Definition 3.6.1.1. We say that a simplicial set $X$ is finite if it satisfies the following pair of conditions:
For every integer $n \geq 0$, the set of $n$-simplices $X_{n} \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, X)$ is finite.
The simplicial set $X$ is finite-dimensional (Definition 1.1.3.1): that is, there exists an integer $m$ such that every nondegenerate simplex has dimension $\leq m$.