## Extending the definition of “pure of dimension n” from simplicial complexes to simplicial sets?

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 $\operatorname{Sk}^n S \subseteq S$ of the $n$-skeleton of $S$ 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?

The property that I'm hoping for in such an extension is sort of a generalization of the "prism decomposition" for products of simplices. That is, for the product of two simplicial sets $S$ and $T$, pure of dimension $s$ and $t$ respectively, I would like the product $S\times T$ to be pure of dimension $s+t$. It is certainly the case that the product has dimension $s+t$ (this can be seen working simplex by simplex), but without a proper definition of purity, we can't give a real generalization.

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