bundles

cohomology

# Contents

## Definition

Let $H$ be an ambient (∞,1)-topos. Let $V,X$ be two objects of $H$. Then a $V$-fiber bundle over $X$ in $marthbfH$ is a morphism $E\to X$ such that there is an effective epimorphism $U\to X$ and an (∞,1)-pullback square of the form

$\begin{array}{ccc}U×V& \to & E\\ ↓& & ↓\\ U& \to & X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ U \times V &\to& E \\ \downarrow && \downarrow \\ U &\to& X } \,.

Externally this is a $V$-fiber $\infty$-bundle.

See at associated ∞-bundle for more.

## Example

A fiber $\infty$-bundle whose typical fiber $V$ is a pointed connected object, hence a delooping $BG$ of an ∞-group $G$

$V\simeq BG$V \simeq \mathbf{B}G

is a $G$-∞-gerbe.

## Properties

Every $V$-fiber $\infty$-bundle is the associated ∞-bundle to an automorphism ∞-group-principal ∞-bundle.

## References

See the references at associated ∞-bundle.

The explicit general definition appears as def. 4.1 in part I of

Revised on January 22, 2013 15:20:50 by Urs Schreiber (89.204.138.238)