nLab
fiber infinity-bundle

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

$$\begin{array}{ccc}U\times V& \to & E\\ \downarrow & & \downarrow \\ U& \to & X\end{array}.$$\array{ U \times V &\to& E \\ \downarrow && \downarrow \\ U &\to& X } \,.

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

See at associated ∞-bundle for more.

Example

A fiber $\mathrm{\infty }$-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 $\mathrm{\infty }$-bundle is the associated ∞-bundle to an automorphism ∞-group-principal ∞-bundle.

Related concepts

• associated ∞-bundle, principal ∞-bundle

• ∞-gerbe

• local coefficient ∞-bundle

References

See the references at associated ∞-bundle.

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

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal infinity-bundles