Proposition 6.3.3.13 (02G3)

Proposition 6.3.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a localizing collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}'$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. Then:

$(1)$: The full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective (Definition 6.2.2.6).
$(2)$: The inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$.
$(3)$: A morphism $w$ of $\operatorname{\mathcal{C}}$ is contained in $W$ if and only if $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$.
$(4)$: The functor $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. Assertion $(1)$ is Example 6.2.3.10 and $(2)$ is a formal consequence (Proposition 6.2.2.23). Assertion $(3)$ follows from Proposition 6.2.3.12 and Remark 6.2.2.21, and the implication $(3) \Rightarrow (4)$ follows from Example 6.3.3.10. $\square$