Categorified trace for module tensor categories over braided tensor categories

  • André Henriques

  • David Penneys

  • James Tener

    Department of Mathematics University of California, Santa Barbara Santa Barbara, CA 93106 USA
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Abstract

Given a braided pivotal category \cC\cC and a pivotal module tensor category \cM\cM, we define a functor \Tr\cC:\cM\cC\Tr_\cC:\cM \to \cC, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor \Tr\cC\Tr_\cC comes equipped with natural isomorphisms τx,y:\Tr\cC(xy)\Tr\cC(yx)\tau_{x,y}:\Tr_\cC(x \otimes y) \to \Tr_\cC(y \otimes x), which we call the traciators. This situation lends itself to a diagramatic calculus of 'strings on cylinders', where the traciator corresponds to wrapping a string around the back of a cylinder. We show that \Tr\cC\Tr_\cC in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects AA and BB, we prove that \Tr\cC(A)\Tr_\cC(A) and \Tr\cC(AB)\Tr_\cC(A \otimes B) are again algebra objects. Moreover, provided certain mild assumptions are satisfied, \Tr\cC(A)\Tr_\cC(A) and \Tr\cC(AB)\Tr_\cC(A \otimes B) are semisimple whenever AA and BB are semisimple.

Cite this article

André Henriques, David Penneys, James Tener, Categorified trace for module tensor categories over braided tensor categories. Doc. Math. 21 (2016), pp. 1089–1149

DOI 10.4171/DM/553