We prove the universal lifting theorem: for an α-simply connected and α-connected Lie groupoid Γ with Lie algebroid A, the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on Γ. As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular, we prove that a group pair (D; G) associated to a quasi-Manin triple (d; g; h) induces a quasi-Poisson groupoid on the transformation groupoid G × D/G ? D/G. Its momentum map corresponds exactly with the D/G-momentum map of Alekseev and Kosmann-Schwarzbach.
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