Kerodon

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

Remark 1.1.6.23. Let $S_{\bullet }$ be a simplicial set. Proposition 1.1.6.22 supplies a coequalizer diagram of sets

\[ \xymatrix { S_1 \ar@ <.4ex>[r]^-{d_0} \ar@ <-.4ex>[r]_-{d_1} & S_0 \ar [r] & \pi _0(S_{\bullet }).} \]

In other words, it allows us to identify $\pi _0( S_{\bullet } )$ with the quotient of $S_0 / \sim $, where $\sim $ is the equivalence relation generated by the set of edges of $S_{\bullet }$ (that is, the smallest equivalence relation with the property that $d_0(e) \sim d_1(e)$, for every edge $e \in S_1$). In particular, the set $\pi _0( S_{\bullet } )$ depends only on the $1$-skeleton of $S_{\bullet }$.