# Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets

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.

• 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 18:07
• 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 18:15
• 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 18:18
• 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 19:43
• @TimCampion: Yes, I forgot to include “fibrant” in the original statement, but you already added it. Sep 11, 2020 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.

• 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). Sep 12, 2020 at 17:47