Given a braided pivotal category C and a pivotal module tensor
category M, 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 τx,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.Comment: 49 pages, many figure