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 $BG$?
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.
