Kerodon

Proof. Since the final object $\Delta ^{0} \in \operatorname{Set_{\Delta }}$ is connected (Example 1.2.1.7), it will suffice to show that the collection of connected simplicial sets is closed under pairwise products. Let $S_{\bullet }$ and $T_{\bullet }$ be connected simplicial sets; we wish to show that $S_{} \times T_{}$ is connected. Equivalently, we wish to show that $\pi _0( S_{\bullet } \times T_{\bullet } )$ consists of a single element (Corollary 1.2.1.15). By virtue of Proposition 1.2.1.22, the component map supplies a surjection

It will therefore suffice to show that every pair of vertices $(s,t), (s', t') \in S_0 \times T_0$ belong to the same connected component of $S_{\bullet } \times T_{\bullet }$. Let $K_{\bullet } \subseteq S_{\bullet } \times T_{\bullet }$ be the connected component which contains the vertex $(s', t )$. Since $S_{\bullet }$ is connected, the map

factors through a unique connected component of $S_{\bullet } \times T_{\bullet }$, which must be equal to $K_{\bullet }$. It follows that $K_{\bullet }$ contains the vertex $(s,t)$. A similar argument (with the roles of $S_{\bullet }$ and $T_{\bullet }$ reversed) shows that $K_{\bullet }$ contains $(s',t')$. $\square$