Categorified trace for module tensor categories over braided tensor categories

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