Where a principal ∞-bundle is the “geometric” incarnation of a cocycle in cohomology, more generally a twisted -bundle is the geometric incarnation of a cocycle in twisted cohomology.
Let be an (∞,1)-topos. Let in be a morphism of ∞-groups in , write for the corresponding delooping and for the corresponding homotopy fiber, so that we have a universal coefficient bundle
Then let be a twisting cocycle with corresponding -principal ∞-bundle ; and let be a cocycle in -twisted cohomology. By the pasting law this induces a twisted -equivariant -principal -bundle on the total space of
This is the twisted -bundle classified by
Section I 4.3 in