TITLE: Group cohomology version of Deligne-Beilinson cohomology
QUESTION [10 upvotes]: I appreciate Deligne-Beilinson cohomology as a topological cohomology generalization of de Rham cohomology, which concerns the topological structure of manifolds.
On the other hand, we know that there is Group Cohomology theory, suitable for describing the classifying spaces $BG$ of a group $G$, for example, see
ncatlab: group+cohomology and ncatlab: Dijkgraaf-Witten gauge theory.
Is there a group cohomology version of Deligne-Beilinson cohomology, concerning classifying spaces $B^n G$ of a group $G$? What are some key intro References? Thank you in advance.

REPLY [6 votes]: Sorry, I just saw this question.
One can consider Deligne cohomology for simplicial manifolds in the rather obvious way, namely by adding an additional "simplicial manifold" direction to the ususal double complex, and then take the total cohomology. As resolutions one can use certain hypercovers.
For the simplicial manifold $BG$, I have described and done this is Section 2.2. of my paper
Waldorf, Konrad, Multiplicative bundle gerbes with connection, Differ. Geom. Appl. 28, No. 3, 313-340 (2010). ZBL1191.53022.
As explained there, the underlying version of "smooth group cohomology" is Brylinski's "Differentiable cohomology of gauge groups", which remained unpublished. It is equivalent to the Segal-Mitchison smooth group cohomology.
References on general simplicial Deligne cohomology are:
Brylinski, J.-L.; McLaughlin, D. A., The geometry of degree-four characteristic classes and of line bundles on loop spaces. I, Duke Math. J. 75, No. 3, 603-638 (1994). ZBL0844.57025.
Gomi, Kiyonori, Equivariant smooth Deligne cohomology, Osaka J. Math. 42, No. 2, 309-337 (2005). ZBL1081.14030.