Remark 3.5.6.11. In the situation of Construction 3.5.6.10, the relation $\sigma \sim _{m} \sigma '$ implies that $\sigma = \sigma '$ whenever $m < n$. It follows that the tautological map $v: X \rightarrow \pi _{\leq n}(X)$ is bijective on simplices of dimension $< n$, and surjective on simplices of dimension $n$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$