For $G$ a group without topology (or a discrete topological group), let $BG$ denote the groupoid with one object and morphisms given by $G$.
Then, as described at this nLab page, the geometric realization of $BG$ produces a classifying space $\mathscr{B}G$ for continuous principal $G$-bundles (for $G$ discrete this means a covering together with an action of $G$ by deck transformations which is free and transitive on fibers).
Here, I assume we compute the geometric realization by first taking the nerve of $BG$ as a category (w.r.t. the standard "cosimplicial category" given by Definition 3.4 here), then taking the geometric realization of the resulting simplicial set.
I was wondering how to upgrade the above procedure for topological groups $G$?
E.g., do we place a topology on $BG$, making it a topological groupoid?
Then, to compute the geometric realization, do we take the nerve of $BG$ as a topological category? (After choosing some topology on the "cosimplicial category"? Ie lifting the corresponding functor $\Delta \rightarrow \mathrm{Cat}$ to a functor $\Delta \rightarrow \mathrm{TopCat}$?)
Is this the correct procedure?
Edit: I realized we need the "cosimplicial category" to have a topology; ie it should be a cosimplicial Top-internal category, hence a functor $\Delta \rightarrow \mathrm{TopCat}$. As described at the page linked above, the cosimplicial category is the fully faithful functor embedding $\Delta$ in $\mathrm{Cat}$ as a full subcategory consisting of non-empty, finite total orders (regarded as objects in Cat), one for each positive integer, and the (non-strict) poset maps between them.