Kerodon

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

Proposition 1.1.4.1. 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 1.1.4.1. By virtue of Proposition 1.1.4.13, 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 1.1.4.10. $\square$