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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.

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    $\begingroup$ The nerve functor $N : \mathbf{Grpd} \to \mathbf{sSet}$ sends equivalences of groupoids to homotopy equivalences of Kan complexes. In particular, the nerve of a contractible groupoid is contractible. $\endgroup$ – Zhen Lin Mar 1 '14 at 19:46
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    $\begingroup$ The classifying space of a groupoid is a disjoint union of classifying spaces of groups. In this case the group is trivial. $\endgroup$ – Qiaochu Yuan Mar 1 '14 at 20:39
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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...).

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    $\begingroup$ It's in Quillen's higher K-theory I right at the beginning. Essentially: the localisation of a category C at all arrows is the fundamental groupoid of the classifying space BC of C: thus its geometric realisation is the first Postnikov stage/1-truncation of BC. $\endgroup$ – David Roberts Jan 8 '15 at 5:52
  • $\begingroup$ Thanks! This will take some grokking since he's doing things in terms of covering spaces, and refers to Gabriel-Zisman in a few places. I'll have to look at it in the morning! $\endgroup$ – Tim Campion Jan 8 '15 at 6:17

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