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5.6.6 Application: Extending Cocartesian Fibrations

In ยง3.3.8, we showed that every Kan fibration of simplicial sets $f: X \rightarrow S$ can be obtained as the pullback of a Kan fibration between Kan complexes. Our goal in this section is to prove an analogous result for cocartesian fibrations of simplicial sets (Corollary 5.6.6.3). Our starting point is the following:

Lemma 5.6.6.1. Suppose we are given a commutative diagram of simplicial sets

5.42
\begin{equation} \begin{gathered}\label{equation:isomorphism-extension-small-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r]^-{G_0} \ar [d]^{U_0} & \operatorname{\mathcal{E}}' \ar [d]^{V} \\ \operatorname{\mathcal{C}}_0 \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

where the vertical maps are inner fibrations, the bottom horizontal map exhibits $\operatorname{\mathcal{C}}_0$ as a simplicial subset of $\operatorname{\mathcal{C}}$, and $G_0$ induces an equivalence $\operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ of inner fibrations over $\operatorname{\mathcal{C}}_0$. Then (5.42) can be extended to a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{G} & \operatorname{\mathcal{E}}' \ar [d]^{V} \\ \operatorname{\mathcal{C}}_0 \ar [r] & \operatorname{\mathcal{C}}\ar@ {=}[r] & \operatorname{\mathcal{C}}, } \]

where $U$ is an inner fibration, $G$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$, and the square on the left induces an isomorphism of simplicial sets $\operatorname{\mathcal{E}}_0 \simeq \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

Proof. Choose a monomorphism of simplicial sets $\operatorname{\mathcal{E}}_0 \hookrightarrow \operatorname{\mathcal{Q}}$, where $\operatorname{\mathcal{Q}}$ is a contractible Kan complex (see Exercise 3.1.7.11). Replacing $\operatorname{\mathcal{E}}'$ with the product $\operatorname{\mathcal{E}}' \times \operatorname{\mathcal{Q}}$, we can reduce to the case where $G_0$ is a monomorphism of simplicial sets. Let $\operatorname{\mathcal{E}}$ denote the simplicial subset of $\operatorname{\mathcal{E}}'$ consisting of those simplices $\sigma : \Delta ^ m \rightarrow \operatorname{\mathcal{E}}'$ for which the induced map $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \Delta ^ m \rightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ factors through $G_0$. To complete the proof, it will suffice to verify the following:

$(a)$

The morphism $U = V|_{\operatorname{\mathcal{E}}}$ is an inner fibration from $\operatorname{\mathcal{E}}$ to $\operatorname{\mathcal{C}}$.

$(b)$

The inclusion $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$.

By virtue of Remark 4.1.1.13 and Proposition 5.6.2.8, it suffices to prove $(a)$ and $(b)$ in the special case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In this case, the morphism $V: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ is an isofibration (Example 4.4.1.6).

Let $\operatorname{\mathcal{E}}'_{0}$ denote the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$. Applying Lemma 5.6.2.9 to the morphism $G_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$ (which is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}_{0}$), we deduce that there exists a morphism $R_0: \operatorname{\mathcal{E}}'_{0} \rightarrow \operatorname{\mathcal{E}}_0$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}_0}$ such that $R_0 \circ G_0 = \operatorname{id}_{\operatorname{\mathcal{E}}_0}$, and an isomorphism $\alpha _0: \operatorname{id}_{ \operatorname{\mathcal{E}}'_0 } \rightarrow G_0 \circ R_0$ in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0}( \operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}'_0 )$ whose image in $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0}( \operatorname{\mathcal{E}}_0, \operatorname{\mathcal{E}}'_0 )$ is degenerate. Applying Proposition 4.4.5.6 (and the criterion of Proposition 4.4.4.9), we can choose a morphism $R: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}'$ in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ such that $R|_{\operatorname{\mathcal{E}}'_0} = G_0 \circ R_0$ and an isomorphism $\alpha : \operatorname{id}_{\operatorname{\mathcal{E}}'} \rightarrow R$ in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}')$ whose image in $\operatorname{Fun}_{/\operatorname{\mathcal{C}}_0}(\operatorname{\mathcal{E}}'_0, \operatorname{\mathcal{E}}'_0)$ is equal to $\alpha _0$.

