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6.3.5 Fiberwise Localization

The formation of localizations is compatible with products:

Proposition 6.3.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to a collection of edges $W$. Let $K$ be any simplicial set, and let $W_{K}$ denote the collection of edges $e = (e', e'')$ of the product $K \times \operatorname{\mathcal{C}}$ for which $e'$ is a degenerate edge of $K$ and $e''$ belongs to $W$. Then the induced map $F_{K}: K \times \operatorname{\mathcal{C}}\rightarrow K \times \operatorname{\mathcal{D}}$ exhibits $K \times \operatorname{\mathcal{D}}$ as the localization of $K \times \operatorname{\mathcal{C}}$ with respect to $W_{K}$.

Proof. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category, and let

\[ \theta : \operatorname{Fun}( K \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(K \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

be the functor given by precomposition with $F_ K$. We wish to show that $F_{K}$ is fully faithful, and that its essential image is the full subcategory $\operatorname{Fun}( (K \times \operatorname{\mathcal{C}})[W_{K}^{-1}], \operatorname{\mathcal{E}})$ of Notation 6.3.1.1. Unwinding the definitions, we can identify $\theta $ with the functor

\[ \theta ': \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Fun}(K,\operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Fun}(K, \operatorname{\mathcal{E}})) \]

given by precomposition with $F$. Under this identification $\operatorname{Fun}( (K \times \operatorname{\mathcal{C}})[W_{K}^{-1}], \operatorname{\mathcal{E}})$ corresponds to the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{Fun}(K,\operatorname{\mathcal{E}}) ) \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Fun}(K,\operatorname{\mathcal{E}}) )$ (see Theorem 4.4.4.4), so that the desired result follows from our assumption on the functor $F$. $\square$

Our goal in this section is to establish the following variant of Proposition 6.3.5.1:

Proposition 6.3.5.2. Suppose we are given a commutative diagram of simplicial sets

6.2
\begin{equation} \begin{gathered}\label{equation:cocartesian-fiberwise-localization} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{F} \ar [dr]^{U} & & \operatorname{\mathcal{E}}' \ar [dl]_{U'} \\ & \operatorname{\mathcal{C}}& } \end{gathered} \end{equation}

with the following properties:

$(1)$

The morphisms $U$ and $U'$ are cocartesian fibrations.

$(2)$

The morphism $F$ carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to $U'$-cocartesian edges of $\operatorname{\mathcal{E}}'$.

$(3)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the induced functor of $\infty $-categories $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ exhibits $\operatorname{\mathcal{E}}'_{C}$ as the localization of $\operatorname{\mathcal{E}}_{C}$ with respect to some collection of morphisms $W_{C}$.

Set $W = \bigcup _{C \in \operatorname{\mathcal{C}}} W_{C}$, which we regard as a collection of edges of the simplicial set $\operatorname{\mathcal{E}}$. Then $F$ exhibits $\operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$.

Corollary 6.3.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is weakly contractible. Let $W$ be the collection of all edges $e$ of $\operatorname{\mathcal{E}}$ having the property that $U(e)$ is a degenerate edge of $\operatorname{\mathcal{C}}$. Then $U$ exhibits $\operatorname{\mathcal{C}}$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$.

Proof. For each vertex $C \in \operatorname{\mathcal{C}}$, let $W_{C}$ be the collection of all morphisms in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. Since $\operatorname{\mathcal{E}}_{C}$ is weakly contractible, the projection map $\operatorname{\mathcal{E}}_{C} \rightarrow \{ C\} $ exhibits $\{ C\} $ as a localization of $\operatorname{\mathcal{E}}_{C}$ with respect to $W_{C}$ (Proposition 6.3.1.19). The desired result now follows by applying Proposition 6.3.5.2 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]^{U} \ar [rr]^{U} & & \operatorname{\mathcal{C}}\ar [dl]_{\operatorname{id}_{\operatorname{\mathcal{C}}} } \\ & \operatorname{\mathcal{C}}. & } \]
$\square$

