Citation: Naidu, Deepak. "Categorical Morita equivalence for group-theoretical categories." Communications in Algebra® 35.11 (2007): 3544-3565.
Web: https://arxiv.org/abs/math/0605530
Tags: Mathematical, Modular-tensor-categories, Kitaev-quantum-double, Monoidal-categories
This is the paper proved that non-isomorphic groups can give the same quantum double category. This phenomenon is important - it is the general fact that different inputs to the Levin-Wen construction can give equivalent topological phases, just with different microscopic Hamiltonians.
It is good to be aware of this original reference because of its additional discussion which adds useful context to the main result.