Categorified trace for module tensor categories over braided tensor categories

Given a braided pivotal category C and a pivotal module tensor categoryM, we define a functor TrC:M C, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor TrC comes equipped with natural isomorphisms Τ_{x, y}: TrC(x⊗ y) TrC(y ⊗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 TrC 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 TrC(A) and TrC(A ⊗ B) are again algebra objects. Moreover, provided certain mild assumptions are satisfied, TrC(A) and TrC(A⊗B) are semisimple whenever A and B are semisimple.