Classifying space commutes with geometric realization - reference request Let $G$ be a nice topological group, say a compact connected Liegroup. 
Then one can construct a model of its classifying space as $EG/G$ where $EG$ is any contractible space with free $G$ action. 
On the other hand, one could take the geometric realization of the simplicial construction of the classifying space of the singular simplices of $G$:
$|\bar{W} (sing G)|$ 
Is it written down somewhere that these two spaces are weakly homotopy equivalent (or some variation of this statement)?
 A: I think you should be able to prove this roughly as follows: first consider the loop space of your construction.  For nice simplicial spaces, the loop space can be calculated level-wise (see May's Geometry of Iterated Loop Spaces, for instance), and hence the loop space of your construction is homotopy equivalent to the (realization of the) simplicial space $[n] \mapsto \Omega |\overline{W} Sing_n (G)|$.  But this space is level-wise weakly equivalent to $[n] \mapsto Sing_n (G)$, whose realization is homotopy equivalent to $G$.  So the loop space of your construction yields $G$, and now you need to apply some form of the statement: $B\Omega X \simeq X$.
Also, note that it's a standard fact (due to Segal's "Classifying spaces and spectral sequences" and/or May's "Classifying spaces and fibrations") that the simplicial bar construction applied directly to your topological group $G$ gives a model for $BG$ (this requires the inclusion of the identity of $G$ to be a cofibration, which is of course true for Lie groups).
