  We describe topological T-duality and Poisson-Lie T-duality in terms of QP
(differential graded symplectic) manifolds and their canonical transformations.
Duality is mediated by a QP-manifold on doubled non-abelian "correspondence"
space, from which we can perform mutually dual symplectic reductions, where
certain canonical transformations play a vital role. In the presence of
spectator coordinates, we show how the introduction of "bibundle" structure on
correspondence space realises changes in the global fibration structure under
Poisson-Lie duality. Our approach can be directly translated to the worldsheet
to derive dual string current algebras. Finally, the canonical transformations
appearing in our reduction procedure naturally suggest a Fourier-Mukai integral
transformation for Poisson-Lie T-duality.
