Kerodon

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

Corollary 1.1.6.26. The functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ preserves finite products.

Proof. Since $\pi _0( \Delta ^0)$ is a singleton (Example 1.1.6.7), it will suffice to show that for every pair of simplicial sets $S_{\bullet }$ and $T_{\bullet }$, the canonical map

\[ \pi _0( S_{\bullet } \times T_{\bullet } ) \rightarrow \pi _0( S_{\bullet } ) \times \pi _0( T_{\bullet } ) \]

is bijective. Writing $S_{\bullet }$ and $T_{\bullet }$ as a disjoint union of connected components (Proposition 1.1.6.13), we can reduce to the case where $S_{\bullet }$ and $T_{\bullet }$ are connected, in which case the desired result follows from Proposition 1.1.6.25. $\square$