The classifying space of the groupoid $\pmb\Delta^n$ Consider the groupoid generated by the category $\{0\to 1\to\cdots\to n\}$; let's call this category $\pmb\Delta^n$ opposed to the category $\triangle^n$, which is "thinner".
I'm trying to figure out what the classifying space (i.e. the geometric realization of the nerve $N\pmb\Delta^n$) looks like, given that the classifying space of $\triangle^n$ is fairly "simple" to draw.
More generally, is there any hope to relate the classifying space $\mathbb B\mathcal C$ of a category to that of its groupoidification, in some non-trivial cases? I believe that the most general problem is considerably more difficult to attack (describe $\mathbb B(\mathcal C[W^{-1}])$ given $\mathbb B\mathcal C$ and $W\subset \text{Mor }\mathcal C$), but any clue is accepted.
 A: As to the more general question you mention at the end: What is the relationship between the geometric realization of a category and the geometric realization of its fundamental gropuoid?
I believe that the classifying space of the groupoidification of a category should be the 1-truncation of the classifying space of the category, equivalently the classifying space of the fundamental groupoid of the classifying space of the category. Let me say that again:
$|N(\Pi_1C)| = \tau_1|N(C)| = \mathbf{B}\Pi_1|N(C)|$
where 


*

*$N(C)$ is the nerve of a category $C$

*$|S|$ is the geometric realization of a simplicial set $S$

*$\tau_1 X$ is the 1-truncation of a space $X$, i.e. the space obtained by killing all higher homotopy groups.

*$\Pi_1C = C[C^{-1}]$ is the fundamental groupoid / localization-everywhere of a category $C$

*$\Pi_1X$ is also the fundamental groupoid of a space $X$

*$\mathbf{B}G = K(G,1) = |N(G)|$ is the classifying space of a groupoid $G$


But I don't have a reference. It might be somewhere in Gabriel-Zisman. (I'm secretly hoping that someone will see this and provide a reference or else set me straight...).
