Question 9.6.9.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_ L$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. When does there exist a collection of morphisms $S_{R}$ of $\operatorname{\mathcal{C}}$ such that $(S_ L, S_ R)$ is a factorization system?
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.6.9 Accessible Factorization Systems
Our goal in this section is to (partially) address the following:
We begin by recording some necessary conditions.
Definition 9.6.9.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $S$ is saturated if it contains all isomorphisms, is closed under composition, and is closed under small levelwise colimits (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
Example 9.6.9.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $T$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $S$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ which are left orthogonal to $T$. Then $S$ is saturated. This follows from Remark 9.6.6.9, Proposition 9.6.6.10, and Proposition 9.6.6.12.
Example 9.6.9.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $(S_ L, S_ R)$ be a factorization system on $\operatorname{\mathcal{C}}$. Then $S_{L}$ is saturated. This is a special case of Example 9.6.9.3, since a morphism of $\operatorname{\mathcal{C}}$ belongs to $S_{L}$ if and only if it is left orthogonal to $S_{R}$ (Proposition 9.6.8.8).
Remark 9.6.9.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: A \rightarrow B$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a relative codiagonal $\gamma _{A/B}: B \coprod _{A} B \rightarrow B$ (Variant 7.6.2.23). If $S$ is a saturated collection of morphisms which contains $f$, then it also contains $\gamma _{A/B}$. This follows from the observation that there is a (levelwise) pushout diagram in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$.
\[ \xymatrix@R =50pt@C=50pt{ f \ar [r] \ar [d] & \operatorname{id}_ B \ar [d] \\ \operatorname{id}_{B} \ar [r] & \gamma _{A/B} } \]
Remark 9.6.9.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Any intersection of saturated collections of morphisms of $\operatorname{\mathcal{C}}$ is also saturated. In particular, for any collection $W$ of morphisms of $\operatorname{\mathcal{C}}$, there is a smallest collection $\overline{W}$ which is saturated and contains $W$. We will refer to $\overline{W}$ as the saturated collection generated by $W$. We will say that a collection of morphisms is accessibly saturated if it has the form $\overline{W}$, where $W$ is a small collection of morphisms of $\operatorname{\mathcal{C}}$.
Remark 9.6.9.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. If $S$ is saturated (in the sense of Definition 9.6.9.2), then it is weakly saturated (in the sense of Definition 9.6.5.6). That is:
For any pushout diagram
in the $\infty $-category $\operatorname{\mathcal{C}}$, if $w$ belongs to $S$, then $w'$ also belongs to $S$. This follows from the observation that $w'$ is a pushout of $w$ and $\operatorname{id}_{C'}$ along $\operatorname{id}_{C}$.
The collection $S$ is closed under the formation of retracts (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$), since every retract can be written as a colimit indexed by the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{Idem})$ (Remark 8.5.3.9).
The collection $S$ is closed under small transfinite composition: see Proposition 9.6.1.10.
\[ \xymatrix@C =50pt@R=50pt{ C \ar [d]^{w} \ar [r] & C' \ar [d]^{w'} \\ D \ar [r] & D' } \]
Theorem 9.6.9.8 (Existence of Factorization Systems). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{C}}$ admits a factorization system $(S_ L, S_ R)$, where $S_ L$ is the saturated collection of morphisms generated by $W$ (Remark 9.6.9.6) and $S_{R}$ is the collection of morphisms which are right orthogonal to $W$.
