invertible bimodule categories

a simple question

Given a pair of vector spaces \(V,\,W\), we often want to know if they are isomorphic. One way to witness such an isomorphism is by providing an invertible matrix \(M\) witnessing the isomorphism. This leads to the question:

Given an \(m\times n\) matrix \(M\), how can we determine whether it is invertible?

This question is answered in a first course on linear algebra. \(M\) is invertible if and only if:

1. \[m = n\]
2. \[\det M \neq 0\]

an analogous question

Given a pair of fusion categories \(\cat{C},\, \cat{D}\), we commonly want to know when they give rise to equivalent Turaev-Viro 3-manifold invariants (or equivalent Levin-Wen theories). This equivalence, commonly called Morita equivalence can be witnessed by providing an invertible bimodule category \({}_{\cat{C}}\cat{M}_{\cat{D}}\). This leads to the question:

Given (unitary) fusion categories \(\cat{C},\, \cat{D}\), and a bimodule category \({}_{\cat{C}}\cat{M}_{\cat{D}}\), how can we determine whether it is invertible.

In arXiv:2211.01947, we give an easily checkable, iff condition for invertiblility in terms of associator data for \(\cat{M}\). This condition arises from the representation theory of a certain algebra, called the annular algebra, associated to \({}_{\cat{C}}\cat{M}\) as a module category.