Definition 9.2.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A weak 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 $h: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$, there exists a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$.
- $(2)$
Every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y } \]in $\operatorname{\mathcal{C}}$ admits a solution, provided that $f \in S_{L}$ and $g \in S_{R}$.
- $(3)$
The collections $S_{L}$ and $S_{R}$ are closed under retracts (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).