$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. Then a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}^{\mathrm{short}}$ if and only if, for every integer $0 \leq i < n$, the morphism $\sigma (i) \rightarrow \sigma (n)$ belongs to $S$. Condition $(\ast )$ of Notation can be deduced from this a priori weaker assumption by applying assumption $(2)$ of Definition to the diagrams

\[ \xymatrix@R =50pt@C=50pt{ & \sigma (j) \ar [dr] & \\ \sigma (i) \ar [rr] \ar [ur] & & \sigma (n) } \]

for $i \leq j \leq n$.