For an ∞-topos and for an ∞-group in , the higher analog of group cohomology of is the cohomology of the delooping object .
Given a cocycle in the -group cohomology of , then its homotopy fiber (the principal ∞-bundle over that it modulates) is the corresponding ∞-group extension.
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):
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