nLab fiber infinity-bundle

Contents

Context

Bundles

bundles

Context

Classes of bundles

Universal bundles

Presentations

Examples

Constructions

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

Last revised on November 25, 2014 at 19:39:29. See the history of this page for a list of all contributions to it.