A Trace for Bimodule Categories | Applied Categorical Structures Skip to main content
Springer Nature Link
Log in

A Trace for Bimodule Categories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We study a 2-functor that assigns to a bimodule category over a finite \(\Bbbk \)-linear tensor category a \(\Bbbk \)-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Witten topological field theories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ayala, D., Francis, J., Tanaka, H.L.: Local structures on stratified spaces. arXiv:1409.0501

  2. Barrett, J.W., Westbury, B.W.: Invariants of piecewise-linear 3-manifolds. Trans. Am. Math. Soc 348, 3997–4022 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Davydov, A., Nikshych, D.: The Picard crossed module of a braided tensor category. Algebra & Number Theory 7, 1365–1403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deligne, P.: Catégories tannakiennes. In: Cartier, P., et al. (eds.) The Grothendieck Festschrift, vol. II, pp 111–195. Birkhäuser, Boston (1990)

  5. Douglas, C.L., Schommer-Pries, C., Snyder, N.: Dualizable tensor categories. arXiv:1312.7188

  6. Douglas, C.L., Schommer-Pries, C., Snyder, N.: The balanced tensor product of module categories. arXiv:1406.4204

  7. Etingof, P., Nikshych, D., Ostrik, V.: An analogue of Radford’s S 4 formula for finite tensor categories. Intl. Math. Res. Notices 54, 2915–2933 (2004)

    Article  MATH  Google Scholar 

  8. Etingof, P., Nikshych, D., Ostrik, V.: (With an appendix by E. Meir): Fusion categories and homotopy theory. Quantum Topol. 1, 209–273 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Etingof, P., Ostrik, V.: Finite tensor categories. Mosc. Math. J. 4, 627–654 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Correspondences of ribbon categories. Adv. Math 199, 192–329 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fuchs, J., Ganchev, A.C., Szlachányi, K., Vecsernyés, P.: S 4-symmetry of 6j-symbols and Frobenius-Schur indicators in rigid monoidal C ∗-categories. J. Math. Phys. 40, 408–426 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Freyd, P.J.: Abelian Categories. Harper & Row, New York (1964)

    MATH  Google Scholar 

  13. Freed, D.S.: Classical Chern-Simons theory, part 1. Adv. Math 113, 237–303 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fuchs, J., Schweigert, C., Valentino, A.: Bicategories for boundary conditions and for surface defects in 3-d TFT. Comm. Math. Phys. 321, 543–575 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fuchs, J., Schweigert, C., Valentino, A.: A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories. Commun. Math. Phys. 332, 981–1015 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Greenough, J.: Bimodule Categories and monoidal 2-structures. PhD thesis, University of New Hampshire (2010)

    MATH  Google Scholar 

  17. Ginot, G., Tradler, T., Zeinalian, M.: Higher Hochschild homology, topological chiral homology and factorization algebras. Comm. Math. Phys. 326, 635–686 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gurski, M.N.: An algebraic theory of tricategories. PhD thesis, University of Chicago (2006)

  19. Lurie, J.: On the classification of topological field theories. In: Current developments in mathematics, pp 129–280. Intl. Press, Somerville (2009)

  20. Morton, J.C.: Cohomological twisting of 2-linearization and extended TQFT. J. Homotopy Relat. Struct. (2013)

  21. Ostrik, V.: Module categories over the Drinfeld double of a finite group. Intl. Math. Res. Notices No. 27, 1507–1520 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ponto, K., Shulman, M.: Shadows and traces in bicategories. J. Homotopy Related Struct. 8, 151–200 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schaumann, G.: Duals in tricategories and in the tricategory of bimodule categories. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2013). http://opus4.kobv.de/opus4-fau/frontdoor/index/index/docId/3732

  25. Schaumann, G.: Pivotal tricategories and a categorification of inner-product modules. arXiv:1405.5667

  26. Street, R.: Limits indexed by category-valued 2-functors. J. Pure Appl. Algebra 8, 149–181 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Willerton, S.: The twisted Drinfeld double of a finite group via gerbes and finite groupoids. Alg. Geom. Topol 8, 1419–1457 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Teoretisk fysik, Karlstads Universitet, Universitetsgatan 21, S– 651 88, Karlstad, Sweden

    Jürgen Fuchs

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, D–53 111, Bonn, Germany

    Gregor Schaumann

  3. Fachbereich Mathematik, Universität Hamburg Bereich Algebra und Zahlentheorie, Bundesstraße 55, D–20 146, Hamburg, Germany

    Christoph Schweigert

Authors
  1. Jürgen Fuchs
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Gregor Schaumann
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Christoph Schweigert
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Christoph Schweigert.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fuchs, J., Schaumann, G. & Schweigert, C. A Trace for Bimodule Categories. Appl Categor Struct 25, 227–268 (2017). https://doi.org/10.1007/s10485-016-9425-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-016-9425-3

Keywords