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Construction (The Component Map). Let $S_{\bullet }$ be a simplicial set. For every $n$-simplex $\sigma $ of $S_{\bullet }$, Proposition implies that there is a unique connected component $S'_{\bullet } \subseteq S_{\bullet }$ which contains $\sigma $. The construction $\sigma \mapsto S'_{\bullet }$ then determines a map of simplicial sets

\[ u: S_{\bullet } \rightarrow \underline{ \pi _0( S_{\bullet })}_{\bullet }, \]

where $\underline{ \pi _0( S_{\bullet })}_{\bullet }$ denotes the constant simplicial set associated to $\pi _0(S_{\bullet })$ (Construction We will refer to $u$ as the component map.