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Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$). What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$ is not Joyal weakly equivalent to a 2-coskeletal quasicategory? (Equivalently, mapping simplicial sets between objects of $Y$ have nontrivial homotopy groups in degree 1 or higher.)

If $X$ satisfies the Segal conditions, then $X$ is the nerve of a 1-category, hence is Joyal fibrant, so such $X$ cannot be an example.

In the Kan model structure on simplicial sets, examples are easy to construct: the Kan fibrant replacement of the nerve of the delooping of a monoid $M$ is the homotopy group completion of $M$, which can have nontrivial higher homotopy groups.

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  • $\begingroup$ 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?" $\endgroup$
    – Tim Campion
    Sep 11 '20 at 18:07
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    $\begingroup$ 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. $\endgroup$ Sep 11 '20 at 18:15
  • $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Sep 11 '20 at 18:18
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    $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Sep 11 '20 at 19:43
  • $\begingroup$ @TimCampion: Yes, I forgot to include “fibrant” in the original statement, but you already added it. $\endgroup$ Sep 11 '20 at 22:11
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Let $P$ be the poset $(\partial \Delta[1]) \star (\partial \Delta[1])$ (where $\star$ means "join"). Note that the classifying space of $P$ is $S^1$. Moreover, as a poset, (the nerve of) $P$ is 1-coskeletal.

There is a "suspension" $\Sigma P$ of $P$, like Phil Tosteson suggests, but constructed in a more hands-on way: $\Sigma P$ has

  • two objects $\{-,+\}$,

  • 4 nondegenerate 1-cells, all going from $-$ to $+$, corresponding to the 4 elements of $P$, and

  • 4 nondegenerate 2-cells corresponding to the 4 1-cells of (the nerve of) $P$. (in each of these one of the 1-faces is degenerate; there's a choice to make of which one -- let's say that the $\partial_0$ face is degenerate)

An exhaustive (but not too bad) search reveals that $\Sigma P$ is 2-coskeletal -- this is essentially because $P$ is 1-coskeletal and has no nontrivial "composable pairs". But clearly the Joyal fibrant replacement of $\Sigma P$ is not 2-coskeletal -- we have $Hom_{\Sigma P}(-,+) \simeq S^1$ which is not essentially discrete.

To be a bit more careful about that last claim, think about it this way. If we apply $\mathfrak C$ to $\Sigma P$, then I think it's pretty clear that we get the simplicial category which I'd also denote $\Sigma P$, with two objects $\{-,+\}$, and with the homspace $Hom(-,+)$ given by (the nerve of) $P$. Since every simplicial set is Joyal-cofibrant and $\mathfrak C$ is left Quillen, we haven't messed up the $\infty$-categorical equivalence class of $\Sigma P$.

Then, a Bergner-fibrant replacement of this simplicial category can be found by simply Kan-fibrantly replacing the homspaces levelwise, and we find that indeed we have an $\infty$-category with two objects $-,+$ and the only nontrivial homspace being $Hom(-,+) \simeq S^1$. This is a model-independent statement, so the Joyal-fibrant replacement of $\Sigma P$ likewise has this property, which shows it's not equivalent to an ordinary 1-category, and hence not equivalent to anything Joyal-fibrant and 2-coskeletal.

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  • $\begingroup$ After a bit more thought, I'm pretty sure that the above construction works in greater generality: let $P$ be an arbitrary 1-coskeletal simplicial set (note that every homotopy type is modeled by a poset, which is in particular 1-coskeletal). Construct $\Sigma P$ as indicated above. Then there is still a close relationship between $n$-simplices of $P$ and $(n+1)$-simplices of $\Sigma P$, with the result that $\Sigma P$ is 2-coskeletal. Moreover, the same argument as above shows that $Hom_{\Sigma P}(-,+) \simeq P$ (which can be an arbitrary homotopy type). $\endgroup$
    – Tim Campion
    Sep 12 '20 at 17:47

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