By P. Hoffman, V. Snaith

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Gk ) = (g1 , . . , gi gi+1 , . . , gk ), if 0 < i < k if i = k. (g1 , . . , gk−1 ) deﬁnes a simplicial manifold G• . The bar-de Rham complex of G is the total complex of the double complex Ω• (G• ), where the boundary maps are d : Ωq (Gk ) → Ωq+1 (Gk ), the usual de Rham diﬀerential, and ∂ : Ωq (Gk ) → Ωq (Gk+1 ), the alternating sum of the pull-back of the k + 1 maps Gk → Gk+1 , as in group cohomology. For example, if ω ∈ Ω2 (G), then ∂ω = p∗1 ω − m∗ ω + p∗2 ω. ) It follows that a 2-form ω is a 3-cocycle in the total complex if and only if it is multiplicative and closed; in particular, a symplectic groupoid can be deﬁned as a Lie groupoid G together with a nondegenerate 2-form ω which is a 3-cocycle.

The previous exercise is the ﬁrst step in establishing that complete Poisson maps with symplectic target must be ﬁbrations. In fact, if P1 is symplectic and dim(P1 ) = dim(P2 ), then a complete Poisson map ψ : P1 → P2 is a covering map. In general, a complete Poisson map ψ : P1 → P2 , where P2 is symplectic, is a locally trivial symplectic ﬁbration with a ﬂat Ehresmann connection: the horizontal lift in Tx P1 of a vector X in Tψ(x) P2 is deﬁned as Π1 ((Tx ψ)∗ Π−1 2 (X)). The horizontal subspaces deﬁne a foliation whose leaves are coverings of P2 , and P1 and ψ are completely determined, up to isomorphism, by the holonomy π1 (P2 , x) → Aut(ψ −1 (x)), see [20, Sec.

Right actions and torsors are deﬁned in the obvious analogous way. If groupoids G1 and G2 act on S from the left and right, respectively, and the actions commute, then we call S a (G1 , G2 )-bibundle. A bibundle is left principal when the left G1 -action is principal with respect to the moment map for the right action of G2 . J J If S is a (G1 , G2 )-bibundle with moments P1 ←1 S →2 P2 , and if S J J is a (G2 , G3 )-bibundle with moments P2 ←2 S →3 P3 , then their “tensor product” is the orbit space S ∗ S := (S ×(J2 ,J2 ) S )/G2 , (55) where G2 acts on S ×(J2 ,J2 ) S diagonally.