Contents

Idea

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

Definition

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

$\array{ \mathbf{B}A &\to& \mathbf{B}\hat G \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } \,.$

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

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

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

Section I 4.3 in