For strongly self-absorbing C*-algebras, the classifying spaces for cocycle actions and Γ-kernels are infinite loop spaces, so H^1 obstruction sets take values in cohomology groups.
The classifying space of a topological 2- group
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Infinite Loop Spaces and Group Actions on Strongly Self-Absorbing C*-Algebras
For strongly self-absorbing C*-algebras, the classifying spaces for cocycle actions and Γ-kernels are infinite loop spaces, so H^1 obstruction sets take values in cohomology groups.