Given a simplicial manifold $\,X^{\mathbf{\cdot}}$ (say a classifying spae $BG$ of a Lie group $G$) we have a differential given by $d_n^*=\sum_i (-1)^id^*_{n,i}\,,$ acting on functions $f_n:X^n\to A\,,$ $A$ a discrete abelian grop and where $d_{n,i}$ are the face maps on $X^n\,.$ Is there a relation between this cohomology and the singular cohomology of the geometric realization?
There seems to be some relation from what I've read but I'm having trouble finding exactly what it is. I read that if $G$ is discrete they are the same but it seems this can't be true in general.
Edit: Here there is answer saying it is true for homology:Group (co)homology and classyfing spaces
Edit: Here is an answer suggesting it is true for cohomology, but I don't know if the group is discrete or if the maps are continuous: Group (co)homology and classyfing spaces