nLab
universal principal infinity-bundle

Context

Bundles

Cohomology

cohomology

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Special notions

Variants

Extra structure

Operations

Theorems

Yoneda lemma

(,1)(\infty,1)-Topos theory

(∞,1)-topos theory

Background

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Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

A universal principal ∞-bundle over an ∞-group-object in an (∞,1)-topos H\mathbf{H} is a morphism EGBG\mathbf{E}G \to \mathbf{B}G in a 1-categorical model CC for H\mathbf{H} (a homotopical category) such that every GG-principal ∞-bundle PXP \to X in H\mathbf{H} is modeled in CC by an (ordinary) pullback of EGBG\mathbf{E}G \to \mathbf{B}G.

Notice that in the proper (∞,1)-topos-context the universal GG-principal ∞-bundle for an ∞-group GG is nothing but the point inclusion *BG* \to \mathbf{B}G into the delooping of GG: every GG-principal \infty-bundle PXP \to X is the (∞,1)-pullback

P * X BG \array{ P &\to& * \\ \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G }

of the point in H\mathbf{H}, namely the homotopy kernel of its classifying map gg. In other words, in a full (,1)(\infty,1)-categorical context the notion of universal bundle disappears. It is a notion genuinely associated with 1-categorical models for H\mathbf{H}.

Standard models

Assume that we have a homotopical category model CC for H\mathbf{H} that has the structure of a category of fibrant objects. Notably this can be the full subcategory on fibrant objects of a model structure on simplicial presheaves.

By fibrations

By standard results on homotopy pullbacks every morphism EGBG\mathbf{E}G \to \mathbf{B}'G that

  1. is a fibration

  2. fits into a diagram

    EG * BG BG \array{ \mathbf{E}G &\stackrel{\simeq}{\to}& * \\ \downarrow && \downarrow \\ \mathbf{B}'G &\stackrel{\simeq}{\to}& \mathbf{B}G }

    with the horizontal morphisms being weak equivalences;

is a model for the universal GG-principal \infty-bundle.

By path fibrations

A standard construction of a fibration EGBG\mathbf{E}G \to \mathbf{B}G is above is obtained as follows:

by standard results on homotopy pullbacks, we have that the bundle PXP \to X classified by a morphism XX^BGX \stackrel{\simeq}{\leftarrow} \hat X\to \mathbf{B}G is given by the limit

P * (BG) I BG * BG, \array{ P &\to& &\to& * \\ \downarrow && && \downarrow \\ && (\mathbf{B}G)^I &\stackrel{\simeq}{\to}& \mathbf{B}G \\ \downarrow && \downarrow^{\mathrlap{\simeq}} \\ * &\to& \mathbf{B}G } \,,

where (BG) I(\mathbf{B}G)^I is a path space object for BG\mathbf{B}G.

This limit may be computed as two consecutive pullbacks

P EG * (BG) I BG * BG. \array{ P &\to& \mathbf{E}G &\to& * \\ \downarrow && && \downarrow \\ && (\mathbf{B}G)^I &\to& \mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G } \,.

The intermediate pullback

EG:=(BG) I× BG* \mathbf{E}G := (\mathbf{B}G)^I \times_{\mathbf{B}G} *

is the path fibration over BG\mathbf{B}G. By the factorization lemma we have that the projecton EGBG\mathbf{E}G \to \mathbf{B}G is indeed a fibration and by the fact that the acyclic fibration (BG) IBG(\mathbf{B}G)^I \to \mathbf{B}G is preserved under pullback that indeed EG*\mathbf{E}G \to * is a weak equivalence.

By decalage

For XX a Kan complex with a single vertex, the decalage construction DecXXDec X \to X is a Kan fibration that fits into a diagram

DecX * X to= X. \array{ Dec X &\stackrel{\simeq}{\to}& * \\ \downarrow && \downarrow \\ X &\stackrel{=}{to}& X } \,.

For GG a simplicial group the standard simplicial model for the delooping of GG in H=\mathbf{H} = ∞Grpd is denoted W¯G\bar W G. This is a Kan complex with a single vertex and DecW¯GDec \bar W G is the standard model for the universal simplicial principal bundle, traditionally written WGW G.

DecW¯G=WGW¯G. Dec \bar W G = W G \to \bar W G \,.

These constructions are functorial and hence extend to models for (∞,1)-toposes by a model structure on simplicial presheaves.

The model WGW G for the universal GG-principal bundle has the special property that it is a groupal model for universal principal ∞-bundles.

Revised on June 25, 2012 22:15:58 by Urs Schreiber (89.204.139.149)