(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
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.