Recall that a (combinatorial) simplicial complex $X$ is said to be $n$-dimensional if it contains at least one face of dimension $n$ and no faces of dimension $n+1$. Further, an $n$-dimensional finite simplicial complex $X$ is said to be pure if every face $\delta$ of dimension $k\lt n$ is a face of some $n$-dimensional face of $X$.

For simplicial sets, we have a different definition for dimension given as:

A simplicial set $S$ is said to be $n$-skeletal if the inclusion $\operatorname{Sk}_n S \subseteq S$ is an isomorphism. We say that $S$ is $m$-dimensional if $m$ is the smallest number for which $S$ is $m$-skeletal.

Given a simplicial complex $X$ as a simplicial set, we see that the associated simplicial set $\Delta[X]$ is $n$-dimensional if and only if $X$ is $n$-dimensional.

Is there a useful notion extending the definition of purity to those simplicial sets with only finitely many nondegenerate simplices?