Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 5.3.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Suppose that there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(a)$

The functor $F$ carries each element of $W$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

$(b)$

The composition $F \circ G$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$).

$(c)$

There exists a natural transformation $u: \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, the morphism $u_{C}: C \rightarrow (G \circ F)(C)$ belongs to $W$.

Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. It follows from $(a)$ that, for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a map

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

By virtue of Proposition 5.3.1.12, it will suffice to show that $\theta _{\operatorname{\mathcal{E}}}$ is bijective for every $\infty $-category $\operatorname{\mathcal{E}}$. Let $\rho : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1} ], \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } )$ be the map given by precomposition with $G$. It follows from $(b)$ that the composition $\rho \circ \theta _{\operatorname{\mathcal{E}}}$ is the identity. We will complete the proof by showing that $\theta _{\operatorname{\mathcal{E}}} \circ \rho $ is also the identity. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor which carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$. By virtue of Theorem 4.4.4.4, $H$ carries the natural transformation $u$ to an isomorphism $H = H \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow H \circ G \circ F$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, so that the isomorphism class of $H$ is fixed by the composition $\theta _{\operatorname{\mathcal{E}}} \circ \rho $. $\square$