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.