Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended to the following situations: bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized: bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated; and the base manifold parameterizing the bundles may be replaced by a topological stack. Some methods developed for the construction may be of independent interest: these are a Pontryagin type duality between commutative principal bundles and gerbes, nonabelian Takai duality for groupoids, and the computation of certain equivariant Brauer groups.
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