Kerodon

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

Proposition 1.1.3.10. Let $S_{\bullet }$ be a simplicial set and let $k \geq -1$ be an integer. Then:

$(a)$

The simplicial set $\operatorname{sk}_{k}( S_{\bullet } )$ has dimension $\leq k$.

$(b)$

For every simplicial set $T_{\bullet }$ of dimension $\leq k$, composition with the inclusion map $\operatorname{sk}_{k}( S_{\bullet } ) \hookrightarrow S_{\bullet }$ induces a bijection

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( T_{\bullet }, \operatorname{sk}_{k}( S_{\bullet } ) ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( T_{\bullet }, S_{\bullet } ). \]

In other words, the image of any map $T_{\bullet } \rightarrow S_{\bullet }$ is contained in $\operatorname{sk}_{k}( S_{\bullet } )$.

Proof. Assertion $(a)$ follows from Remark 1.1.3.7. To prove $(b)$, suppose that $f: T_{\bullet } \rightarrow S_{\bullet }$ is a map of simplicial sets, where $T_{\bullet }$ has dimension $\leq k$. We wish to show that $f$ carries every $n$-simplex $\sigma $ of $T_{\bullet }$ to an $n$-simplex of $\operatorname{sk}_{k}( S_{\bullet } )$. Using Proposition 1.1.3.4, we can reduce to the case where $\sigma $ is a nondegenerate $n$-simplex of $T_{\bullet }$. In this case, our assumption that $T_{\bullet }$ has dimension $\leq k$ guarantees that $n \leq k$, so that $f( \sigma )$ belongs to $\operatorname{sk}_{k}( S_{\bullet } )$ by virtue of Remark 1.1.3.6. $\square$