Kerodon

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

Corollary 1.1.4.8. Let $k$ be an integer, let $S$ be a simplicial set, and let $\operatorname{{\bf \Delta }}_{S}^{ \leq k}$ denote the category of simplices of $S$ having dimension $\leq k$ (see Construction 1.1.3.9). Then the tautological map

\[ \varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S}^{ \leq k} } \Delta ^ n \rightarrow S \]

is a monomorphism, whose image is the $k$-skeleton $\operatorname{sk}_{k}(S) \subseteq S$.

Proof. By virtue of Remark 1.1.4.4, replacing $S$ by the $k$-skeleton $\operatorname{sk}_{k}(S)$ does not change the category $\operatorname{{\bf \Delta }}_{S}^{ \leq k}$. We may therefore assume without loss of generality that $S$ has dimension $\leq k$, in which case the desired result follows from Proposition 1.1.3.11. $\square$