Proof of Proposition 6.3.5.2. For every simplicial set $S$ equipped with a morphism $S \rightarrow \operatorname{\mathcal{C}}$, set $\operatorname{\mathcal{E}}_{S} = S \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_{S} = S \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$, so that $F$ induces a morphism $F_{S}: \operatorname{\mathcal{E}}_{S} \rightarrow \operatorname{\mathcal{E}}'_{S}$. Let $W_{S}$ denote the inverse image of $W$ in $\operatorname{\mathcal{E}}_{S}$. Let us say that $S$ is good if the morphism $F_{S}$ exhibits $\operatorname{\mathcal{E}}'_{S}$ as a localization of $\operatorname{\mathcal{E}}_{S}$ with respect to $W_{S}$. To prove Proposition 6.3.5.2, it will suffice to show that every simplicial set $S$ (equipped with a morphism $S \rightarrow \operatorname{\mathcal{C}}$) is good. Writing $S$ as a union of finite simplicial subsets (Remark 3.5.1.8) and applying Proposition 6.3.4.1, we may assume that the simplicial set $S$ is finite. If $S$ is empty, there is nothing to prove; we may therefore assume without loss of generality that $S$ has dimension $n \geq 0$. We proceed by induction on $n$ and on the number of nondegenerate $n$-simplices of $S$. Fix a nondegenerate $n$-simplex $\sigma : \Delta ^{n} \rightarrow S$, so that Proposition 1.1.3.13 supplies a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \Delta ^{n} \ar [d]^{\sigma } \\ S' \ar [r] & S, } \]

where $S'$ is a simplicial subset of $S$ having fewer nondegenerate $n$-simplices. It follows from our inductive hypothesis that $S'$ and $\operatorname{\partial \Delta }^ n$ are good. By virtue of Proposition 6.3.4.2, to show that $S$ is good, it will suffice to show that $\Delta ^ n$ is good. Replacing $\operatorname{\mathcal{C}}$ by $\Delta ^ n$, we are reduced to proving Proposition 6.3.5.2 in the special case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In particular, this guarantees that $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'$ are $\infty $-categories.

If $n = 0$, the result is immediate. We may therefore assume without loss of generality that $n \geq 1$. Let $V: \Delta ^ n \twoheadrightarrow \Delta ^1$ denote the morphism given on vertices by the formula

\[ V(i) = \begin{cases} 0 & \text{ if $i < n$} \\ 1 & \text{ if $i=n$ } \end{cases}. \]

It follows from Proposition 5.1.4.13 that the composite morphisms $V \circ U$ and $V \circ U'$ are cocartesian fibrations, and that the functor $F$ carries $(V \circ U)$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ to $(V \circ U')$-cocartesian morphisms of $\operatorname{\mathcal{E}}'$. Moreover, our inductive hypothesis guarantees that the simplicial subsets $V^{-1} \{ 0\} , V^{-1} \{ 1\} \subseteq \Delta ^ n$ are good. We may therefore replace (6.2) by the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{F} \ar [dr]^{V \circ U} & & \operatorname{\mathcal{E}}' \ar [dl]_{V \circ U'} \\ & \Delta ^1, & } \]

and thereby reduce to the special case where $\operatorname{\mathcal{C}}= \Delta ^1$.

For $i \in \{ 0,1\} $, we let $\operatorname{\mathcal{E}}(i)$ denote the fiber $\{ i\} \times _{\operatorname{\mathcal{C}}} \{ \operatorname{\mathcal{E}}\} $ and define $\operatorname{\mathcal{E}}'(i)$ similarly. Let $W'_{i} \supseteq W_{i}$ be the collection of all morphisms $e$ of $\operatorname{\mathcal{E}}(i)$ for which $F(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}'(i)$. Suppose that $\operatorname{\mathcal{D}}$ an $\infty $-category and $G: \operatorname{\mathcal{E}}(i) \rightarrow \operatorname{\mathcal{D}}$ is a functor which carries each morphism of $W_{i}$ to an isomorphism in $\operatorname{\mathcal{D}}$. Since $F_{i}$ exhibits $\operatorname{\mathcal{E}}'(i)$ as a localization of $\operatorname{\mathcal{E}}(i)$ with respect to $W_{i}$, it follows that there is a functor $G': \operatorname{\mathcal{E}}'(i) \rightarrow \operatorname{\mathcal{D}}$ such that $G$ is isomorphic to the composition $\operatorname{\mathcal{E}}(i) \xrightarrow {F_ i} \operatorname{\mathcal{E}}'(i) \xrightarrow {G'} \operatorname{\mathcal{D}}$ as an object of the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}(i), \operatorname{\mathcal{D}})$. It follows that the functor $G$ carries each morphism of $W'_{i}$ to an isomorphism in $\operatorname{\mathcal{D}}$. Allowing $G$ to vary, we deduce that the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}(i)[ W_ i^{-1} ], \operatorname{\mathcal{D}})$ and $\operatorname{Fun}( \operatorname{\mathcal{E}}(i)[ W'^{-1}_{i} ], \operatorname{\mathcal{D}})$ coincide. It follows that the functor $F_{i}$ exhibits $\operatorname{\mathcal{E}}'(i)$ as a localization of $\operatorname{\mathcal{E}}(i)$ with respect to $W'_{i}$. Setting $W' = W'_0 \cup W'_{1}$, the same argument shows that the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}[W^{-1}], \operatorname{\mathcal{D}})$ and $\operatorname{Fun}( \operatorname{\mathcal{E}}[ W'^{-1}], \operatorname{\mathcal{D}})$ coincide (as full subcategories of $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}})$). Consequently, to prove Proposition 6.3.5.2, we can replace $W$ by $W'$ and thereby reduce to the case where $W$ contains every morphism $e$ of $\operatorname{\mathcal{E}}$ for which $F(e)$ is an isomorphism in $\operatorname{\mathcal{E}}'$.

