A trace for bimodule categories

TL;DR

This paper introduces a categorical trace for bimodule categories over finite tensor categories, generalizing Hochschild homology and with applications to topological field theories, especially Dijkgraaf-Witten models.

Contribution

It defines a universal 2-functor as a categorification of Hochschild homology, providing multiple realizations and demonstrating cyclic invariance.

Findings

Defined a universal 2-functor for bimodule categories

Established cyclic invariance of the trace

Applied results to 3D topological field theories, including Dijkgraaf-Witten models

Abstract

We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Witten topological field theories.