Skeletality of a simplicial set $X$ vs. highest degree of a non-degenerate simplex

Question

Let $X$ be a simplicial set with a non-degenetate simplex in degree $n$ and suppose that all simplices in higher degrees are degenerate.

Is $X$ an $n$-skeletal simplicial set and not $i$-skeletal for $i<n$? If not, is there an inequality between the two numbers (i.e. $n$ and the ''skeletality'' $k$, if it exists)?

If this is not true, I wonder how one may practically test if a simplicial set $X$ is $k$-skeletal. How about the concrete example of the nerve of $\mathbb{N}$ or the simplex category $\Delta$? Are they $k$-skeletal for some $k$?

Answer 1

Indeed, the following are equivalent:

• $X$ is $n$-skeletal.
• Every non-degenerate $k$-simplex of $X$ satisfies $k \le n$.

The downward direction is clear, and for the upward direction, simply note that the inclusion $\operatorname{sk}_n X \hookrightarrow X$ is a bijection on $k$-simplices for $k \le n$.

The nerve of any category with arbitrarily long composable sequences of non-identity morphisms is never $n$-skeletal for any $n$: indeed, any composable sequence of non-identity morphisms of length $k$ gives rise to a non-degenerate $k$-simplex.