Categorified Trace for Module Tensor Categories over Braided Tensor Categories

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.

2010 Mathematics Subject Classification: 18D10

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