Notation 3.3.1.4. Let $X_{\bullet }$ be a simplicial set. For each nonnegative integer $n$, we let $X_{n}^{\mathrm{nd}} \subseteq X_{n}$ denote the collection of nondegenerate $n$-simplices of $X_{\bullet }$. If $X_{\bullet }$ is braced (Definition 3.3.1.1), then the face operators $\{ d^{n}_ i: X_{n} \rightarrow X_{n-1} \} _{0 \leq i \leq n}$ carry $X_{n}^{\mathrm{nd}}$ into $X_{n-1}^{\mathrm{nd}}$. In this case, the construction $[n] \mapsto X_{n}^{\mathrm{nd}}$ determines a semisimplicial set, which we will denote by $X_{\bullet }^{\mathrm{nd}}$.
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