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Does anyone has a simple example of a 1-category $\mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $\mathcal{C}\left[W^{-1}\right]$ is not a 1-category?

Of course there are obvious “big” examples like CW complexes / derived categories, I’m looking for a small example that I’ll be able to understand combinatorially.

Thanks!

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For any $1$-category $C$ the localization $C[C^{-1}]$ at all arrows is an $\infty$-groupoid homotopy equivalent to the nerve of $C$, so it can be any $\infty$-groupoid.

For example take $C$ to be the poset with 6 elements ordered as a,b < c,d < e,f and when you localize at all arrows you get the $2$-sphere $\mathbb{S}^2$. So the simplicial localization will have all objects isomorphic and having a simplicial set of endomorphisms for each object equivalent to $\Omega(\mathbb{S}^2)$.

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  • $\begingroup$ Awesome, thank you! $\endgroup$
    – E. KOW
    Commented Jun 18, 2021 at 1:03
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    $\begingroup$ If I recall correctly, it's even better (or worse ?) than that : any $\infty$-groupoid is $C[C^{-1}]$ for some poset $C$ (viewed as a $1$-category) ! So your posetal example is in some sense prototypical $\endgroup$
    – Maxime Ramzi
    Commented Jun 18, 2021 at 7:37
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    $\begingroup$ On a variant of what Maxime said, there's also a theorem of McDuff saying that every connected homotopy type is of the form $|BM|$ for $M$ a discrete monoid. $\endgroup$
    – Denis Nardin
    Commented Jun 18, 2021 at 12:46
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    $\begingroup$ @MaximeRamzi Yes, that's right. Thomason cofibrant categories are all posets (though not conversely.) arxiv.org/abs/1603.05448 You can start with the category of simplices of a simplicial set and subdivide a couple of times to get the right poset, I believe. $\endgroup$
    – Kevin Carlson
    Commented Jun 18, 2021 at 16:05
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    $\begingroup$ More generally, any $(\infty,1)$-category is the localization of a $1$-category. For instance, if $C$ is a quasi-category with category of simplices $\Delta/C$, there is a canonical map $\tau:N(\Delta/C)\to C$ sending a simplex $\sigma:\Delta^n\to C$ to $\sigma(n)$, say, and this map exhibits $C$ as the localization of $N(\Delta/C)$ by the maps that are sent to identities in $C$. $\endgroup$
    – D.-C. Cisinski
    Commented Jun 27, 2021 at 8:38

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