  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.
