Kerodon

Example 9.5.6.4 (Density and Bousfield Localization). Let $\operatorname{\mathcal{D}}$ be a presentable $\infty $-category. Suppose we are given an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$ and a diagram $f: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$. Set $\operatorname{\mathcal{C}}= \operatorname{Fun}( \operatorname{\mathcal{C}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}})$, so that $f$ can be identified with a cocontinuous functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (see Theorem 8.4.0.3). In this case, $F$ is a Bousfield localization functor (in the sense of Definition 9.5.6.1) if and only if the functor $f$ is dense (in the sense of Definition 8.4.1.15). See Propositions 8.4.1.22 and 8.4.4.1.