Proof. For each morphism $w: A \rightarrow B$ of $\operatorname{\mathcal{C}}$, choose a relative codiagonal $\gamma _{w}: B \coprod _{A} B \rightarrow B$ (see Variant 7.6.2.23). Let $W^{+}$ be the smallest collection of morphisms which contains $W$ and is closed under the operation $w \mapsto \gamma _{w}$. It follows from Theorem 9.6.5.12 that $\operatorname{\mathcal{C}}$ admits a weak factorization system $(S_{L}, S_{R})$, where $S_{L}$ is the weakly saturated collection of morphisms generated by $W^{+}$ and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W^{+}$. Using Corollaries 9.6.6.15 and Corollary 9.6.6.26, we see that a morphism of $\operatorname{\mathcal{C}}$ belongs to $S_{R}$ if and only if it is right orthogonal to $W$. Applying Corollary 9.6.6.15 again, we see that $S_{R}$ is closed under the formation of relative diagonals, so that $(S_ L, S_ R)$ is a factorization system on $\operatorname{\mathcal{C}}$ (Corollary 9.6.8.14). In particular, $S_{L}$ is a saturated collection of morphisms of $\operatorname{\mathcal{C}}$ (Example 9.6.9.4). To complete the proof, it will suffice to show that if $S$ is another saturated collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $W$, then it contains $S_{L}$. This is clear, since $S$ is weakly saturated (Remark 9.6.9.7) and contains $W^{+}$ (Remark 9.6.9.5). $\square$
Remark 9.6.9.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, let $W_{\perp }$ denote the collection of morphisms of $\operatorname{\mathcal{C}}$ which are right orthogonal to every morphism of $W$, and let $\overline{W}$ be the collection of all morphisms which are left orthogonal to $W_{\perp }$. It follows from Example 9.6.9.3 that $\overline{W}$ is a saturated collection of morphisms which contains $W$. If $\operatorname{\mathcal{C}}$ is presentable and $W$ is small, then Theorem 9.6.9.8 guarantees that $\overline{W}$ is the smallest saturated collection of morphisms which contains $W$.
We now characterize the factorization systems which can be obtained from Theorem 9.6.9.8.
Proposition 9.6.9.10. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category and let $(S_ L, S_ R)$ be a factorization system on $\operatorname{\mathcal{C}}$. Let $\mathcal{L}$ and $\mathcal{R}$ denote the full subcategories of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the elements of $S_{L}$ and $S_{R}$, respectively. The following conditions are equivalent:
The $\infty $-category $\mathcal{L}$ is accessibly embedded in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (see Definition 9.4.7.8).
The $\infty $-category $\mathcal{R}$ is accessibly embedded in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$.
Let $D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ be the trivial Kan fibration of Theorem 9.6.8.15 and let $Q: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ be a section of $D$. Then $Q$ is accessible (when regarded as a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$).
Proof. We will prove that $(1) \Leftrightarrow (3)$; the equivalence $(2) \Leftrightarrow (3)$ follows by a similar argument. Assume first that $\mathcal{L}$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. As in Notation 9.6.7.1, we let $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ spanned by those $2$-simplices
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr] & \\ X \ar [ur]^{f} \ar [rr] & & Z } \]
where $f$ belongs to $S_{L}$. Then $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ is the inverse image of $\mathcal{L}$ under the restriction functor $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ and is therefore an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ (Corollary 9.4.8.12). In particular, $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ is an accessible $\infty $-category, and $Q$ is accessible when regarded a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ if and only if it is accessible as a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$. Let $D_{+}: \operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ be the functor given on objects by the formula $D_{+}(\sigma ) = d^{2}_{1}(\sigma )$. By construction, the composition $D_{+} \circ Q$ is the identity functor on $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. It follows from Proposition 9.6.7.4 (and the criterion of Corollary 6.2.6.5) that the identity transformation $\operatorname{id}\rightarrow D_{+} \circ Q$ is the unit of an adjunction. In particular, the functor $Q$ is a left adjoint (when regarded as a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$) and is therefore accessible (Corollary 9.4.7.18).
We now prove the converse. Assume that $Q$ is an accessible functor. Then the composition of $Q$ with the restriction map
\[ T: \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \{ 1 < 2\} ), \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \]
is also an accessible functor. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms (Example 4.4.1.14). It follows from the proof of Corollary 9.6.8.10 that $\mathcal{L}$ is the inverse image of $\operatorname{Isom}( \operatorname{\mathcal{C}})$ under $T \circ Q$. Since $\operatorname{Isom}(\operatorname{\mathcal{C}})$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (Remark 9.4.8.3), it follows that $\mathcal{L}$ is also an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (Corollary 9.4.8.12). $\square$
Definition 9.6.9.11. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category. We will say that a factorization system $(S_ L, S_ R)$ on $\operatorname{\mathcal{C}}$ is accessible if it satisfies the equivalent conditions of Proposition 9.6.9.10.
