Definition 9.2.9.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A factorization system on $\operatorname{\mathcal{C}}$ is a pair $(S_{L}, S_{R})$, where $S_{L}$ and $S_{R}$ are collections of morphisms of $\operatorname{\mathcal{C}}$ which satisfy the following conditions:
- $(1)$
For every morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$, there exists a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{f_{R}} & \\ X \ar [ur]^{f_{L}} \ar [rr]^{f} & & Z } \]where $f_{L}$ belongs to $S_{L}$ and $f_{R}$ belongs to $S_{R}$.
- $(2)$
Every morphism of $S_{L}$ is left orthogonal to every morphism of $S_{R}$ (Definition 9.2.7.4).
- $(3)$
The collections $S_{L}$ and $S_{R}$ are closed under isomorphism (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).