Kerodon

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

Corollary 1.1.9.11. Let $\{ S_{\alpha \bullet } \} _{\alpha \in A}$ be a collection of Kan complexes parametrized by a set $A$, and let $S_{\bullet } = \prod _{\alpha \in A} S_{\alpha \bullet }$ denote their product. Then the canonical map

\[ \pi _0( S_{\bullet } ) \rightarrow \prod _{\alpha \in A} \pi _0 ( S_{\alpha \bullet } ) \]

is bijective. In particular, $S_{\bullet }$ is connected if and only if each factor $S_{\alpha \bullet }$ is connected.