Remark 1.1.0.14 (Simplicial Subsets). Let $S_{\bullet }$ be a simplicial set. Suppose that:
For every integer $n \geq 0$, we are given a subset $T_{n} \subseteq S_{n}$,
For every morphism $\alpha : [m] \rightarrow [n]$ in the simplex category $\operatorname{{\bf \Delta }}$, the associated map $S_{n} \rightarrow S_{m}$ carries $T_{n}$ into $T_{m}$.
Then we the construction $[n] \mapsto T_{n}$ determines another simplicial set $T_{\bullet }$. In this case, we will say that $T_{\bullet }$ is a simplicial subset of $S_{\bullet }$ and write $T_{\bullet } \subseteq S_{\bullet }$.