nLab
centrally extended groupoid

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Bundles

Cohomology

cohomology

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Contents

Idea

For GG a smooth group, and AA an abelian smooth group, a central extension G^\hat G of GG by AA is equivalently a homotopy fiber sequence of smooth groupoid moduli stacks of the form

BABG^BGcB 2A. \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}^2 A \,.

Here cc is the smooth group cohomology cocycle that classifies the extension.

If here we allow the connected smooth groupoid BG\mathbf{B}G by any smooth groupoid 𝒢 \mathcal{G}_\bullet, then a homotopy fiber sequence of the form

BA𝒢^𝒢cB 2A \mathbf{B}A \to \widehat {\mathcal{G}} \to \mathcal{G} \stackrel{c}{\to} \mathbf{B}^2 A

exhibits 𝒢^\widehat \mathcal{G} as a central AA-extension of 𝒢\mathcal{G}.

Equivalently, such a central extension 𝒢^𝒢\widehat {\mathcal{G}} \to \mathcal{G} is a (BA)(\mathbf{B}A)-principal 2-bundle.

In traditional literature this is mostly considered for Lie groupoids. Specifically, for AA a Lie group and for C(𝒰)C(\mathcal{U}) the Cech groupoid of a good open cover 𝒰\mathcal{U} of a smooth manifold XX, morphisms of smooth stacks XB 2AX \to \mathbf{B}^2 A are equivalently given by actual morphism of Lie groupoids C(𝒰)B 2AC(\mathcal{U}) \to \mathbf{B}^2 A, which are equivalently degree-2 AA-cocycles in the Cech cohomology of XX. The corresponding incarnation 𝒢^\widehat \mathcal{G} of the BA\mathbf{B}A-principal 2-bundle classified by this is known as the AA-bundle gerbe over C(𝒰)C(\mathcal{U}).

Properties

Twisted convolution algebra and twisted K-theory

A central extension of a Lie groupoid induces a twisted groupoid convolution algebra. The corresponding operator K-theory is the twisted K-theory of the differentiable stack of the base groupoid. See at KK-theory for more on this.

Last revised on May 16, 2014 at 21:59:53. See the history of this page for a list of all contributions to it.