$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 1.1.5.1. The evaluation functor
\[ \operatorname{ev}_0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad X_{\bullet } \mapsto X_0 \]
restricts to an equivalence of categories
\[ \{ \textnormal{Simplicial sets of dimension $\leq 0$} \} \simeq \operatorname{Set}. \]
Proof of Proposition 1.1.5.1.
By virtue of Proposition 1.1.5.14, it will suffice to show that the construction $X_{\bullet } \mapsto X_0$ induces an equivalence of categories
\[ \{ \text{Discrete simplicial sets} \} \rightarrow \operatorname{Set}. \]
This follows immediately from Corollary 1.1.5.11.
$\square$