Timeline for Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 12, 2020 at 17:21 | vote | accept | Dmitri Pavlov | ||
| Sep 11, 2020 at 22:11 | comment | added | Dmitri Pavlov | @TimCampion: Yes, I forgot to include “fibrant” in the original statement, but you already added it. | |
| Sep 11, 2020 at 20:02 | history | edited | Tim Campion | CC BY-SA 4.0 |
As written, the answer would trivially be that this is impossible.
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| Sep 11, 2020 at 19:43 | comment | added | Tim Campion | Regarding the Kan model structure: since every homotopy type is the classifying space of a category, every simplicial is in particular weakly equivalent to a 2-coskeletal simplicial set. Better yet, every homotopy type is the classifying space of a poset, so every simplicial set is weakly equivalent to a 1-coskeletal simplicial set. But of course, a Kan complex which is $n$-coskeletal is $n$-truncated. | |
| Sep 11, 2020 at 18:51 | answer | added | Tim Campion | timeline score: 5 | |
| Sep 11, 2020 at 18:18 | comment | added | Tim Campion | Emily Riehl has shown that Dugger-Spivak mapping spaces are always 3-coskeletal, and that the homspaces of $\mathfrak C X$ are 2-coskeletal when $X$ is a 1-category. | |
| Sep 11, 2020 at 18:15 | comment | added | Phil Tosteson | Maybe you can promote your Kan example into a Joyal example by taking the simplicial category with two objects $a,b$ and $Hom(a,b) = N(M)$ (self maps are just the identity), and then applying the simplicial nerve to get a simplicial set. Haven't checked whether this works. | |
| Sep 11, 2020 at 18:07 | comment | added | Tim Campion | Up to $\infty$-categorical equivalence, the $n$-coskeletal quasicategories are precisely the quasicategories with $(n-1)$-truncated mapping spaces. So a good place to start would be to ask to slightly weaker question "What are some 2-coskeletal simplicial sets which are not the nerves of 1-categories?" | |
| Sep 11, 2020 at 17:34 | history | asked | Dmitri Pavlov | CC BY-SA 4.0 |