Skip to main content Accesibility Help
×
×
Home

Monoidal Categories, 2-Traces, and Cyclic Cohomology

  • Mohammad Hassanzadeh (a1), Masoud Khalkhali (a2) and Ilya Shapiro (a1)
Abstract

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ( $\mathscr{C},\otimes$ ) endowed with a symmetric 2-trace, i.e., an $F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in $F$ ”. Furthermore, we observe that if $\mathscr{M}$ is a $\mathscr{C}$ -bimodule category and $(F,M)$ is a stable central pair, i.e., $F\in \text{Fun}(\mathscr{M},\text{Vec})$ and $M\in \mathscr{M}$ satisfy certain conditions, then $\mathscr{C}$ acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

Copyright
References
Hide All
[BFO] Bezrukavnikov, R., Finkelberg, M., and Ostrik, V., On tensor categories attached to cells in affine Weyl groups. III . Israel J. Math. 170(2009), 207234. https://doi.org/10.1007/s11856-009-0026-9.
[BS] Bohm, G. and Stefan, D., A categorical approach to cyclic duality . J. Noncommut. Geom. 6(2012), no. 3, 481538. https://doi.org/10.4171/JNCG/98.
[Co] Connes, A., Cohomologie cyclique et foncteurs Ext n . C. R. Acad. Sci. Paris Sér. I Math. 296(1983), no. 23, 953958.
[CM1] Connes, A. and Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem . Commun. Math. Phys. 198(1998), 199246. https://doi.org/10.1007/s002200050477.
[CM2] Connes, A. and Moscovici, H., Cyclic cohomology and Hopf algebras . Lett. Math. Phys. 48(1999), 97108. https://doi.org/10.1023/A:1007527510226.
[DP] Dold, A. and Puppe, D., Duality, trace, and transfer . Proceedings of the Steklov Institute of Mathematics, 154(1984), 85103.
[DSPS] Douglas, C. L., Schommer-Pries, C., and Snyder, N., Dualizable tensor categories. 2018. arxiv:1312.7188.
[EGNO] Etingof, P., Gelaki, S., Nikshych, D., and Ostrik, V., Tensor categories. Mathematical Surveys and Monographs, 205, American Mathematical Society, Providence, RI, 2015. https://doi.org/10.1090/surv/205.
[FSS] Fuchs, J., Schaumann, G., and Schweigert, C., A trace for bimodule categories . Appl. Categ. Structures 25(2017), no. 2, 227268. https://doi.org/10.1007/s10485-016-9425-3.
[HKRS1] Hajac, P. M., Khalkhali, M., Rangipour, B., and Sommerhauser, Y., Hopf-cyclic homology and cohomology with coefficients . C. R. Math. Acad. Sci. Paris 338(2004), no. 9, 667672. https://doi.org/10.1016/j.crma.2003.11.036.
[HKRS2] Hajac, P. M., Khalkhali, M., Rangipour, B., and Sommerhauser, Y., Stable anti-Yetter–Drinfeld modules . C. R. Acad. Sci. Paris Ser. I 338(2004), 587590. https://doi.org/10.1016/j.crma.2003.11.037.
[HPT] Henriques, A., Penneys, D., and Tener, J., Categorified trace for module tensor categories over braided tensor categories . Doc. Math. 21(2016), 10891149.
[JSV] Joyal, A., Street, R., and Verity, D., Traced monoidal categories . Math. Proc. Cambridge Philos. Soc. 119(1996), 447468. https://doi.org/10.1017/S0305004100074338.
[K1] Kaledin, D., Cyclic homology with coefficients . In: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., 270, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 2347. https://doi.org/10.1007/978-0-8176-4747-6_2.
[K2] Kaledin, D., Trace theories and localization . In: Stacks and categories in geometry, topology, and algebra, Contemp. Math., 643, American Mathematical Society, Providence, RI, 2015, pp. 227262. https://doi.org/10.1090/conm/643/12900.
[Kassel] Kassel, C., Quantum groups. Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-0783-2.
[KP] Khalkhali, M. and Pourkia, A., Hopf cyclic cohomology in braided monoidal categories . Homology Homotopy Appl. 12(2010), no. 1, 111155. https://doi.org/10.4310/HHA.2010.v12.n1.a9.
[KS1] Kobyzev, I. and Shapiro, I., A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids. 2018. arxiv:1803.09194.
[KS2] Kobyzev, I. and Shapiro, I., Anti-Yetter–Drinfeld modules for quasi-Hopf algebras. arxiv:1804.02031. 2018.
[Lo] Loday, J. L., Cyclic homology. Grundlehren der Mathematischen Wissenschaften, 301, Springer-Verlag, Berlin, 1992. https://doi.org/10.1007/978-3-662-21739-9.
[M1] Majid, S., Foundations of quantum group theory. Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511613104.
[M2] Majid, S., Quantum double for quasi-Hopf algebras . Lett. Math. Phys. 45(1998), no. 1, 19. https://doi.org/10.1023/A:1007450123281.
[P] Ponto, K., Relative fixed point theory . Algebr. Geom. Topol. 11(2011), 839886. https://doi.org/10.2140/agt.2011.11.839.
[PS] Ponto, K. and Shulman, M., Shadows and traces in bicategories . J. Homotopy Relat. Struct. 8(2013), 151200. https://doi.org/10.1007/s40062-012-0017-0.
[Sch] Schauenburg, P., Hopf modules and Yetter–Drinfel’d modules . J. Algebra 169(1994), 874890. https://doi.org/10.1006/jabr.1994.1314.
[Sh1] Shapiro, I., Some invariance properties of cyclic cohomology with coefficients. 2016. arxiv:1611.01425.
[Sh2] Shapiro, I., On the anti-Yetter–Drinfeld module-contramodule correspondence. 2017. arxiv:1704.06552.
[Y] Yetter, D., Quantum groups and representations of monoidal categories . Math. Proc. Cambridge Philos. Soc. 108(1990), no. 2, 261290. https://doi.org/10.1017/S0305004100069139.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed