Proposition 9.6.9.13 (06XE)

Proposition 9.6.9.13. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\overline{W}$ be a saturated collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\mathcal{W}$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the elements of $\overline{W}$. The following conditions are equivalent:

$(1)$: The collection $\overline{W}$ is accessibly saturated: that is, it is the smallest saturated collection which contains some small subset $W \subseteq \overline{W}$.
$(2)$: There exists an accessible factorization system $(S_ L, S_ R)$ on $\operatorname{\mathcal{C}}$ satisfying $S_ L = \overline{W}$.
$(3)$: The $\infty $-category $\mathcal{W}$ is accessible.
$(4)$: There exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ under small colimits.

Proof. The implication $(1) \Rightarrow (2)$ follows from Theorem 9.6.9.8 and Proposition 9.6.9.12, the implications $(2) \Rightarrow (3)$ and $(4) \Rightarrow (1)$ are trivial, and the implication $(3) \Rightarrow (4)$ follows from the observation that $\mathcal{W}$ is closed under small colimits in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. $\square$