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.



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 $\Omega(\mathbb{S}^2)$.

  • $\begingroup$ Awesome, thank you! $\endgroup$ – E. KOW Jun 18 at 1:03
  • 3
    $\begingroup$ 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 Jun 18 at 7:37
  • 3
    $\begingroup$ 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 Jun 18 at 12:46
  • 1
    $\begingroup$ @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 Arlin Jun 18 at 16:05
  • 2
    $\begingroup$ 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 $C$. $\endgroup$ – Denis-Charles Cisinski Jun 27 at 8:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.