# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 6.3.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is localizing, 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).

$(2)$

Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Then $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, in the sense of Definition 6.3.1.9.

$(3)$

A morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ belongs to $W$ if and only if $L(f)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}'$.

Proof. Let $X$ be an object of $\operatorname{\mathcal{C}}$. Our assumption that $W$ is localizing guarantees that there exists a morphism $w: X \rightarrow Y$ which belongs to $W$, where $Y$ is $W$-local. Note that, if $Z$ is any $W$-local object of $\operatorname{\mathcal{C}}$, then composition with the homotopy class $[w]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [w] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. It follows that $w$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, in the sense of Definition 6.2.2.1. This proves $(1)$.

It follows from $(1)$ that the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint (Proposition 6.2.2.7). Choose a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \iota \circ L$ which is the unit of an adjunction between $L$ and $\iota$. Then, for every object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _ X: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$ (Proposition 6.2.2.11). Since $W$ is localizing, we can also choose a morphism $w: X \rightarrow Y$ of $W$, where $Y$ belongs to $\operatorname{\mathcal{C}}'$. Arguing as above, we see that $w$ also exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. It follows from Remark 6.2.2.3 that we can realize $\eta _{X}$ as a composition of $w$ with an isomorphism $Y \rightarrow L(X)$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $W$ contains all isomorphisms and is closed under composition, it follows that $\eta _{X}$ also belongs to $W$.

We now prove $(3)$. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We then have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{\eta _ X} & Y \ar [d]^{\eta _ Y} \\ L(X) \ar [r]^-{ L(f) } & L(Y), }$

where $\eta _ X$ and $\eta _ Y$ belong to $W$. Since $W$ satisfies the two-out-of-three property, it follows that $f$ belongs to $W$ if and only if $L(f)$ belongs to $W$. Since $L(X)$ and $L(Y)$ are $W$-local objects of $\operatorname{\mathcal{C}}$, this is equivalent to the requirement that $L(f)$ is an isomorphism (Remark 6.3.3.4).

We now prove $(2)$ using the criterion of Proposition 6.3.1.13. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category and let $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those functors which carry each morphism of $W$ to an isomorphism of $\operatorname{\mathcal{E}}$ (Notation 6.3.1.1). It follows from $(2)$ that precomposition with the functor $L$ induces a map

$\theta : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ).$

We wish to show that $\theta$ is bijective. To prove injectivity, we observe that the construction $[F] \mapsto [F|_{\operatorname{\mathcal{C}}} ]$ determines a left inverse to $\theta$ (any functor $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{E}}$ is isomorphic to the restriction $(F' \circ L)|_{\operatorname{\mathcal{C}}'}$ via the natural transformation $\eta$). To prove surjectivity, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be any functor. Then $\eta$ induces a natural transformation $\eta ': F \rightarrow F|_{\operatorname{\mathcal{C}}'} \circ L$, which carries each object $X \in \operatorname{\mathcal{C}}$ to the morphism $F(\eta _ X): F(X) \rightarrow (F \circ L)(X)$. If $F$ carries each morphism of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$, then $\eta '$ is a natural isomorphism (Theorem 4.4.4.4). In particular, $F$ is isomorphic to $F|_{\operatorname{\mathcal{C}}'} \circ L$ so that the isomorphism class $[F]$ belongs to the image of $\theta$. $\square$