# Categorified trace for module tensor categories over braided tensor categories

### André Henriques

University of Oxford, United Kingdom### David Penneys

The Ohio State University, Columbus, United States of America### James Tener

University of California, Santa Barbara, USA

## Abstract

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 $τ_{x,y}:Tr_{C}(x⊗y)→Tr_{C}(y⊗x)$, 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 $Tr_{C}$ 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 $Tr_{C}(A⊗B)$ are again algebra objects. Moreover, provided certain mild assumptions are satisfied, $Tr_{C}(A)$ and $Tr_{C}(A⊗B)$ are semisimple whenever $A$ and $B$ are semisimple.

## Cite this article

André Henriques, David Penneys, James Tener, Categorified trace for module tensor categories over braided tensor categories. Doc. Math. 21 (2016), pp. 1089–1149

DOI 10.4171/DM/553