CATEGORIFIED TRACE FOR MODULE TENSOR CATEGORIES OVER BRAIDED TENSOR CATEGORIES

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