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Let $C$ be a category. The groupoid completion of $C$ is the free groupoid on $C$, i.e. the category $C[C^{-1}]$ obtained by localizing at everything. Recall that the classifying space $\mathbf{B}C$ of $C$ is the geometric realization of the nerve of $C$. It's well-known that there is an equivalence of groupoids $C[C^{-1}] \simeq \Pi_1(\mathbf{B}C)$ between the groupoid completion of $C$ and the fundamental groupoid of the classifying space of $C$.

Now let $\mathcal{C}$ be a topologically-enriched category. Now the groupoid completion $\mathcal{C}[C^{-1}]$ is a topological groupoid. There is also a classifying space $\mathbb{B}\mathcal{C}$, which is the geometric realization of the nerve of $\mathcal{C}$ (where the nerve of $\mathcal{C}$ is a simplicial space). But the fundamental groupoid $\Pi_1(\mathbb{B}\mathcal{C})$ is discrete, so it is generally not equivalent to $\mathcal{C}[\mathcal{C}^{-1}]$.

Question.

Can we recover $\mathcal{C}[C^{-1}]$ up to equivalence of topological groupoids from $\mathbb{B}\mathcal{C}$ in some other way? More generally, what kind of homotopical significance does $\mathcal{C}[\mathcal{C}^{-1}]$ have? Does it fit into a sequence of topological groups analogous to the usual homotopy groups, and is there perhaps a natural model structure on topologically-enriched categories or simplicial spaces where the weak equivalences are the morphisms inverted by these functors?

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  • $\begingroup$ I've accepted Karol's answer because I suspect that it's the closest thing I'm going to find to what I'm looking for. But I'm very much interested if anyone happens to know anything more about the genuinely topological case here, or even in the simplicial case if there are conditions more general than actual cofibrancy where $\mathcal{C}[\mathcal{C}^{-1}] \simeq \Omega N \mathcal{C}$. $\endgroup$ – Tim Campion Mar 17 '16 at 17:46
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I'm not sure how well-posed this question is, what exactly is the topological structure of $\mathcal{C}[\mathcal{C}^{-1}]$?

However, the question certainly makes good sense for simplicial categories and the answer is affirmative. Dwyer and Kan proved that if $\mathcal{C}$ is a cofibrant simplicial category, then $\mathcal{C}[\mathcal{C}^{-1}](X, Y)$ is the loop space of $N \mathcal{C}$ (at $X$). See the proof of Proposition 9.3 in Simplicial Localizations of Categories (this part of their results really goes back to Segal and his delooping machine).

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  • $\begingroup$ Thanks, I'll take a look! I think there should be no problem (assuming size is handled appropriately and we use a nice category of spaces) in defining the topology on the hom-spaces as a quotient of the space of zigzags. The resulting category should have the appropriate enriched analogue of the universal property of the groupoid completion. I haven't checked the details, but is there some part of the process I should be worried about? I'm really interested in the genuinely topological case; I have weird examples in mind where the hom-spaces may be profinite. $\endgroup$ – Tim Campion Mar 17 '16 at 1:38
  • $\begingroup$ On reflection, I agree that the localization of a topological category should be defined exactly as you described. I would still worry about its homotopical meaningfulness though. The simplicial counterpart of the colimit you mention is always a homotopy colimit, but I wouldn't expect it to be the case in the topological setting. $\endgroup$ – Karol Szumiło Mar 17 '16 at 1:48
  • $\begingroup$ Oh -- right, this is pretty subtle. And even Bergner-cofibrancy is a pretty strong condition. $\endgroup$ – Tim Campion Mar 17 '16 at 2:05

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