  Dijkgraaf, Pasquier and Roche introduced twisted quantum doubles of a finite
group in the context of conformal field theory. We study equivalences that
arise among the braided monoidal categories associated to these quantum
doubles, especially in the commutative case. This involves a close study of the
cohomology of various complexes introduced by Eilenberg-MacLane. We show among
other things that an equivalence of braided monoidal categories is the same as
gauge equivalence of the corresponding quantum doubles, and that an invariant
for an equivalence class is given by a metabolic quadratic space.
