nLab fiber infinity-bundle

Definition

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

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

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 $\mathbf{B}G$ of an ∞-group $G$

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

is a $G$-∞-gerbe.

Properties

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

For let $Type$ be the object classifier. Then any bundle $E \to X$ is classified by a morphism

$$X \longrightarrow Type$$

On the other hand, since the pullback to the bundle on some $U$ is trivializable, that bundle over $U$ is classsified by a map that factors through the point which is the name of the fiber $V$

$$U \longrightarrow \ast \stackrel{\vdash V}{\longrightarrow} Type \,.$$

The 1-image-of this point inclusion is the delooping of the automorphism ∞-group of $V$ :

$$U \longrightarrow \ast \longrightarrow \mathbf{B}\mathbf{Aut}(V) \hookrightarrow Type \,.$$

Therefore the fact that $E$ is trivialized over $U$ means that there the classifying maps fit into a commuting diagram of the form

$$\array{ U &\longrightarrow& \mathbf{B}\mathbf{Aut}(V) \\ \downarrow && \downarrow \\ X &\longrightarrow& Type }$$

By assumption the left morphism is a 1-epimorphism and by the above construction the right morphism is a 1-monomorphism. Therefore by the (n-connected, n-truncated) factorization system this diagram has an essentially unique lift

$$\array{ U &\longrightarrow& \mathbf{B}\mathbf{Aut}(V) \\ \downarrow &\nearrow& \downarrow \\ X &\longrightarrow& Type }$$

This diagonal lift classifies an $\mathbf{Aut}(V)$-principal ∞-bundle and the commutativity of the bottom right triangle exhibits the original bundle $E \to X$ as the associated ∞-bundle to that.

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