nLab
twisted ∞-bundles

Contents

Idea

The notion of twisted \infty-bundle is the generalization to higher geometry of the notion of twisted bundle.

Definition.

Let H\mathbf{H} be an (∞,1)-topos, GGrp(H)G \in Grp(\mathbf{H}) an ∞-group, VHV \in \mathbf{H} an object, and ρ\rho an ∞-representation of GG on VV, exhibited by a fiber sequence

V V//G ρ BG, \array{ V &\to& V//G \\ && \downarrow^{\mathrlap{\rho}} \\ && \mathbf{B}G } \,,

which is identified with the universal VV-associated ∞-bundle.

Then for PXP \to X a GG-principal ∞-bundle, correspoding to a cocycle g:XBGg : X \to \mathbf{B}G with coefficients in the moduli ∞-stack of GG-principal \infty-bundles, the ∞-groupoid of sections of the associated ∞-bundle Ptimes GVP times_G V is equivalently the cocycle \infty-groupoid of gg-twisted cohomology relative to ρ\rho:

Γ X(P× GV)H /BG(g,ρ). \Gamma_X(P\times_G V) \simeq \mathbf{H}_{/\mathbf{B}G}(g,\rho) \,.

Each such cocycle may be understood as locally having coefficients in ΩV\Omega V, but globally being twisted by PP.

The geometric structure canonically associated to this is a (twisted GG-equivariant) ΩV\Omega V-principal ∞-bundle QPQ \to P on the total space PP of the twisting \infty-bundle, seen by applying the pasting law of (∞,1)-pullbacks to the pasting diagram

P * P V * X σ V//G BG. \array{ P &\to& * \\ \downarrow && \downarrow \\ P &\to& V &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{\sigma}{\to}& V//G &\to& \mathbf{B}G } \,.

Examples

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References

Created on April 23, 2012 13:01:08 by Urs Schreiber (89.204.154.84)