Kerodon

Remark 9.5.6.3 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is presentable and that $F$ is a Bousfield localization functor (so that $\operatorname{\mathcal{D}}$ is also presentable). Then $G$ is a Bousfield localization functor if and only if the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is a Bousfield localization functor.

Remark 9.5.6.6. Let $\operatorname{\mathcal{D}}$ be a presentable $\infty $-category. We can always find Bousfield localization functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{C}}= \operatorname{Fun}( \operatorname{\mathcal{C}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}})$ arises as the cocompletion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$. By virtue of Example 9.5.6.4, this is equivalent to the problem of finding a dense functor $f: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{C}}_0$ is essentially small. If $\operatorname{\mathcal{D}}$ is $\kappa $-presentable, we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{D}}_{< \kappa }$ to be the full subcategory of $\operatorname{\mathcal{D}}$ spanned by the $\kappa $-compact objects (and $f$ to be the inclusion functor).

Remark 9.5.6.7 (Colimits in Bousfield Localizations). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Bousfield localization functor between presentable $\infty $-categories and let $K$ be a small simplicial set. Then a morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram if and only if it is isomorphic to $F \circ \overline{p}$, where $\overline{p}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. The “if” direction is immediate (since the functor $F$ preserves small colimits). For the converse, it suffices to show that $q = \overline{q}|_{K}$ is isomorphic to $F \circ p$ for some diagram $K \rightarrow \operatorname{\mathcal{C}}$ (we can then take $\overline{p}$ to be a colimit diagram extending $p$). For example, we can take $p = G \circ q$, where $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is right adjoint to $F$.

Remark 9.5.6.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Bousfield localization functor between presentable $\infty $-categories, and suppose we are given another functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then $G$ is cocontinuous if and only if $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is cocontinuous. This follows immediately from Remark 9.5.6.7.

Remark 9.5.6.10. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory. It follows from Corollary 7.1.4.33 that $\operatorname{\mathcal{C}}_0$ is automatically complete and cocomplete. Consequently, if $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$, then it is a presentable $\infty $-category.

Remark 9.5.6.11. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. If a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.5.6.9), then the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ which is a Bousfield localization functor (in the sense of Definition 9.5.6.1). Using Remark 6.3.3.12, we see that this construction determines a bijection

The inverse bijection carries a Bousfield localization functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to the essential image its right adjoint.

Remark 9.5.6.13 (Products). Let $\{ \operatorname{\mathcal{C}}(s) \} _{s \in S}$ be a collection of presentable $\infty $-categories indexed by a small set $S$, so that the product $\operatorname{\mathcal{C}}= \prod _{s \in S} \operatorname{\mathcal{C}}(s)$ is also presentable (Example 9.5.4.8). Suppose that, for each $s \in S$, we are given a Bousfield localization $\operatorname{\mathcal{C}}_0(s) \subseteq \operatorname{\mathcal{C}}(s)$. Then the product $\operatorname{\mathcal{C}}_0 = \prod _{s \in S} \operatorname{\mathcal{C}}_0(s)$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.

Remark 9.5.6.14. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\operatorname{\mathcal{C}}_0$ be a replete full subcategory. Then $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$ if and only if the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ which is accessible when viewed as a functor from $\operatorname{\mathcal{C}}$ to itself. See Proposition 9.4.7.9.