Kerodon

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

Proposition 5.3.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which is given as the colimit (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$) of a filtered diagram of morphisms $\{ F_{\alpha }: \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{D}}_{\alpha } \} $. Assume that:

  • Each morphism $F_{\alpha }$ exhibits $\operatorname{\mathcal{D}}_{\alpha }$ as a localization of $\operatorname{\mathcal{C}}_{\alpha }$ with respect to some collection of edges $W_{\alpha }$.

  • Each of the transition maps $\operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{C}}_{\beta }$ of the diagram carries $W_{\alpha }$ into $W_{\beta }$.

Let us regard $W = \varinjlim W_{\alpha }$ as a collection of edges of the simplicial set $\operatorname{\mathcal{C}}$. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. Using Corollary 4.1.3.3, we can choose a compatible family of inner anodyne morphisms $G_{\alpha }: \operatorname{\mathcal{D}}_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\alpha }$, where each $\operatorname{\mathcal{E}}_{\alpha }$ is an $\infty $-category. Set $\operatorname{\mathcal{E}}= \varinjlim \operatorname{\mathcal{E}}_{\alpha }$, so that the morphisms $G_{\alpha }$ determine a map of simplicial sets $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Since each $G_{\alpha }$ is a categorical equivalence of simplicial sets, each of the composite maps $(G_{\alpha } \circ F_{\alpha }): \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\alpha }$ exhibits $\operatorname{\mathcal{E}}_{\alpha }$ as a localization of $\operatorname{\mathcal{C}}_{\alpha }$ with respect to $W_{\alpha }$. In particular, each of the morphisms $G_{\alpha } \circ F_{\alpha }$ carries edges of $W_{\alpha }$ to isomorphisms in the $\infty $-category $\operatorname{\mathcal{E}}_{\alpha }$ (Remark 5.3.1.10). Applying Proposition 5.3.2.5, we can (functorially) factor each of the morphisms $G_{\alpha } \circ F_{\alpha }$ as a composition

\[ \operatorname{\mathcal{C}}_{\alpha } \xrightarrow { G'_{\alpha } } \operatorname{\mathcal{C}}_{\alpha }[ W^{-1}_{\alpha } ] \xrightarrow { F'_{\alpha } } \operatorname{\mathcal{E}}_{\alpha }, \]

where each $\operatorname{\mathcal{C}}_{\alpha }[ W^{-1}_{\alpha } ]$ is an $\infty $-category, each of the morphisms $G'_{\alpha }$ exhibits $\operatorname{\mathcal{C}}_{\alpha }[ W_{\alpha }^{-1} ]$ as a localization of $\operatorname{\mathcal{C}}_{\alpha }$ with respect to $W_{\alpha }$, and the colimit map $G': \operatorname{\mathcal{C}}\rightarrow \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }[ W^{-1}_{\alpha } ]$ exhibits $\operatorname{\mathcal{C}}[W^{-1}] = \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }[ W^{-1}_{\alpha } ]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. We then have a filtered diagram of commutative squares

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\alpha } \ar [r]^-{ F_{\alpha } } \ar [d]^{ G'_{\alpha } } & \operatorname{\mathcal{D}}_{\alpha } \ar [d]^{ G_{\alpha } } \\ \operatorname{\mathcal{C}}_{\alpha }[ W_{\alpha }^{-1} ] \ar [r]^-{F'_{\alpha }} & \operatorname{\mathcal{E}}_{\alpha } } \]

having colimit

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{G'} & \operatorname{\mathcal{D}}\ar [d]^{F'} \\ \operatorname{\mathcal{C}}[W^{-1}] \ar [r]^-{F'} & \operatorname{\mathcal{E}}. } \]

Applying Remark 5.3.1.17, we deduce that each of the morphisms $F'_{\alpha }$ is a categorical equivalence of simplicial sets. Since the collection of categorical equivalences is stable under filtered colimits (Corollary 4.5.4.2), the morphism $F'$ is also a categorical equivalence of simplicial sets. Applying Remark 5.3.1.17 again, we deduce that $F' \circ G'$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Since each $G_{\alpha }$ is a categorical equivalence, Corollary 4.5.4.2 also guarantees that $G$ is a categorical equivalence. Using the equality $G \circ F = F' \circ G'$ and applying Remark 5.3.1.17 again, we conclude that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, as desired. $\square$