nLab
twisted ∞-bundles

Idea

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

Definition.

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

$$\begin{array}{ccc}V& \to & V//G\\ & & {\downarrow }^{\rho }\\ & & BG\end{array},$$\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 $g:X\to BG$ with coefficients in the moduli ∞-stack of $G$-principal $\mathrm{\infty }$-bundles, the ∞-groupoid of sections of the associated ∞-bundle $P{\mathrm{times}}_{G}V$ is equivalently the cocycle $\mathrm{\infty }$-groupoid of $g$-twisted cohomology relative to $\rho$:

$${\Gamma }_X(P{\times }_{G}V)\simeq H_{/BG}(g,\rho ).$$\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 $\mathrm{\infty }$-bundle, seen by applying the pasting law of (∞,1)-pullbacks to the pasting diagram

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ P& \to & V& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ X& \stackrel{\sigma }{\to }& V//G& \to & BG\end{array}.$$\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