Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.5.1.8. Every simplicial set $X$ can be realized as a union $\bigcup _{X' \subseteq X} X'$, where $X'$ ranges over the collection of finite simplicial subsets of $X$ (to prove this, we observe that every $n$-simplex $\sigma $ is contained in a finite simplicial subset $X' \subseteq X$: in fact, we can take $X'$ to be the image of $\sigma : \Delta ^{n} \rightarrow X$). Moreover, the collection of finite simplicial subsets of $X$ is closed under finite unions. It follows that realization $X \simeq \bigcup _{X' \subseteq X} X'$ exhibits $X$ as a filtered direct limit of its finite simplicial subsets.