nLab
fiber infinity-bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

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

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

Externally this is a VV-fiber \infty-bundle.

See at associated ∞-bundle for more.

Example

A fiber \infty-bundle whose typical fiber VV is a pointed connected object, hence a delooping BG\mathbf{B}G of an ∞-group GG

VBG V \simeq \mathbf{B}G

is a GG-∞-gerbe.

Properties

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

For let TypeType be the object classifier. Then any bundle EXE \to X is classified by a morphism

XType X \longrightarrow Type

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

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

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

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

Therefore the fact that EE is trivialized over UU means that there the classifying maps fit into a commuting diagram of the form

U BAut(V) X Type \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

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

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

References

See the references at associated ∞-bundle.

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

Revised on November 25, 2014 19:39:29 by Urs Schreiber (82.224.164.72)