It is well known that a measured groupoid G defines a von Neu-mann algebra W * (G), and that a Lie groupoid G canonically defines both a C *-algebra C * (G) and a Poisson manifold A * (G). We show that the maps (G) are functorial with respect to suitable categories. In these categories Morita equivalence is isomorphism of objects, so that these maps preserve Morita equivalence.

We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) → C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

In this paper we define twisted equivariantK-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, this defines a periodic cohomology theory on the category of finite G-CW-complexes with G-stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras 1 , in the theory of integrable system 2 , and in field theory 3 . But for instance, in 2 , the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In 4, 5 we have defined such a test functional space in the case of a linear Poisson...

Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible. We also consider the problem of integrability by closed subgroupoids, and we give conditions...

We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.

The notion of local equivalence relation on a topological space is generalised to that of local subgroupoid. Properties of coherence are considered. The main result is notions of holonomy and monodromy groupoid for certain Lie local subgroupoids.

Androulidakis–Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of assignment: if a foliated manifold $(M,\mathcal{F}\_M)$ is the quotient $(P,\mathcal{F}\_P)$ along surjective submersion with connected fibers, then same true for corresponding groupoids. For quotients by Lie...

For a Lie groupoid G over smooth manifold M we construct the adjoint action of étale # germs local bisections on algebroid g . With this action, form associated convolution C c ∞ ( ) / R -bialgebra , We represent in algebra transversal distributions This construction extends Cartier-Gabriel decomposition Hopf with finite support group.