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Remark Let $S_{\bullet }$ be a simplicial set. Suppose that, for every integer $n \geq 0$, we are given a subset $T_{n} \subseteq S_{n}$, and that the face and degeneracy maps

\[ d_{i}: S_{n} \rightarrow S_{n-1} \quad \quad s_{i}: S_{n} \rightarrow S_{n+1} \]

carry $T_{n}$ into $T_{n-1}$ and $T_{n+1}$, respectively. Then the collection $\{ T_{n} \} _{n \geq 0}$ inherits the structure of a simplicial set $T_{\bullet }$. In this case, we will say that $T_{\bullet }$ is a simplicial subset of $S_{\bullet }$ and write $T_{\bullet } \subseteq S_{\bullet }$.