# Kerodon

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### 6.3.4 Stability Properties of Localizations

Our goal in this section is to record some basic formal properties of the localization construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}[W^{-1}]$ introduced in ยง6.3.2. We first show that localization commutes with the formation of filtered colimits. More precisely, we have the following:

Proposition 6.3.4.1. 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 6.3.1.10). Applying Proposition 6.3.2.5, 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

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

Applying Remark 6.3.1.19, 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 4.5.7.2), the morphism $F'$ is also a categorical equivalence of simplicial sets. Applying Remark 6.3.1.19 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 4.5.7.2 also guarantees that $G$ is a categorical equivalence. Using the equality $G \circ F = F' \circ G'$ and applying Remark 6.3.1.19 again, we conclude that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, as desired. $\square$

We now show that localization is compatible with the formation of categorical pushout squares.

Proposition 6.3.4.2. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [dr]^{ F_{01} } \ar [rr]^(.55){ G } \ar [dd]^(.55){H} & & \operatorname{\mathcal{C}}_{0} \ar [dd]^(.55){H'} \ar [dr]^{ F_{0} } & \\ & \operatorname{\mathcal{D}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{D}}_{0} \ar [dd] \\ \operatorname{\mathcal{C}}_{1} \ar [rr]^(.6){G'} \ar [dr]^{ F_{1} } & & \operatorname{\mathcal{C}}\ar [dr]^{ F } & \\ & \operatorname{\mathcal{D}}_{1} \ar [rr] & & \operatorname{\mathcal{D}}}$

with the following properties:

$(a)$

The back face

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^-{G} \ar [d]^{H} & \operatorname{\mathcal{C}}_0 \ar [d]^{H'} \\ \operatorname{\mathcal{C}}_{1} \ar [r]^-{G'} & \operatorname{\mathcal{C}}}$

is a categorical pushout square of simplicial sets.

$(b)$

The morphism of simplicial sets $F_{01}: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{D}}_{01}$ exhibits $\operatorname{\mathcal{D}}_{01}$ as a localization of $\operatorname{\mathcal{C}}_{01}$ with respect to some collection of edges $W_{01}$.

$(c)$

The morphism of simplicial sets $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ exhibits $\operatorname{\mathcal{D}}_0$ as a localization of $\operatorname{\mathcal{C}}_0$ with respect to some collection of edges $W_0$ containing $G(W_{01})$.

$(d)$

The morphism of simplicial sets $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}_1$ exhibits $\operatorname{\mathcal{D}}_1$ as a localization of $\operatorname{\mathcal{C}}_1$ with respect to some collection of edges $W_1$ containing $H(W_{01})$.

Then the following conditions are equivalent:

$(1)$

The front face

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{D}}_0 \ar [d] \\ \operatorname{\mathcal{D}}_1 \ar [r] & \operatorname{\mathcal{D}}}$

is a categorical pushout square of simplicial sets.

$(2)$

The morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to the collection of edges $W = H'( W_0 ) \cup G'( W_1)$.

Proof. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category. Assumption $(a)$ guarantees that the diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{E}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}_1, \operatorname{\mathcal{E}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{01}, \operatorname{\mathcal{E}})^{\simeq } }$

is a homotopy pullback square. Applying Proposition 3.4.1.14, we deduce that the diagram of summands

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}_0[W^{-1}_0], \operatorname{\mathcal{E}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}_1[W_1^{-1}], \operatorname{\mathcal{E}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{01}[W_{01}^{-1}], \operatorname{\mathcal{E}})^{\simeq } }$

is also a homotopy pullback square. Invoking Corollary 3.4.1.12, we conclude that the following conditions are equivalent:

$(1_{\operatorname{\mathcal{E}}})$

The diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}})^{\simeq } \ar [d] \\ \operatorname{Fun}(\operatorname{\mathcal{D}}_1, \operatorname{\mathcal{E}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } }$

is a homotopy pullback square.

$(2_{\operatorname{\mathcal{E}}})$

Precomposition with $F$ induces a homotopy equivalence of Kan complexes

$\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \xrightarrow {\circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq }.$

We now observe that condition $(1)$ is equivalent to the requirement that $(1_{\operatorname{\mathcal{E}}} )$ holds for every $\infty$-category $\operatorname{\mathcal{E}}$ (by definition), and condition $(2)$ is equivalent to the requirement that $(2_{\operatorname{\mathcal{E}}})$ holds for every $\infty$-category $\operatorname{\mathcal{E}}$ (Proposition 6.3.1.13). $\square$

Corollary 6.3.4.3. Suppose we are given a categorical pushout diagram of simplicial sets

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

where $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of edges $W$. Then $F'$ exhibits $\operatorname{\mathcal{D}}'$ as a localization of $\operatorname{\mathcal{C}}'$ with respect to $G(W)$.

Proof. Apply Proposition 6.3.4.2 to the cubical diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [dr] \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}' \ar [dd] \ar [dr] & \\ & \operatorname{\mathcal{C}}\ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}' \ar [dd] \\ \operatorname{\mathcal{C}}\ar [rr] \ar [dr] & & \operatorname{\mathcal{C}}' \ar [dr] & \\ & \operatorname{\mathcal{D}}\ar [rr] & & \operatorname{\mathcal{D}}'. }$
$\square$

Example 6.3.4.4 (Contracting an Edge). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $e$ be an edge of $\operatorname{\mathcal{C}}$ which corresponds to a monomorphism of simplicial sets $\Delta ^1 \hookrightarrow \operatorname{\mathcal{C}}$ (that is, the source and target of $e$ are distinct when regarded as vertices of $\operatorname{\mathcal{C}}$). Let $\operatorname{\mathcal{C}}'$ denote the simplicial set obtained from $\operatorname{\mathcal{C}}$ by collapsing the edge $e$, so that we have a pushout square of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \ar [r]^{e} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{T} \\ \Delta ^{0} \ar [r] & \operatorname{\mathcal{C}}'. }$

Since the horizontal maps in this diagram are monomorphisms, it is also a categorical pushout square (Example 4.5.4.12). Combining Corollary 6.3.4.3 with Example 6.3.1.14, we see that $T$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to the singleton $W = \{ e \}$.