Categorified trace for module tensor categories over braided tensor categories

Abstract

Given a braided pivotal category $\cC$ and a pivotal module tensor category $\cM$, we define a functor $\Tr_\cC:\cM \to \cC$, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor $\Tr_\cC$ comes equipped with natural isomorphisms $\tau_{x,y}:\Tr_\cC(x \otimes y) \to \Tr_\cC(y \otimes x)$, 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 $\Tr_\cC$ 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 $\Tr_\cC(A)$ and $\Tr_\cC(A \otimes B)$ are again algebra objects. Moreover, provided certain mild assumptions are satisfied, $\Tr_\cC(A)$ and $\Tr_\cC(A \otimes B)$ are semisimple whenever $A$ and $B$ are semisimple.