To connect Definition 9.6.9.11 with Theorem 9.6.9.8, it will be convenient to give another characterization of accessible factorization systems:
Proposition 9.6.9.12. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category and let $(S_ L, S_ R)$ be a factorization system on $\operatorname{\mathcal{C}}$. Then $(S_ L, S_ R)$ is an accessible factorization system if and only if it satisfies the following condition:
There exists a small collection $W$ of morphisms of $\operatorname{\mathcal{C}}$ such that a morphism $g$ of $\operatorname{\mathcal{C}}$ belongs to $S_{R}$ if and only if it is right orthogonal to $W$.
Proof. Let $\mathcal{L}, \mathcal{R} \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ be as in Proposition 9.6.9.10. If the factorization system $(S_ L, S_ R)$ is accessible, then $\mathcal{L}$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Let $W \subseteq S_{L}$ be the collection of morphisms which are $\kappa $-compact when viewed as objects of $\mathcal{L}$. Then every morphism belonging to $S_{L}$ can be realized as a small $\kappa $-filtered colimit of morphisms belonging to $W$. Applying Proposition 9.6.6.12. we see that a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is right orthogonal to $W$ if and only if it is right orthogonal to $S_{L}$: that is, if and only if it belongs to $S_{R}$ (Proposition 9.6.8.13).
We now prove the converse. For every collection $W$ of morphisms of $\operatorname{\mathcal{C}}$, let $\mathcal{R}_{W}$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by those morphisms which are right orthogonal to $W$. We will show that if $W$ is small, then $\mathcal{R}_{W}$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. In particular, if $W$ satisfies condition $(\ast )$, then $\mathcal{R} = \mathcal{R}_{W}$ is an accessible embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, so that $(S_ L, S_ R)$ is an accessible factorization system.
Using Corollary 9.4.8.13, we can reduce to the case where $W = \{ w\} $ consists of a single morphism $w: C \rightarrow D$ of $\operatorname{\mathcal{C}}$. Let $T: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{S}})$ denote the functor which carries each morphism $g: X \rightarrow Y$ to the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( D, X ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, X ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( D, Y) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y ). } \]
Note that the functor $T$ is accessible (see Example 9.4.7.16 and Remark 9.4.8.2), and that a morphism $g$ of $\operatorname{\mathcal{C}}$ is right orthogonal to $W$ if and only if $T(g)$ is a pullback diagram in $\operatorname{\mathcal{C}}$. The desired result now follows from Corollary 9.4.8.12. $\square$
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:
The collection $\overline{W}$ is accessibly saturated: that is, it is the smallest saturated collection which contains some small subset $W \subseteq \overline{W}$.
There exists an accessible factorization system $(S_ L, S_ R)$ on $\operatorname{\mathcal{C}}$ satisfying $S_ L = \overline{W}$.
The $\infty $-category $\mathcal{W}$ is accessible.
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 $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. $\square$
Corollary 9.6.9.14 (Classification of Accessible Factorization Systems). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Then the construction $(S_ L, S_ R) \mapsto S_{L}$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Accessible factorization systems $(S_ L,S_ R)$ on $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Accessibly saturated collections of morphisms $\overline{W}$ of $\operatorname{\mathcal{C}}$} \} . } \]
Variant 9.6.9.15 (Accessible Weak Factorization Systems). Corollary 9.6.9.14 has a counterpart for weak factorization systems. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Let us say that a weak factorization system $(S_ L, S_ R)$ on $\operatorname{\mathcal{C}}$ is accessible if there exists a small collection of morphisms $W$ of $\operatorname{\mathcal{C}}$ such that $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$, and let us say that a collection of morphisms $\overline{W}$ of $\operatorname{\mathcal{C}}$ is accessibly weakly saturated if it is the weakly saturated collection of morphisms generated by some small subset $W \subseteq \overline{W}$. It follows from Theorem 9.6.5.12 that the construction $(S_ L, S_ R) \mapsto S_{L}$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Accessible weak factorization systems $(S_ L,S_ R)$ on $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Accessibly weakly saturated collections of morphisms $\overline{W}$ of $\operatorname{\mathcal{C}}$} \} . } \]
Warning 9.6.9.16. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $(S_{L},S_ R)$ be an accessible weak factorization system on $\operatorname{\mathcal{C}}$ (Variant 9.6.9.15). Then $S_{R}$ spans an accessible subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. However, $S_{L}$ does not necessarily span an accessible subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$.
