Kerodon

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

Variant 1.1.6.24. Let $S_{\bullet }$ be a simplicial set. Then the set of connected components $\pi _0( S_{\bullet } )$ can also be described as the coequalizer of the pair of maps $d_0, d_1: S_{1}^{ \mathrm{nd} } \rightrightarrows S_0$, where $S_{1}^{\mathrm{nd} } \subseteq S_{1}$ denotes the set of nondegenerate edges of $S_{\bullet }$ (since every degenerate edge $e \in S_{1}$ automatically satisfies $d_0(e) = d_1(e)$). We therefore have a coequalizer diagram of sets

\[ \xymatrix { \operatorname{Edge}(G) \ar@ <.4ex>[r]^-{s} \ar@ <-.4ex>[r]_-{t} & \operatorname{Vert}(G) \ar [r] & \pi _0(S_{\bullet }),} \]

where $G = \mathrm{Gr}(S_{\bullet } )$ is the directed graph of Example 1.1.5.4. In other words, we can identify $\pi _0( S_{\bullet } )$ with the set of connected components of $G$, in the usual graph-theoretic sense.