Kerodon

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Proposition 1.1.6.13. Let $S_{\bullet }$ be a simplicial set. Then $S_{\bullet }$ is the disjoint union of its connected components.

Proof. Let $\sigma$ be an $n$-simplex of $S_{\bullet }$; we wish to show that there is a unique connected component of $S_{\bullet }$ which contains $\sigma$. This follows from Proposition 1.1.6.11, applied to the map $\Delta ^ n \rightarrow S_{\bullet }$ classified by $\sigma$ (since the standard $n$-simplex $\Delta ^ n$ is connected; see Example 1.1.6.7). $\square$