Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric realization functor. 

Is an analogous statement true for *topological groups*? That is, given a topological group $G$, regarded as a topological category with one object, does the geometric realization of its homotopy coherent nerve model the classifying space $BG$? 


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**Edit**: I should have been clear about what I meant by the "homotopy coherent nerve" of a topological category; what I meant was the composition 

$$\mathsf{Cat}_{\mathsf{Top}}\xrightarrow{\operatorname{Sing}}\mathsf{Cat}_{\mathsf{sSet}}\xrightarrow{N}\mathsf{sSet},$$

where $\mathsf{Cat}_{\mathsf{Top}}$ and $\mathsf{Cat_{sSet}}$ are the category of small topologically/simplicially enriched categories, $\operatorname{Sing}$ is the base change along the singular complex functor, and $N$ is the [homotopy coheret nerve functor][1].


  [1]: https://ncatlab.org/nlab/show/homotopy+coherent+nerve#the_homotopy_coherent_nerve