nLab centrally extended groupoid

Redirected from "centrally extended Lie groupoid".
Contents

Context

Bundles

bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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.