nLab
twisted infinity-bundle
Contents
Contents
Idea
Where a principal ∞-bundle is the “geometric” incarnation of a cocycle in cohomology , more generally a twisted ∞ \infty -bundle is the geometric incarnation of a cocycle in twisted cohomology .
Definition
Let H \mathbf{H} be an (∞,1)-topos . Let G ^ → G \hat G \to G in Grp ( H ) Grp(\mathbf{H}) be a morphism of ∞-groups in H \mathbf{H} , write c : B G ^ → B G \mathbf{c} : \mathbf{B}\hat G \to \mathbf{B}G for the corresponding delooping and B A → B G ^ \mathbf{B}A \to \mathbf{B}\hat G for the corresponding homotopy fiber , so that we have a universal coefficient bundle
B A → B G ^ ↓ c B G .
\array{
\mathbf{B}A &\to& \mathbf{B}\hat G
\\
&& \downarrow^{\mathrlap{\mathbf{c}}}
\\
&& \mathbf{B}G
}
\,.
Then let ϕ : X → B G \phi : X \to \mathbf{B}G be a twisting cocycle with corresponding G G -principal ∞-bundle P → X P \to X ; and let σ : X → B G ^ \sigma : X \to \mathbf{B}\hat G be a cocycle in ϕ \phi -twisted cohomology . By the pasting law this induces a twisted G G -equivariant A A -principal ∞ \infty -bundle P ˜ \tilde P on the total space of P P
P ˜ → * ↓ ↓ P → σ ˜ B A → * ↓ ↓ ↓ X → σ B G ^ → c B G .
\array{
\tilde P &\to& *
\\
\downarrow && \downarrow
\\
P &\stackrel{\tilde \sigma}{\to}& \mathbf{B}A &\to& *
\\
\downarrow && \downarrow && \downarrow
\\
X
&\stackrel{\sigma}{\to}&
\mathbf{B}\hat G
&\stackrel{\mathbf{c}}{\to}&
\mathbf{B}G
}
\,.
This is the twisted ∞ \infty -bundle classified by σ \sigma
Examples
References
Section I 4.3 in
Last revised on June 28, 2016 at 14:00:31.
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