$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 1.1.4.6. Let $S$ be a simplicial set and let $k$ be an integer. Then:
- $(a)$
The simplicial set $\operatorname{sk}_{k}( S )$ has dimension $\leq k$.
- $(b)$
For every simplicial set $T$ of dimension $\leq k$, composition with the inclusion map $\operatorname{sk}_{k}( S ) \hookrightarrow S$ induces a bijection
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( T, \operatorname{sk}_{k}( S ) ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( T, S). \]
In other words, the image of any map $T \rightarrow S$ is contained in $\operatorname{sk}_{k}( S )$.
Proof.
Assertion $(a)$ follows from Remark 1.1.4.5. To prove $(b)$, suppose that $f: T \rightarrow S$ is a map of simplicial sets, where $T$ has dimension $\leq k$. We wish to show that $f$ carries every $n$-simplex $\sigma $ of $T$ to an $n$-simplex of $\operatorname{sk}_{k}( S )$. Using Proposition 1.1.3.8, we can reduce to the case where $\sigma $ is a nondegenerate $n$-simplex of $T$. In this case, our assumption that $T$ has dimension $\leq k$ guarantees that $n \leq k$, so that $f( \sigma )$ belongs to $\operatorname{sk}_{k}( S)$ by virtue of Remark 1.1.4.4.
$\square$