Construction 1.1.4.1. Let $S = S_{\bullet }$ be a simplicial set and let $k$ be an integer. For every integer $n$, we let $\operatorname{sk}_{k}(S)_{n}$ denote the subset of $S_{n}$ consisting of those $n$-simplices $\sigma : \Delta ^ n \rightarrow S$ which satisfy the following condition:

- $(\ast )$
In the category of simplicial sets, $\sigma $ admits a factorizaton

\[ \Delta ^{n} \rightarrow \Delta ^{m} \xrightarrow { \tau } S \]where $m \leq k$.

It follows immediately from the definitions that the collection of subsets $\{ \operatorname{sk}_{k}(S)_{n} \subseteq S_{n} \} _{n \geq 0}$ is stable under the face and degeneracy operators for the simplicial set $S_{\bullet }$, and therefore defines a simplicial subset $\operatorname{sk}_{k}(S) \subseteq S$. We will refer to $\operatorname{sk}_{k}(S)$ as the *$k$-skeleton* of $S$.