Categorified trace for module tensor categories over braided tensor categories

Abstract

Given a braided pivotal category \mathcal C and a pivotal module tensor category \mathcal M , we define a functor \operatorname{Tr}_\mathcal C:\mathcal M \to \mathcal C , called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor \operatorname{Tr}_{\mathcal C} comes equipped with natural isomorphisms \tau_{x,y}:\operatorname{Tr}_{\mathcal C}(x \otimes y) \to \operatorname{Tr}_{\mathcal C}(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 \operatorname{Tr}_{\mathcal C} in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects A and B , we prove that \operatorname{Tr}_{\mathcal C}(A) and \operatorname{Tr}_{\mathcal C}(A \otimes B) are again algebra objects. Moreover, provided certain mild assumptions are satisfied, \operatorname{Tr}_{\mathcal C}(A) and \operatorname{Tr}_{\mathcal C}(A \otimes B) are semisimple whenever A and B are semisimple.