Remark 9.6.9.17. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\overline{W}$ be a saturated collection of morphisms of $\operatorname{\mathcal{C}}$. Then $\overline{W}$ is accessibly saturated (in the sense of Remark 9.6.9.6) if and only if it is accessibly weakly saturated (in the sense of Variant 9.6.9.15). In other words, if there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a saturated collection of morphisms, then there is another small subset $W^{+} \subseteq \overline{W}$ which generates $\overline{W}$ as a weakly saturated class of morphisms. For example, we can take $W^{+}$ to be the collection of morphisms appearing in the proof of Theorem 9.6.9.8.
We conclude this section with another variant of Corollary 9.6.9.14
Definition 9.6.9.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{W}$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $\overline{W}$ is strongly saturated if it contains all isomorphisms, has the two-out-of-three property, and is closed under small levelwise colimits (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
Remark 9.6.9.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{W}$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. If $\overline{W}$ is strongly saturated (in the sense of Definition 9.6.9.18), then it is saturated (in the sense of Definition 9.6.9.2).
Example 9.6.9.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocontinuous functor between cocomplete $\infty $-categories, and let $\overline{W}$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $\overline{W}$ is strongly saturated.
Example 9.6.9.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\overline{W}$ be the collection of $\operatorname{\mathcal{C}}_0$-local equivalences in $\operatorname{\mathcal{C}}$ (Definition 6.2.2.1). Then $\overline{W}$ is strongly saturated.
Remark 9.6.9.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Any intersection of strongly saturated collections of morphisms of $\operatorname{\mathcal{C}}$ is also strongly saturated. In particular, for any collection $W$ of morphisms of $\operatorname{\mathcal{C}}$, there is a smallest collection $\overline{W}$ which is strongly saturated and contains $W$. We will refer to $\overline{W}$ as the strongly saturated collection generated by $W$. We will say that a collection of morphisms is accessibly strongly saturated if it has the form $\overline{W}$, where $W$ is a small collection of morphisms of $\operatorname{\mathcal{C}}$.
Proposition 9.6.9.23. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\overline{W}$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The collection $\overline{W}$ is accessibly strongly saturated.
There exists a Bousfield localization $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ such that $\overline{W}$ is the collection of $\operatorname{\mathcal{C}}_0$-local equivalences (see Definition 6.2.2.1).
The collection $\overline{W}$ is strongly saturated and generated under small colimits by a small subset $W \subseteq \overline{W}$.
Proof. We first show that $(1) \Rightarrow (2)$. Assume that $\overline{W}$ is accessibly strongly saturated: that is, it is generated (as a strongly saturated collection of morphisms) by a small subset $W \subseteq \overline{W}$. As in the proof of Theorem 9.6.9.8, we can arrange that $W$ is closed under the formation of relative codiagonals. It follows that an object of $\operatorname{\mathcal{C}}$ is $W$-local if and only if it is weakly $W$-local (Proposition 9.6.2.14).
Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. It follows from Theorem 9.6.9.8 that $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. By construction, every element of $W$ is a $\operatorname{\mathcal{C}}_0$-local equivalence. Since the collection of $\operatorname{\mathcal{C}}_0$-local equivalences is strongly saturated (Example 9.6.9.21), it follows that every element of $\overline{W}$ is a $\operatorname{\mathcal{C}}_0$-local equivalence. For every object $X \in \operatorname{\mathcal{C}}$, Theorem 9.6.3.3 guarantees that there is a morphism $f: X \rightarrow Y$, where $Y \in \operatorname{\mathcal{C}}_0$ and $f$ is a (small) transfinite pushout of morphisms of $W$, and therefore belongs to $\overline{W}$. Since $\overline{W}$ has the two-out-of-three property, Lemma 6.2.4.1 guarantees that $\overline{W}$ is the collection of all $\operatorname{\mathcal{C}}_0$-local equivalences in $\operatorname{\mathcal{C}}$.