By virtue of Exercise 3.1.7.11, the functor $F$ factors as a composition $\operatorname{\mathcal{E}}\xrightarrow {F'} \operatorname{\mathcal{Q}}\xrightarrow {F''} \operatorname{\mathcal{E}}'$, where $F'$ is a monomorphism of simplicial sets and $F''$ is a trivial Kan fibration. By virtue of Remark 6.3.1.18, we can replace $\operatorname{\mathcal{E}}'$ by $\operatorname{\mathcal{Q}}$ and thereby reduce to the case where $F$ is a monomorphism of simplicial sets. Let $T: \operatorname{\mathcal{E}}(0) \rightarrow \operatorname{\mathcal{E}}(1)$ be given by covariant transport along the nondegenerate edge of $\operatorname{\mathcal{C}}= \Delta ^1$, so that there exists a natural transformation $h: \operatorname{id}_{\operatorname{\mathcal{E}}(0)} \rightarrow T$ in $\operatorname{Fun}( \operatorname{\mathcal{E}}(0), \operatorname{\mathcal{E}})$ carrying each object $X \in \operatorname{\mathcal{E}}(0)$ to a $U$-cocartesian morphism $h_ X: X \rightarrow T(X)$ of $\operatorname{\mathcal{E}}$. In this case, $F(h_ X)$ is also a $U'$-cocartesian morphism of $\operatorname{\mathcal{E}}'$. Applying Proposition 5.2.1.3, we deduce that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ (\{ 0\} \times \operatorname{\mathcal{E}}'(0) ) \coprod _{ (\{ 0\} \times \operatorname{\mathcal{E}}(0) )} (\Delta ^1 \times \operatorname{\mathcal{E}}(0) ) \ar [r]^-{ (\operatorname{id}, F \circ h) } \ar [d] & \operatorname{\mathcal{E}}' \ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{E}}'(0) \ar@ {-->}[ur]^{h'} \ar [r] & \Delta ^1 } \]

admits a solution $h'$ which carries each object $X' \in \operatorname{\mathcal{E}}'(0)$ to a $U'$-cocartesian morphism of $\operatorname{\mathcal{E}}'$. The restriction of $h'$ to $\{ 1\} \times \operatorname{\mathcal{E}}'(0)$ can be identified with a functor $T': \operatorname{\mathcal{E}}'(0) \rightarrow \operatorname{\mathcal{E}}'(1)$, so that $h'$ is a natural transformation $\operatorname{id}_{\operatorname{\mathcal{E}}'(0)} \rightarrow T'$ which exhibits $T'$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$. Note that the commutativity of the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}(0) \ar [r]^-{T} \ar [d]^{F_0} & \operatorname{\mathcal{E}}(1) \ar [d] \\ \operatorname{\mathcal{E}}'(0) \ar [r]^-{T'} & \operatorname{\mathcal{E}}'(1) } \]

guarantees that the functor $T$ carries each morphism of $W_0$ to a morphism of $W_1$. Applying Proposition 6.3.4.2 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{E}}(0) \ar [dr]^{ F_{01} } \ar [rr]^(.55){T} \ar [dd] & & \operatorname{\mathcal{E}}(1) \ar [dd]^(.55){H'} \ar [dr]^{ F_{0} } & \\ & \{ 1\} \times \operatorname{\mathcal{E}}'(0) \ar [rr]^{T'} \ar [dd] & & \operatorname{\mathcal{E}}'(1) \ar [dd] \\ \Delta ^1 \times \operatorname{\mathcal{E}}(0) \ar [rr]^(.6){h} \ar [dr]^{ F_{1} } & & \operatorname{\mathcal{E}}\ar [dr]^{ F } & \\ & \Delta ^1 \times \operatorname{\mathcal{E}}'(0) \ar [rr]^{h'} & & \operatorname{\mathcal{E}}' } \]

(where the front and back faces are categorical pushout squares by virtue of Theorem 5.2.5.1), we are reduced to proving Proposition 6.3.5.2 for the diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{\mathcal{E}}(0) \ar [rr]^-{\operatorname{id}\times F_0} \ar [dr] & & \Delta ^1 \times \operatorname{\mathcal{E}}'(0) \ar [dl] \\ & \Delta ^1, & } \]

which is a special case of Proposition 6.3.5.1. $\square$