The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) $G$-module, where $G$-modules are structures, which differentiate to representations. Many geometric structures involving Lie groupoids and stacks are classified by the cohomology of sheaves taking values in $G$-modules and not in representations, including $S^1$-groupoid extensions and equivariant gerbes. Examples of such sheaves are $\mathcal{O}^*$ and $\mathcal{O}^*(*D)\,,$ where the latter is the sheaf of invertible meromorphic functions with poles along a divisor $D\,.$ We show that there is an infinitesimal description of $G$-modules and a corresponding Lie algebroid cohomology. We then define a generalized van Est map relating these Lie groupoid and Lie algebroid cohomologies and study its kernel and image. Applications include the integration of several infinitesimal geometric structures, including Lie algebroid extensions, Lie algebroid actions on gerbes, and certain Lie $\infty $-algebroids.

We generalize the van Est map and isomorphism theorem in three ways, and we propose a category of Lie algebroids and LA-groupoids (with equivalences). First, we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack G and the foliated cohomology of a stack H Ñ G mapping into it. At the level of Lie groupoids, this amounts to describing the van Est map as a map from Lie groupoid cohomology to the cohomology of a particular LA-groupoid. We do this by associating to any (nice enough) homomorphism of Lie groupoids f : H Ñ G a natural foliation of the stack rH 0 {Hs . In the case of a wide subgroupoid H ãÝ Ñ G , this foliation can be thought of as equipping the normal bundle of H with the structure of an LA-groupoid. In particular, this generalization allows us to derive classical results, including van Est's isomorphism theorem about the maximal compact subgroup, which we generalize to proper subgroupoids, as well as the Poincaré lemma; it also gives a new method of computing Lie groupoid cohomology.Secondly, we generalize the functions that we can take cohomology of in the context of the van

Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution algebra of the geometric quantization of a Lie 2-groupoid, one which integrates the Poisson structure. We show that there is an analogue of the Poisson sigma model which is valued in Lie 1-groupoids and which can often be defined non-perturbatively; it can be obtained by symplectic reduction using the quantization of the Lie 2-groupoid. We call these groupoid-valued sigma models and we argue that, when they exist, they can be used to compute correlation functions of gauge invariant observables. This leads to a (possibly non-associative) product on the underlying space of functions on the Poisson manifold, and in several examples we show that we recover strict deformation quantizations, in the sense of Rieffel. Even in the cases when our construction leads to a non-associative product we still obtain a C * -algebra and a "quantization" map. In particular, we construct noncommutative C * -algebras equipped with an SU (2)-action, together with an equivariant "quantization" map from C ∞ (S 2 ) . No polarizations are used in the construction of these algebras.

The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) G-module, where G-modules are structures which differentiate to representations. Many geometric structures involving Lie groupoids and stacks are classified by the cohomology of sheaves taking values in G-modules and not in representations, including S 1 -groupoid extensions and equivariant gerbes. Examples of such sheaves are O * and O * ( * D) , where the latter is the sheaf of invertible meromorphic functions with poles along a divisor D . We show that there is an infinitesimal description of G-modules and a corresponding Lie algebroid cohomology. We then define a generalized van Est map relating these Lie groupoid and Lie algebroid cohomologies, and study its kernel and image. Applications include the integration of several infinitesimal geometric structures, including Lie algebroid extensions, Lie algebroid actions on gerbes, and Lie ∞-algebroids.