Proposition 9.1.1.18. 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 6.2.2.1).
- $(2)$
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.
Let $X$ be an object of $\operatorname{\mathcal{C}}$. Our assumption that $W$ is localizing guarantees that there exists a morphism $w_ X: X \rightarrow X'$ which belongs to $W$, where $X' \in \operatorname{\mathcal{C}}'$. By definition, every object $C \in \operatorname{\mathcal{C}}'$ is $W$-local, so composition with $w_ X$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X',C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, C)$. It follows that $w_ X$ exhibits $X'$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, in the sense of Definition 6.2.2.1. Assertion $(1)$ follows by allowing the object $X$ to vary. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.2.15, and the implication $(3) \Rightarrow (4)$ from Example 6.3.3.7.
It remains to prove $(3)$. Choose a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor (see Definition 6.2.2.12). For each object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _{X}: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, and can therefore be obtained by composing $w_{X}$ with an isomorphism $X' \xrightarrow {\sim } L(X)$. Since $W$ contains all isomorphisms and is closed under composition, it follows that $\eta _{X}$ belongs to $W$.
For every morphism $w: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the natural transformation $\eta $ determines a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{w} \ar [d]^{\eta _ X} & Y \ar [d]^{\eta _ Y} \\ L(X) \ar [r]^-{ L(w) } & L(Y) } \]
where $\eta _ X$ and $\eta _ Y$ belong to $W$. Using the two-out-of-three property, we see that $w$ is contained in $W$ if and only if $L(w)$ is contained in $W$. Since $L(X)$ and $L(Y)$ are $W$-local, this is equivalent to the requirement that $L(w)$ is an isomorphism (Remark 9.1.1.5).
$\square$