# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

Definition 3.5.1.1. 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 1.1.3.9): that is, there exists an integer $m$ such that every nondegenerate simplex has dimension $\leq m$.