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Proposition 4.7.2.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$ such that, for each $w \in W$, the image $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $F$ factors as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{C}}[W^{-1}] \xrightarrow {H} \operatorname{\mathcal{D}}, \]

where $G$ exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ and $H$ is an inner fibration (so that $\operatorname{\mathcal{C}}[W^{-1}]$ is also an $\infty $-category). Moreover, this factorization can be chosen to depend functorially on the diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and the collection of edges $W$, in such a way that the construction $(F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}, W) \mapsto \operatorname{\mathcal{C}}[W^{-1}]$ commutes with filtered colimits.

Proof. For each element $w \in W$, the image $F(w)$ can be regarded as a morphism from $\Delta ^1$ to the core $\operatorname{\mathcal{D}}^{\simeq }$. By virtue of Proposition 3.1.7.1, we can (functorially) choose a factorization of this morphism as a composition

\[ \Delta ^1 \xrightarrow { i_{w} } Q_{w} \xrightarrow { q_{w} } \operatorname{\mathcal{D}}^{\simeq }, \]

where $i_{w}$ is anodyne and $q_{w}$ is a Kan fibration. Since $\operatorname{\mathcal{D}}^{\simeq }$ is a Kan complex, $Q_{w}$ is also a Kan complex, which is contractible by virtue of the fact that $i_{w}$ is anodyne. Form a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \coprod _{w \in W} \Delta ^1 \ar [r] \ar [d]^{\coprod _{w \in W} i_ w} & \operatorname{\mathcal{C}}\ar [d]^{i} \\ \coprod _{w \in W} Q_ w \ar [r] & \operatorname{\mathcal{C}}'. } \]

We first claim that $i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. Note that if $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism of simplicial sets which factors through $\operatorname{\mathcal{C}}'$, then for each $w \in W$ the morphism $G(w)$ belongs to the image of a functor $Q_ w \rightarrow \operatorname{\mathcal{E}}$, and is therefore an isomorphism in $\operatorname{\mathcal{E}}$. It follows that composition with $i$ induces a functor $\theta : \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$, and we wish to show that $\theta $ is an equivalence of $\infty $-categories. This follows by inspecting the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d] \\ \prod _{w \in W} \operatorname{Fun}(Q_ w,\operatorname{\mathcal{E}}) \ar [r]^-{\theta '} & \prod _{w \in W} \operatorname{Isom}(\operatorname{\mathcal{E}}) \ar [r] & \prod _{w \in W} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}}). } \]

The outer rectangle is a pullback square by the definition of $\operatorname{\mathcal{C}}'$, and the right square is a pullback by the definition of $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$. It follows that the left square is also a pullback. Lemma 4.7.2.4 implies that $\theta '$ is a trivial Kan fibration, so that $\theta $ is also a trivial Kan fibration (hence an equivalence of $\infty $-categories by Proposition 4.5.2.9).

Note that the morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and the collection of morphisms $\{ q_{w}: Q_ w \rightarrow \operatorname{\mathcal{D}}^{\simeq } \subseteq \operatorname{\mathcal{D}}\} _{w \in W}$ can be amalgamated to a single morphism of simplicial sets $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$. Applying Proposition 4.1.3.2, we can (functorially) factor $F'$ as a composition $\operatorname{\mathcal{C}}' \xrightarrow {G'} \operatorname{\mathcal{C}}[W^{-1}] \xrightarrow {H} \operatorname{\mathcal{D}}$, where $G'$ is inner anodyne and $H$ is an inner fibration. We conclude by observing that the composite map $G = (G' \circ i): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, by virtue of Remark 4.7.1.17. $\square$