Categorified trace for module tensor categories over braided tensor categories
André Henriques
University of Oxford, United KingdomDavid Penneys
The Ohio State University, Columbus, United States of AmericaJames Tener
University of California, Santa Barbara, USA
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