nLab
twisted infinity-bundle

Idea

Where a principal ∞-bundle is the “geometric” incarnation of a cocycle in cohomology, more generally a twisted $\mathrm{\infty }$-bundle is the geometric incarnation of a cocycle in twisted cohomology.

Definition

Let $H$ be an (∞,1)-topos. Let $\hat{G}\to G$ in $\mathrm{Grp}(H)$ be a morphism of ∞-groups in $H$, write $c:B\hat{G}\to G$ for the corresponding delooping and $BA\to B\hat{G}$ for the corresponding homotopy fiber, so that we have a universal coefficient bundle

$$\begin{array}{ccc}BA& \to & B\hat{G}\\ & & {\downarrow }^c\\ & & BG\end{array}.$$\array{ \mathbf{B}A &\to& \mathbf{B}\hat G \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } \,.

Then let $\varphi :X\to BG$ be a twisting cocycle with corresponding $G$-principal ∞-bundle $P\to X$; and let $\sigma :X\to B\hat{G}$ be a cocycle in $\varphi$-twisted cohomology. By the pasting law this induces a twisted $G$-equivariant $A$-principal $\mathrm{\infty }$-bundle $\tilde{P}$ on the total space of $P$

$$\begin{array}{ccc}\tilde{P}& \to & *\\ \downarrow & & \downarrow \\ P& \stackrel{\tilde{\sigma }}{\to }& BA& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ X& \stackrel{\sigma }{\to }& B\hat{G}& \stackrel{c}{\to }& BG\end{array}.$$\array{ \tilde P &\to& * \\ \downarrow && \downarrow \\ P &\stackrel{\tilde \sigma}{\to}& \mathbf{B}A &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{\sigma}{\to}& \mathbf{B}\hat G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G } \,.

This is the twisted $\mathrm{\infty }$-bundle classified by $\sigma$

Examples

• twisted bundle

• Jandl gerbe

• twisted String 2-bundle

Related concepts

• principal bundle / torsor / associated bundle / twisted bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / 2-gerbe / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle / ∞-gerbe,

• vector bundle

• (∞,1)-vector bundle / (∞,n)-vector bundle

References

Section I 4.3 in

• Principal ∞-bundles -- theory, presentations and applications