For any given topological group $G$ we have Segal's construction/definition of $BG$. I'm recalling it in case the details turn out to be relevant.

Form the disjoint union of $G^n\times\Delta_n$ for $n\geq 0$ and identify points via $(d_i\cdot,\cdot)\sim (\cdot,\partial_i \cdot)$ where $\partial_i:\Delta_{n-1}\to\Delta_n$ are the face maps and $d_i :G^n\to G^{n-1}$ maps $(g_1,\ldots,g_n)\to (g_1,\ldots , g_i g_{i+1},\ldots,g_n)$ for $i\neq 0,n$. When $i=0$ (resp. $n$) the map $d_i$ is the projection on to the first (resp. last) $n-1$ factors.

On the other hand, one can consider $G$ as a topological category with one object and $G$ as its morphisms. One then considers the **topological nerve** $\mathcal{N}(G)$ which has one $0$-simplex, one $1$-simplex for each element of $G$, $2$-simplices are given by triangles labelled by $g_1,g_2,g_3$ such that the interior of the triangle corresponds to a path $h$ joining $g_1 g_2$ and $g_3$. Higher simplices are defined similarly. If one uses the natural gluing on this collection, one arrives at the geometric realization of $\mathcal{N}(G)$. Let's call it $\mathcal{B}G$. So here's my question, which may very well be known.

How are the two constructions $BG$ and $\mathcal{B}G$ related?

In fact, this prompts a more general question :

Given a topological category (in the sense of Segal) $\mathcal{C}$, i.e., $\mathit{Ob}$ and $\mathit{Mor}$ are topological spaces, one can form the topological nerve $\mathcal{N}(\mathcal{C})$, as explained above, and then take its realization. On the other hand, the usual nerve $N(\mathcal{C})$ of $\mathcal{C}$ is a simplicial object in $\mathbf{Top}$ and we can take its realization. How are these two realizations related?

I would love to know the answer to the above question for the usual notion of topological category too, viz., where only morphisms $\mathit{Mor}(x,y)$ is required to be a (compacty generated Hausdorff) space for any $x,y\in\mathit{Ob}$.

EDIT : May be I hadn't explained the topological nerve definition which Harry succintly does in his comment below. It is also the same definition given in *Higher Topos Theory* by J. Lurie.

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