Proposition 1.2.1.13. Let $S_{}$ be a simplicial set. Then $S_{}$ is the disjoint union of its connected components.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\sigma $ be an $n$-simplex of $S_{}$; we wish to show that there is a unique connected component of $S_{}$ which contains $\sigma $. This follows from Proposition 1.2.1.11, applied to the map $\Delta ^ n \rightarrow S_{}$ classified by $\sigma $ (since the standard $n$-simplex $\Delta ^ n$ is connected; see Example 1.2.1.7). $\square$