Remark 4.8.4.10. In the situation of Construction 4.8.4.9, the relation $\sigma \sim _{m} \sigma '$ implies that $\sigma = \sigma '$ whenever $m < n$. It follows that the tautological map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is bijective on simplices of dimension $< n$, and surjective on simplices of dimension $n$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$