Categorified trace for module tensor categories over braided tensor categories

Understand

Given a braided pivotal category $\mathcal C$ and a pivotal module tensor category $\mathcal M$, we define a functor $\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$, called the associated categorified trace.

• By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor $\mathrm{Tr}_{\mathcal C}$ comes equipped with natural isomorphisms $\tau_{x,y}:\mathrm{Tr}_{\mathcal C}(x \otimes y) \to \mathrm{Tr}_{\mathcal C}(y \otimes x)$, 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 $\mathrm{Tr}_{\mathcal C}$ in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid.