(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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
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 $C(\mathcal{U})$.
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.