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Notation Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. We let $\operatorname{\mathcal{C}}^{\mathrm{short}} \subseteq \operatorname{\mathcal{C}}$ denote the simplicial subset consisting of those simplices $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ having the following property:

$(\ast )$

For every pair of integers $0 \leq i \leq j \leq n$, the induced morphism $\sigma (i) \rightarrow \sigma (j)$ belongs to $S$.

Note that, since $S$ contains all identity morphisms of $\operatorname{\mathcal{C}}$, condition $(\ast )$ is automatically satisfied in the case $i = j$. In particular, every vertex of $\operatorname{\mathcal{C}}$ is contained in $\operatorname{\mathcal{C}}^{\mathrm{short}}$, and an edge of $\operatorname{\mathcal{C}}$ is contained in $\operatorname{\mathcal{C}}^{\mathrm{short}}$ if and only if belongs to $S$.