nLab twisted ∞-bundles

Idea

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

Definition.

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

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

Examples

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References

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