categories. We describe the orbits of the action of the Brauer–Picard group on the set of module categories and we propose a categorification of the Rosenberg–Zelinsky sequence for fusion categories.


Fusion categories Brauer–Picard group Module categories Tensor functors 

Mathematics Subject Classification

16T05 18D10 



We thank Paul Bruillard and the referee for useful comments.


  1. 1.
    Bakalov, B., Kirillov Jr., A.: Lectures on Tensor categories and modular functors, University Lecture Series 21. Am. Math. Soc, Providence (2001)zbMATHGoogle Scholar
  2. 2.
    Beggs, E.: Making non-trivially associated tensor categories from left coset representatives. J. Pure Appl. Algebra 177, 5–41 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Davydov, A.: Galois algebras and monoidal fuctor between categories of representations of finite groups. J. Algebra 244(1), 273–301 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Davydov, A.: Modular invariants for group-theoretical modular data. I. J. Algebra 323(1), 1321–1348 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, vol. II, Progr. Math., vol. 87, pp. 111–195. Birkhuser, Boston (1990)Google Scholar
  6. 6.
    Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi Hopf algebra, group cohomology and orbifold models. Nuclear Phys. B Proc. Suppl. 18B, 60–72 (1990)Google Scholar
  7. 7.
    Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories I. Selecta. Math. 16(1), 1–119 (2010)Google Scholar
  8. 8.
    Etingof, P., Gelaki, S.: Isocategorical groups. IMRN 2, 50–76 (2001)zbMATHGoogle Scholar
  9. 9.
    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories, Mathematical Surveys and Monographs. Am. Math. Soc, Providence (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. 162, 581–642 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Etingof, P., Nikshych, D., Ostrik, V.: Weakly group-theoretical and solvable fusion categories. Adv. Math. 226, 176–205 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. With an appendix by Ehud Meir. Quantum Topol. 1(3), 209–273 (2010)Google Scholar
  13. 13.
    Etingof, P., Ostrik, V.: Finite tensor categories. Moscow Math. J. 4, 627–654 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Galindo, C.: Clifford theory for tensor categories. J. Lond. Math. Soc. 2(83), 57–78 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Galindo, C.: Crossed product tensor categories. J. Algebra 1(337), 233–252 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Galindo, C.: Clifford theory for graded fusion categories. Israel J. Math. 2(192), 841–867 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gelaki, S., Naidu, D., Nikshych, D.: Centers of graded fusion categories. Algebra Number Theory 3(8), 959–990 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Greenough, J.: Monoidal 2-structure of Bimodule Categories. J. Algebra 324, 1818–1859 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grossman, P., Snyder, N.: The Brauer-Picard group of the Asaeda–Haagerup fusion categories (preprint). arXiv:1202.4396
  20. 20.
    Izumi, M., Kosaki, H.: On a subfactor analogue of the second cohomology. Rev. Math. Phys. 14(7–8), 733–757 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Joyal, A., Street, R.: An introduction to Tannaka duality and quantum groups. In: Category Theory, Proceeding, Como. Part II of Lecture Notes in Mathematics, vol. 1488, pp. 411–492 (1990)Google Scholar
  22. 22.
    Karpilovsky, G.: Clifford theory for group representations. North-Holland Mathematics Studies, vol. 156 (1989)Google Scholar
  23. 23.
    Lang, S.: Algebra, volume 211 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (2002)Google Scholar
  24. 24.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1997)Google Scholar
  25. 25.
    Mombelli, M.: The Brauer–Picard group of the representation category of finite supergroup algebras. Rev. Unión Mat. Argent (accepted)Google Scholar
  26. 26.
    Naidu, D.: Categorical Morita equivalence for group-theoretical categories. Commun. Algebra 35(11), 3544–3565 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Naidu, D., Nikshych, D.: Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups. Commun. Math. Phys. 279, 845–872 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nikshych, D.: Non group-theoretical semisimple Hopf algebras from group actions on fusion categories. Selecta Math. (N.S.) 14, 145–161 (2008)Google Scholar
  29. 29.
    Nikshych, D.: Morita equivalence methods in classification of fusion categories. Contemp. Math. 582, 289–325 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nikshych, D., Riepel, B.: Categorical Lagrangian Grassmannians and Brauer–Picard groups of pointed fusion categories. J. Algebra 411, 191–214 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ostrik, V.: Module categories over the Drinfeld double of a finite group. Int. Math. Res. Notices 2003, 1507–1520 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tambara, D., Yamagami, S.: Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209, 692–707 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tambara, D.: Invariants and semi-direct products for finite group actions on tensor categories. J. Math. Soc. Japan 53(2), 429–456 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tambara, D.: Representations of tensor categories with fusion rules of self-duality for abelian groups. Israel J. Math. 118, 29–60 (2000)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Departamento de MatematicasUniversidad de los AndesBogotáColombia
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations