Kerodon

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

Construction 1.1.3.5. Let $S_{\bullet }$ be a simplicial set, let $k \geq -1$ be an integer, and let $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$ be an $n$-simplex of $S_{\bullet }$. The proof of Proposition 1.1.3.4 shows that the following conditions are equivalent:

$(a)$

Let $\Delta ^{n} \xrightarrow {\alpha } \Delta ^{m} \xrightarrow {\tau } S_{\bullet }$ be the factorization of Proposition 1.1.3.4 (so that $\alpha $ induces a surjection $[n] \rightarrow [m]$, the map $\tau $ is nondegenerate, and $\sigma = \tau \circ \alpha $). Then $m \leq k$.

$(b)$

There exists a factorization $\Delta ^{n} \rightarrow \Delta ^{m'} \rightarrow S_{\bullet }$ of $\sigma $ for which $m' \leq k$.

For each $n \geq 0$, we let $\operatorname{sk}_{k}( S_{n} )$ denote the subset of $S_{n}$ consisting of those $n$-simplices which satisfy conditions $(a)$ and $(b)$. From characterization $(b)$, we see that the collection of subsets $\{ \operatorname{sk}_ k(S_ n) \subseteq S_ n \} _{n \geq 0}$ is stable under the face and degeneracy operators of $S_{\bullet }$, and therefore determine a simplicial subset of $S_{\bullet }$ (Remark 1.1.2.4). We will denote this simplicial subset by $\operatorname{sk}_{k}( S_{\bullet } )$ and refer to it as the $k$-skeleton of $S_{\bullet }$.