Notation 6.2.5.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We let $\ell (f)$ denote the smallest integer $n$ such that $f$ can be written as the composition of $n$ morphisms of $S$: that is, there exists an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ which carries the spine $\operatorname{Spine}[n]$ into $\operatorname{\mathcal{C}}^{\mathrm{short}}$, for which the composition
coincides with $f$. Note that condition $(4)$ of Definition 6.2.5.4 guarantees that $\ell (f) < \infty $. We will refer to $\ell (f)$ as the $S$-length of $f$. Note that $\ell (f) = 0$ if and only if $f$ is an identity morphism of $\operatorname{\mathcal{C}}$, and $\ell (f) \leq 1$ if and only if $f$ belongs to $S$.