Construction 1.2.1.18 (The Component Map). Let $S_{}$ be a simplicial set. For every $n$-simplex $\sigma $ of $S_{}$, Proposition 1.2.1.13 implies that there is a unique connected component $S'_{} \subseteq S_{}$ which contains $\sigma $. The construction $\sigma \mapsto S'_{}$ then determines a map of simplicial sets
\[ u: S_{} \rightarrow \underline{ \pi _0( S_{})}, \]
where $\underline{ \pi _0( S_{})}$ denotes the constant simplicial set associated to $\pi _0(S_{})$ (Construction 1.1.5.2). We will refer to $u$ as the component map.