twisted ∞-bundles

The notion of *twisted $\infty$-bundle* is the generalization to higher geometry of the notion of twisted bundle.

Let $\mathbf{H}$ be an (∞,1)-topos, $G \in Grp(\mathbf{H})$ an ∞-group, $V \in \mathbf{H}$ an object, and $\rho$ an ∞-representation of $G$ on $V$, exhibited by a fiber sequence

$\array{
V &\to& V//G
\\
&& \downarrow^{\mathrlap{\rho}}
\\
&& \mathbf{B}G
}
\,,$

which is identified with the universal $V$-associated ∞-bundle.

Then for $P \to X$ a $G$-principal ∞-bundle, correspoding to a cocycle $g : X \to \mathbf{B}G$ with coefficients in the moduli ∞-stack of $G$-principal $\infty$-bundles, the ∞-groupoid of sections of the associated ∞-bundle $P times_G V$ is equivalently the cocycle $\infty$-groupoid of $g$-twisted cohomology relative to $\rho$:

$\Gamma_X(P\times_G V) \simeq \mathbf{H}_{/\mathbf{B}G}(g,\rho)
\,.$

Each such cocycle may be understood as locally having coefficients in $\Omega V$, but globally being *twisted* by $P$.

The geometric structure canonically associated to this is a (twisted $G$-equivariant) $\Omega V$-principal ∞-bundle $Q \to P$ on the total space $P$ of the twisting $\infty$-bundle, seen by applying the pasting law of (∞,1)-pullbacks to the pasting diagram

$\array{
P &\to& *
\\
\downarrow && \downarrow
\\
P &\to& V &\to& *
\\
\downarrow && \downarrow && \downarrow
\\
X &\stackrel{\sigma}{\to}& V//G &\to& \mathbf{B}G
}
\,.$

(…)

- Thomas Nikolaus, Urs Schreiber, Danny Stevenson,
*Principal ∞-bundles -- Theory, presentations and applications*

Created on April 23, 2012 13:01:08
by Urs Schreiber
(89.204.154.84)