Definition 6.2.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A class of short morphisms for $\operatorname{\mathcal{C}}$ is a collection $S$ of morphisms of $\operatorname{\mathcal{C}}$ with the following properties:
- $(1)$
Every identity morphism of $\operatorname{\mathcal{C}}$ belongs to $S$.
- $(2)$
For every $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{f'} & \\ X \ar [ur]^{f''} \ar [rr]^{f} & & Z } \]of the $\infty $-category $\operatorname{\mathcal{C}}$, if $f$ and $f'$ belong to $S$, then $f''$ also belongs to $S$.
- $(3)$
Every morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ admits an $S$-optimal factorization (Definition 6.2.5.1).
- $(4)$
Every morphism of $\operatorname{\mathcal{C}}$ can be obtained as a composition of morphisms which belong to $S$.