Timeline for Simple example of nontrivial simplicial localization
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2021 at 8:38 | comment | added | D.-C. Cisinski | 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 $C$. | |
| Jun 27, 2021 at 7:57 | history | edited | David White | CC BY-SA 4.0 |
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| Jun 18, 2021 at 16:05 | comment | added | Kevin Carlson | @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. | |
| Jun 18, 2021 at 12:46 | comment | added | Denis Nardin | 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. | |
| Jun 18, 2021 at 7:37 | comment | added | Maxime Ramzi | 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 | |
| Jun 18, 2021 at 1:03 | comment | added | E. KOW | Awesome, thank you! | |
| Jun 18, 2021 at 1:02 | vote | accept | E. KOW | ||
| Jun 18, 2021 at 0:57 | history | answered | Simon Henry | CC BY-SA 4.0 |