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Quinn’s Formula and Abelian 3-Cocycles for Quadratic Forms

Abstract

In pointed braided fusion categories knowing the self-symmetry braiding of simples is theoretically enough to reconstruct the associator and braiding on the entire category (up to twisting by a braided monoidal auto-equivalence). We address the problem to provide explicit associator formulas given only such input. This problem was solved by Quinn in the case of finitely many simples. We reprove and generalize this in various ways. In particular, we show that extra symmetries of Quinn’s associator can still be arranged to hold in situations where one has infinitely many isoclasses of simples.

References

  1. 1.

    Baues, H.: Combinatorial homotopy and 4-dimensional complexes, De Gruyter Expositions in Mathematics, vol. 2. Walter de Gruyter & Co., Berlin (1991). With a preface by Ronald Brown. MR 1096295

    Book  Google Scholar 

  2. 2.

    Bulacu, D., Caenepeel, S., Torrecillas, B.: The braided monoidal structures on the category of vector spaces graded by the Klein group, Proc. Edinb. Math. Soc. (2) 54(3), 613–641. MR 2837470 (2011)

  3. 3.

    Braunling, O.: Braided categorical groups and strictifying associators. Homol. Homotopy Appl. 22(2), 295–321 MR 4098945 (2020)

  4. 4.

    Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories. I, Selecta Math. (N.S.) 16(1), 1–119. MR 2609644 (2010)

  5. 5.

    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories, Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence (2015). MR 3242743

    MATH  Google Scholar 

  6. 6.

    Eilenberg, S., Mac Lane, S.: On the groups H(π,n). I, Ann. of Math. (2) 58, 55–106. MR 0056295 (1953)

  7. 7.

    Flanders, H.: Tensor and exterior powers. J. Algebra 7, 1–24. MR 212044 (1967)

  8. 8.

    Huang, H.-L., Liu, G., Ye, Y.: The braided monoidal structures on a class of linear Gr-categories. Algebr. Represent. Theory 17(4), 1249–1265. MR 3228486 (2014)

  9. 9.

    Huang, H.-L., Liu, G., Yang, Y., Ye, Y.: Finite quasi-quantum groups of diagonal type, J. Reine Angew. Math. 759, 201–243. MR 4058179 (2020)

  10. 10.

    Huang, H.-L, Wan, Z, Ye, Y: Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf-Witten invariants. Proc. Roy. Soc. Edinburgh Sect. A 150(4), 1937–1964. MR 4122441 (2020)

  11. 11.

    Johnson, N., Osorno, A.: Modeling stable one-types, Theory Appl. Categ. 26(20), 520–537. MR 2981952 (2012)

  12. 12.

    Joyal, A., Street, R.: Braided monoidal categories. Macquarie Mathematical Reports, no 860081 (1986)

  13. 13.

    Joyal, A.: Braided tensor categories. Adv. Math. 102(1), 20–78. MR 1250465 (1993)

  14. 14.

    Kapustin, A., Saulina, N.: Topological boundary conditions in abelian Chern-Simons theory. Nuclear Phys. B 845(3), 393–435. MR 2755172 (2011)

  15. 15.

    Mac Lane, S.: Cohomology theory of Abelian groups. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 8–14. MR 0045115 (1950)

  16. 16.

    Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323. Springer, Berlin. MR 2392026 (2008)

  17. 17.

    Quinn, F.: Group categories and their field theories. Proceedings of the Kirbyfest, Berkeley. Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 407–453. MR 1734419 (1998)

  18. 18.

    Sính, H.X.: Gr-catégories (thesis, handwritten manuscript). Université, Paris 7. https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html (1975)

  19. 19.

    Whitehead, J.H.C.: A certain exact sequencex. Ann. Math. (2) 52, 51–110. MR 35997 (1950)

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Open Access funding enabled and organized by Projekt DEAL.

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Correspondence to Oliver Braunling.

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The author was supported by DFG GK1821 “Cohomological Methods in Geometry”. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Presented by: Alistair Savage

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Braunling, O. Quinn’s Formula and Abelian 3-Cocycles for Quadratic Forms. Algebr Represent Theor (2020). https://doi.org/10.1007/s10468-020-10001-1

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Keywords

  • Braided categorical group
  • Picard groupoid
  • Strictification
  • Skeletalization
  • Associator

Mathematics Subject Classification (2010)

  • 18D10
  • 19D23