Notation 6.2.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$. For every $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, we let $\mathrm{pr}(\sigma )$ denote the smallest nonnegative integer $p$ such that, for $p \leq q \leq n$, the morphism $\sigma (q) \rightarrow \sigma (n)$ belongs to $S$. We will refer to $\mathrm{pr}( \sigma )$ as the priority of $\sigma $. This definition has the following properties:
The simplex $\sigma $ has priority $0$ if and only if it belongs to the simplicial subset $\operatorname{\mathcal{C}}^{\mathrm{short}}$ of Notation 6.2.5.6 (see Remark 6.2.5.7).
For each $0 \leq i \leq n$, the face $\tau = d^{n}_ i(\sigma )$ satisfies $\mathrm{pr}( \tau ) \leq \mathrm{pr}(\sigma )$. The inequality is strict if $i < \mathrm{pr}(\sigma )$, and equality holds if $\mathrm{pr}(\sigma ) \leq i < n$.
For each $0 \leq i \leq n$, the degenerate simplex $\tau = s^{n}_ i(\sigma )$ satisfies
\[ \mathrm{pr}( \tau ) = \begin{cases} \mathrm{pr}( \sigma ) + 1 & \text{ if $0 \leq i < \mathrm{pr}( \sigma )$ } \\ \mathrm{pr}(\sigma ) & \text{ if $\mathrm{pr}(\sigma ) \leq i \leq n$} \end{cases} \]