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.

  • $\begingroup$ 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
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.

  • $\begingroup$ 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

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.