We now complete the proof by showing that $(2) \Rightarrow (3)$ (the implication $(3) \Rightarrow (1)$ is trivial). Suppose that $\overline{W}$ is the collection of $\operatorname{\mathcal{C}}_0$-local equivalences, where $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. Invoking Example 9.6.9.21, we see that $\overline{W}$ is strongly saturated. Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a left adjoint to the inclusion functor and let $\mathcal{W}$ be the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the morphisms of $\operatorname{\mathcal{C}}$ which belong $\overline{W}$. Then we have a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{W}}\ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{L \circ } \\ \operatorname{Isom}( \operatorname{\mathcal{C}}_0 ) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_0 ). } \]
Applying Corollary 9.5.4.6, we conclude that the $\infty $-category $\operatorname{\mathcal{W}}$ is presentable (and closed under colimits in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}$), so that $\overline{W}$ is generated by a small subset under (small) colimits. $\square$
Remark 9.6.9.24. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\overline{W}$ be a strongly saturated collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The collection $\overline{W}$ is accessibly strongly saturated: that is, there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a strongly saturated collection of morphisms.
The collection $\overline{W}$ is accessibly saturated: that is, there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a saturated collection of morphisms.
The collection $\overline{W}$ is acccessibly weakly saturated: that is, there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a weakly saturated collection of morphisms.
The implications $(c) \Rightarrow (b) \Rightarrow (a)$ are trivial, the implication $(b) \Rightarrow (c)$ is a special case of Remark 9.6.9.17, and the implication $(a) \Rightarrow (b)$ follows from Proposition 9.6.9.23.
Corollary 9.6.9.25 (Classification of Bousfield Localizations). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Then there is a bijection which carries a Bousfield localization $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ to the collection of $\operatorname{\mathcal{C}}_0$-local equivalences in $\operatorname{\mathcal{C}}$.
\[ \xymatrix@R =50pt@C=50pt{\{ \textnormal{Bousfield localizations $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$} \} \ar [d]^{\sim }\\ \{ \textnormal{Accessibly strongly saturated collections of morphisms $\overline{W}$} \} } \]
Proof. Combine Corollary 6.2.3.15 with Proposition 9.6.9.23. $\square$
As an application of the preceding ideas, we record a stronger version of Proposition 9.5.7.2:
Corollary 9.6.9.26. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category, let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a left adjoint to the inclusion functor. For every cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $L$ determines a fully faithful functor whose essential image is spanned by the cocontinuous functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$.
\[ L^{\ast }: \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}), \]
Proof. Let $\overline{W}$ be the collection of all $\operatorname{\mathcal{C}}_0$-local equivalences in $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}_0$ is a reflective full subcategory of $\operatorname{\mathcal{C}}$, the functor $L$ exhibits $\operatorname{\mathcal{C}}_0$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $\overline{W}$ (Example 6.3.3.11). It follows that, for any $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $L$ induces a fully faithful functor $\operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ whose essential image is spanned by the functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry every element of $\overline{W}$ to an isomorphism in $\operatorname{\mathcal{D}}$. Note that, if $\operatorname{\mathcal{D}}$ is cocomplete, then a functor $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is cocontinuous if and only if its composition with $L$ is cocontinuous (Remark 9.5.6.7). It will therefore suffice to show that if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cocontinuous functor which carries every element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$, then it carries every element of $\overline{W}$ to an isomorphism in $\operatorname{\mathcal{D}}$. This follows from Example 9.6.9.20, since $\overline{W}$ is generated by $W$ as a strongly saturated collection of morphisms (Proposition 9.6.9.23). $\square$