Kerodon

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

Remark 6.3.7.12. Let $S$ be a simplicial set and $A$ denote the collection of simplicial subsets $S' \subseteq S$ which are isomorphic to a standard simplex. We regard $\operatorname{Sub}_{\operatorname{{\bf \Delta }}}(S)$ as a partially ordered set with respect to inclusion. If $S$ is nonsingular, the construction

\[ ( \sigma : \Delta ^ n \rightarrow S) \mapsto ( \operatorname{im}(\sigma ) \subseteq S ) \]

determines an isomorphism of categories $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \simeq A$, where $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}}$ denotes category of nondegenerate simplices of $S$ (Notation 3.3.3.11). Combining this observation with Proposition 3.3.3.16, we obtain an isomorphism of simplicial sets $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{Sd}(S)$.