Corollary 3.5.2.6. For every nonnegative integer $n$, the simplicial set $\operatorname{\partial \Delta }^{n}$ is $(n-1)$-connective.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^{n}$ is bijective on $k$-simplices for $k < n$, it will suffice to show that the standard simplex $\Delta ^{n}$ is $(n-1)$-connective (Corollary 3.5.2.5). This is clear, since $\Delta ^{n}$ is contractible (Example 3.2.4.2). $\square$