Slightly degenerate categories and $\mathbb{Z}$-modular data

Given a slightly degenerate fusion category $\mathcal{C}$, we explain how it naturally gives rise to a $\mathbb{Z}$-modular data. We do not restrict to spherical categories and work with pivotal categories instead. Finally, we give an interpretation in this framework of the Bonnafe-Rouquier categorification of the $\mathbb{Z}$-modular datum associated to non trivial family of the cyclic complex reflection group.