Kerodon

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

Warning 1.1.6.27. The collection of connected simplicial sets is not closed under infinite products (so the functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ does not commute with infinite products). For example, let $G$ be the directed graph with vertex set $\operatorname{Vert}(G) = \operatorname{\mathbf{Z}}_{\geq 0} = \operatorname{Edge}(G)$, with source and target maps

\[ s,t: \operatorname{Edge}(G) \rightarrow \operatorname{Vert}(G) \quad \quad s(n) = n \quad \quad t(n) = n+1. \]

More informally, $G$ is the directed graph depicted in the diagram

\[ \xymatrix { 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & 3 \ar [r] & 4 \ar [r] & \cdots } \]

The associated $1$-dimensional simplicial set $G_{\bullet }$ is connected. However, the infinite product $S_{\bullet } = \prod _{n \in \operatorname{\mathbf{Z}}_{\geq 0}} G_{\bullet }$ is not connected. By definition, the vertices of $S_{\bullet }$ can be identified with functions $f: \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathbf{Z}}_{\geq 0}$. It is not difficult to see that two such functions $f,g: \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathbf{Z}}_{\geq 0}$ belong to the same connected component of $S_{\bullet }$ if and only if the function $n \mapsto | f(n) - g(n) |$ is bounded. In particular, the identity function $n \mapsto n$ and the zero function $n \mapsto 0$ do not belong to the same connected component of $S_{\bullet }$.