$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.5.7.5. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. The following conditions are equivalent:
- $(1)$
The full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.
- $(2)$
There exists a small collection of morphisms $W$ of $\operatorname{\mathcal{C}}$ such that $\operatorname{\mathcal{C}}_0$ is the full subcategory spanned by the $W$-local objects of $\operatorname{\mathcal{C}}$.
Proof.
We will show that $(1) \Rightarrow (2)$ (the reverse implication is a restatement of Theorem 9.5.7.1). Assume that $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a left adjoint to the inclusion functor and let $\overline{W}$ be the collection of morphisms $f$ of $\operatorname{\mathcal{C}}$ such that $L(f)$ is an isomorphism in $\operatorname{\mathcal{C}}_0$. Then we can regard $\overline{W}$ as the collection of objects of a full subcategory $\mathcal{W} \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ which fits into a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \mathcal{W} \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{L \circ } \\ \operatorname{Isom}( \operatorname{\mathcal{C}}_0 ) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_0 ). } \]
Applying Corollary 9.5.4.6, we deduce that the $\infty $-category $\mathcal{W}$ is presentable and that the inclusion functor $\mathcal{W} \hookrightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ preserves small colimits. In particular, there exists a small subset $W \subseteq \overline{W}$ which generates $\mathcal{W}$ under small colimits. It follows from Remark 9.5.7.4 that an object of $\operatorname{\mathcal{C}}$ is $W$-local if and only if it is $\overline{W}$-local: that is, if and only if it belongs to $\operatorname{\mathcal{C}}_0$ (Corollary 6.2.3.10).
$\square$