nLab centrally extended groupoid

Idea

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

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

Here $c$ is the smooth group cohomology cocycle that classifies the extension.

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

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

exhibits $\widehat \mathcal{G}$ as a central $A$-extension of $\mathcal{G}$.

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

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

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.

Related concepts

• central extension

• infinity-group extension