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Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact etale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles.
We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zungs theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The
In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Serre, Hurewicz stack.
Let $mathcal{G}$ be a Lie groupoid. The category $Bmathcal{G}$ of principal $mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $mathcal{D}$, there exists a Lie groupoid $mathcal{H}$ such that $Bmathca
Let $mathbb{X}=[X_1rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $mathcal{H} subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $mathcal{H}$ is i
We give a concise introduction to (discrete) algebras arising from etale groupoids, (aka Steinberg algebras) and describe their close relationship with groupoid C*-algebras. Their connection to partial group rings via inverse semigroups also explored.