Corollary 1.2.1.25. For $n \geq 2$, the simplicial set $\operatorname{\partial \Delta }^{n}$ is connected.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 1.2.1.25. For $n \geq 2$, the simplicial set $\operatorname{\partial \Delta }^{n}$ is connected.
Proof. Example 1.2.1.7 guarantees that the standard simplex $\Delta ^ n$ is connected. The desired result now follows from Proposition 1.2.1.22, since the inclusion map $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ is bijective on simplices of dimension $\leq 1$. $\square$