Categorified trace for module tensor categories over braided tensor categories

  • André Henriques

    University of Oxford, United Kingdom
  • David Penneys

    The Ohio State University, Columbus, United States of America
  • James Tener

    University of California, Santa Barbara, USA
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Abstract

Given a braided pivotal category and a pivotal module tensor category , we define a functor , called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor comes equipped with natural isomorphisms , 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 in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects and , we prove that and are again algebra objects. Moreover, provided certain mild assumptions are satisfied, and are semisimple whenever and 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