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

$\begin{array}{ccc}BA& \to & B\stackrel{^}{G}\\ & & {↓}^{c}\\ & & BG\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\sigma :X\to B\stackrel{^}{G}$ be a cocycle in $\varphi$-twisted cohomology. By the pasting law this induces a twisted $G$-equivariant $A$-principal $\infty$-bundle $\stackrel{˜}{P}$ on the total space of $P$

$\begin{array}{ccc}\stackrel{˜}{P}& \to & *\\ ↓& & ↓\\ P& \stackrel{\stackrel{˜}{\sigma }}{\to }& BA& \to & *\\ ↓& & ↓& & ↓\\ X& \stackrel{\sigma }{\to }& B\stackrel{^}{G}& \stackrel{c}{\to }& BG\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\sigma$

## References

Section I 4.3 in

Revised on June 28, 2012 21:37:28 by Urs Schreiber (89.204.138.204)