$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 1.1.3.13. Let $S_{\bullet }$ be a simplicial set and let $k \geq 0$. Then the construction outlined above determines a pushout square
\[ \xymatrix { \underset { \sigma \in S_{k}^{\mathrm{nd}} }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { \sigma \in S_{k}^{\mathrm{nd}} }{\coprod } \Delta ^{k} \ar [d] \\ \operatorname{sk}_{k-1}( S_{\bullet } ) \ar [r] & \operatorname{sk}_{k}( S_{\bullet } ) } \]
in the category $\operatorname{Set_{\Delta }}$ of simplicial sets.
Proof.
Unwinding the definitions, we must prove the following:
- $(\ast )$
Let $\tau $ be an $n$-simplex of $\operatorname{sk}_{k}( S_{\bullet } )$ which is not contained in $\operatorname{sk}_{k-1}(S_{\bullet } )$. Then $\tau $ factors uniquely as a composition
\[ \Delta ^{n} \xrightarrow {\alpha } \Delta ^{k} \xrightarrow {\sigma } S_{\bullet }, \]
where $\sigma $ is a nondegenerate simplex of $S_{\bullet }$ and $\alpha $ does not factor through the boundary $\operatorname{\partial \Delta }^{k}$ (in other words, $\alpha $ induces a surjection of linearly ordered sets $[n] \rightarrow [k]$).
Proposition 1.1.3.4 implies that any $n$-simplex of $S_{\bullet }$ admits a unique factorization $\Delta ^{n} \xrightarrow {\alpha } \Delta ^{m} \xrightarrow {\sigma } S_{\bullet }$, where $\alpha $ is surjective and $\sigma $ is nondegenerate. Our assumption that $\tau $ belongs to the $\operatorname{sk}_{k}(S_{\bullet })$ guarantees that $m \leq k$, and our assumption that $\tau $ does not belong to $\operatorname{sk}_{k-1}( S_{\bullet } )$ guarantees that $m \geq k$.
$\square$