  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.
Given algebra objects $A$ and $B$, we prove that $\mathrm{Tr}_{\mathcal C}(A)$
and $\mathrm{Tr}_{\mathcal C}(A \otimes B)$ are again algebra objects.
Moreover, provided certain mild assumptions are satisfied,
$\mathrm{Tr}_{\mathcal C}(A)$ and $\mathrm{Tr}_{\mathcal C}(A \otimes B)$ are
semisimple whenever $A$ and $B$ are semisimple.
