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.
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