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.

References

See the references at associated ∞-bundle.

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

Revised on September 15, 2014 16:31:18 by John Dougherty? (68.101.162.59)