Definition 9.2.3.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that $W$ is weakly saturated if it satisfies the following conditions:
- $(1)$
The collection $W$ is closed under pushouts: that is, for every pushout diagram
\[ \xymatrix@C =50pt@R=50pt{ X \ar [d]^{w} \ar [r] & X' \ar [d]^{w'} \\ Y \ar [r] & Y' } \]in the $\infty $-category $\operatorname{\mathcal{C}}$, if $w$ belongs to $W$, then $w'$ also belongs to $W$.
- $(2)$
The collection $W$ is closed under the formation of retracts (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).
- $(3)$
The collection $W$ is closed under transfinite composition (Definition 9.2.2.1).