Duality for arithmetic Dijkgraaf-Witten theory

A 3-dimensional topological quantum field theory is a symmetric monoidal functor from the oriented cobordism category to vector spaces, with Dijkgraaf-Witten theory being a prominent example using finite gauge groups and 3-cocycles. Naidu classified pairs of finite groups and 3-cocycles that lead to equivalent Dijkgraaf-Witten theories for 3-manifolds. We establish analogous equivalences for arithmetic Dijkgraaf-Witten theory over totally imaginary number fields F containing n-th roots of unity, where n is invertible on X subset spec O_F. For the full ring of integers X = spec O_F, we provide examples where these equivalences fail but also identify sufficient conditions under which they still hold.