# Simple example of nontrivial simplicial localization

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!

## 1 Answer

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

• Awesome, thank you! – E. KOW Jun 18 at 1:03
• 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 – Maxime Ramzi Jun 18 at 7:37
• 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. – Denis Nardin Jun 18 at 12:46
• @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. – Kevin Arlin Jun 18 at 16:05
• 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 $C$. – Denis-Charles Cisinski Jun 27 at 8:38