twisted infinity-bundle



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.


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:BG^BG\mathbf{c} : \mathbf{B}\hat G \to \mathbf{B}G for the corresponding delooping and BABG^\mathbf{B}A \to \mathbf{B}\hat G for the corresponding homotopy fiber, so that we have a universal coefficient bundle

BA BG^ c BG. \array{ \mathbf{B}A &\to& \mathbf{B}\hat G \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G } \,.

Then let ϕ:XBG\phi : X \to \mathbf{B}G be a twisting cocycle with corresponding GG-principal ∞-bundle PXP \to X; and let σ:XBG^\sigma : X \to \mathbf{B}\hat G be a cocycle in ϕ\phi-twisted cohomology. By the pasting law this induces a twisted GG-equivariant AA-principal \infty-bundle P˜\tilde P on the total space of PP

P˜ * P σ˜ BA * X σ BG^ c BG. \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



Section I 4.3 in

Revised on June 28, 2016 10:00:31 by David Corfield (