Theorem 9.5.7.1. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category, let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. Then $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.5.7 Existence of Bousfield Localizations
The following result provides an abundant supply of examples of Bousfield localizations.
Proof. By virtue of Corollary 9.5.6.21, we may assume that $W = \{ w\} $ consists of a single morphism $w: C \rightarrow D$ of $\operatorname{\mathcal{C}}$. Let $h^ C, h^{D}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ denote the functors corepresented by $C$ and $D$, respectively, so that $w$ induces a natural transformation $h^{w}: h^{D} \rightarrow h^{C}$. By definition, an object $X \in \operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}_0$ if and only if $h^{w}$ carries $X$ to an isomorphism in $\operatorname{\mathcal{S}}$. Since the functors $h^{C}$ and $h^{D}$ are continuous and accessible (Proposition 7.4.1.22 and Example 9.4.7.16), Corollary 9.5.6.20 guarantees that $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. $\square$
The Bousfield localizations of Theorem 9.5.7.1 can be characterized by a universal mapping property.
Proposition 9.5.7.2. 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 presentable $\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 $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ be the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ spanned by the continuous accessible functors, and define $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}_0 )$ similarly. Using Theorem 9.5.2.1 and Corollary 8.3.4.10, we can identify $L^{\ast }$ with the opposite of the inclusion functor $\operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}_0 ) \hookrightarrow \operatorname{RFun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$. It follows immediately that $L^{\ast }$ is fully faithful. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocontinuous functor, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a right adjoint of $F$. Then $F$ belongs to the essential image of $L^{\ast }$ if and only if the functor $G$ carries each object $X \in \operatorname{\mathcal{D}}$ to a $W$-local object of $\operatorname{\mathcal{C}}$: that is, for every morphism $w: C \rightarrow D$ of $W$, the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( D, G(X) ) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(X) )$ is a homotopy equivalence of Kan complexes. This is equivalent to the condition that composition with $[F(w)]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(D), X ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), X)$, which is satisfied for every $X \in \operatorname{\mathcal{D}}$ if and only if $F(w)$ is an isomorphism. $\square$
Remark 9.5.7.3. In the formulation of Proposition 9.5.7.2, it is not necessary to assume that $\operatorname{\mathcal{D}}$ is presentable: the conclusion holds more generally when $\operatorname{\mathcal{D}}$ is cocomplete. However, this requires a different proof. See Corollary 9.6.9.26.
We now show every Bousfield localization can be obtained from Theorem 9.5.7.1.
Remark 9.5.7.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ which is the levelwise colimit of a diagram If an object $C \in \operatorname{\mathcal{C}}$ is $f_{v}$-local for each vertex $v \in K$, then it is also $f$-local. This follows from Propositions 7.4.1.22 and 7.1.3.18.
\[ K \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \quad \quad v \mapsto f_{v}. \]
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:
The full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.
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). $\square$
Corollary 9.5.7.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Bousfield localization functor. Then $F$ is an epimorphism in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ (see Variant 4.7.5.3).
Proof. By virtue of Remark 9.5.6.11 and Proposition 9.5.7.5, we may assume without loss of generality that there exists a small collection $W$ of morphisms of $\operatorname{\mathcal{C}}$ such that $\operatorname{\mathcal{D}}$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects and $F$ is left adjoint to the inclusion functor $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$. In this case, the result follows from the universal property of Proposition 9.5.7.2. $\square$
Warning 9.5.7.7. The converse of Corollary 9.5.7.6 is false in general. For example, if $X$ is an acyclic Kan complex which is not contractible, then the colimit functor $\varinjlim : \operatorname{Fun}(X, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ is an epimorphism in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ which is not a Bousfield localization functor (compare with Remark 6.3.3.2).
As an application of Proposition 9.5.7.5, we obtain a structure theorem for presentable $\infty $-categories:
Corollary 9.5.7.8. An $\infty $-category $\operatorname{\mathcal{C}}$ is presentable if and only if there exists a small $\infty $-category $\operatorname{\mathcal{K}}$, a small collection of morphisms $W$ of the cocompletion $\widehat{\operatorname{\mathcal{K}}} = \operatorname{Fun}( \operatorname{\mathcal{K}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, and an equivalence of $\operatorname{\mathcal{C}}$ with the full subcategory of $\widehat{\operatorname{\mathcal{K}}}$ spanned by the $W$-local objects.
Proof. Using Remark 9.5.6.6, we see that there exists a small $\infty $-category $\operatorname{\mathcal{K}}$ such that $\operatorname{\mathcal{C}}$ is a Bousfield localization of $\widehat{\operatorname{\mathcal{K}}}$. The desired result now follows from the classification of Bousfield localizations supplied by Proposition 9.5.7.5. $\square$
Remark 9.5.7.9 (Presentations of Cocomplete $\infty $-Categories). In the situation of Corollary 9.5.7.8, we have a Bousfield localization functor $F: \widehat{\operatorname{\mathcal{K}}} \rightarrow \operatorname{\mathcal{C}}$, which restricts to a dense functor $f: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ (Example 9.5.6.4). For every presentable $\infty $-category $\operatorname{\mathcal{D}}$, Theorem 8.4.0.3 and Proposition 9.5.7.2 imply that the functor is fully faithful, and that its essential image consists of those functors $T: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ for which the cocontinuous extension $\widehat{T}: \widehat{\operatorname{\mathcal{K}}} \rightarrow \operatorname{\mathcal{D}}$ carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$. We will see later that this holds more generally when the $\infty $-category $\operatorname{\mathcal{D}}$ is cocomplete (Corollary 9.6.9.26). Stated more informally, an $\infty $-category $\operatorname{\mathcal{C}}$ is presentable if and only if it can be obtained from a small $\infty $-category $\operatorname{\mathcal{K}}$ by freely adjoining small colimits (to obtain the $\infty $-category $\widehat{\operatorname{\mathcal{K}}}$) and then forcing a small collection of morphisms $W$ to become isomorphisms (while remaining in the setting of cocomplete $\infty $-categories).
\[ \operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {\circ f} \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) \]
Proposition 9.5.7.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be presentable $\infty $-categories and let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a cocontinuous functor. Then $H$ factors (up to isomorphism) as a composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{E}}$ where $\operatorname{\mathcal{D}}$ is a presentable $\infty $-category, $F$ is a Bousfield localization functor, and $G$ is conservative and cocontinuous.
We will see later that the factorization of Proposition 9.5.7.10 is essentially unique (see Example 9.6.8.7).
Proof of Proposition 9.5.7.10. Let $\overline{W}$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $H(w)$ is an isomorphism in $\operatorname{\mathcal{E}}$. As in the proof of Proposition 9.5.7.5, we see that $\overline{W}$ is the collection of objects of a presentable full subcategory $\operatorname{\mathcal{W}}\subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, and is therefore generated under small colimits by a small subset $W \subseteq \overline{W}$. Let $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. It follows from Theorem 9.5.7.1 that $\operatorname{\mathcal{D}}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$, so the inclusion functor $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Applying Proposition 9.5.7.2, we see that there is an essentially unique cocontinuous functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ such that $G \circ F$ is isomorphic to $H$. Concretely, we can identify $G$ with the restriction $H|_{\operatorname{\mathcal{D}}}$. To complete the proof, it will suffice to show that $G$ is conservative. In other words, we wish to show that if $w$ is a morphism of $\operatorname{\mathcal{D}}$ which belongs to $\overline{W}$, then $w$ is an isomorphism. This is a special case of Remark 6.2.3.8, since every object of $\operatorname{\mathcal{D}}$ is $\overline{W}$-local. $\square$