We now prove $(a)$. Suppose we are given a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r]^-{f_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ B \ar [r]^-{\overline{f}} \ar@ {-->}[ur]^{f} & \operatorname{\mathcal{C}}, } \]

where the left vertical map is inner anodyne. Since $V: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ is an inner fibration, we can extend $f_0$ to a morphism $f': B \rightarrow \operatorname{\mathcal{E}}$ satisfying $V \circ f' = \overline{f}$. Set $B_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} B$ and $A_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} A$, and define

\[ f_{1}: (A \coprod _{A_0} B_0) \rightarrow \operatorname{\mathcal{E}} \]

by the formula $f_1|_{A} = f_0$ and $f_1|_{B_0} = R \circ f'|_{B_0}$. Note that there is an isomorphism

\[ \beta : f'|_{ A \coprod _{A_0} B_0} \rightarrow f_{1} \]

in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( A \coprod _{A_0} B_0, \operatorname{\mathcal{E}}' )$, whose image in $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( A, \operatorname{\mathcal{E}}')$ is degenerate and whose image in $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B_0, \operatorname{\mathcal{E}}' )$ is the restriction of $\alpha $. Applying Proposition 4.4.5.6, we deduce that $f_{1}$ admits an extension $f: B \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ f = \overline{f}$.

To prove $(b)$, we observe that the morphism $R: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ is a homotopy inverse of the inclusion $\iota : \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ relative to $\operatorname{\mathcal{C}}$. By construction, $\alpha $ determines an isomorphism from $\operatorname{id}_{\operatorname{\mathcal{E}}'}$ to the composition $\iota \circ R$ in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}}')$, and the restriction of $\alpha $ determines an isomorphism from $\operatorname{id}_{\operatorname{\mathcal{E}}}$ to $R \circ \iota $ in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}})$. $\square$

Proposition 5.6.6.2 (Extending Cocartesian Fibrations). Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset, and let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be a cocartesian fibration of simplicial sets. Suppose that the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is a categorical equivalence of simplicial sets. Then there exists a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \ar [r] & \operatorname{\mathcal{C}}, } \]

where $U$ is a cocartesian fibration.

Proof. By virtue of Corollary 5.6.3.12, there exists a morphism of simplicial sets $\mathscr {F}_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{QC}}$ and an equivalence $G_0: \operatorname{\mathcal{E}}_0 \rightarrow \int _{\operatorname{\mathcal{C}}_0} \mathscr {F}_0$ of cocartesian fibrations over $\operatorname{\mathcal{C}}_0$. Since $\operatorname{\mathcal{QC}}$ is an $\infty $-category (Proposition 5.4.4.3), our assumption that the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is a categorical equivalence guarantees that we can extend $\mathscr {F}_0$ to a morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. We can then identify $G_0$ with an equivalence $\operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of cocartesian fibrations over $\operatorname{\mathcal{C}}_0$. Applying Lemma 5.6.6.1, we can write $U_0$ as the pullback of an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is equivalent to the projection map $V: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ as an inner fibration over $\operatorname{\mathcal{C}}$. Since $V$ is a cocartesian fibration (Proposition 5.5.4.2), it follows that $U$ is also a cocartesian fibration (Proposition 5.6.2.10). $\square$

Corollary 5.6.6.3. Let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be a cocartesian fibration of simplicial sets. Then there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [r] \ar [d]^{U_0} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{F} & \operatorname{\mathcal{C}}, } \]

where $U$ is a cocartesian fibration of $\infty $-categories and $F$ is inner anodyne.

Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne map $F: \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}$ is an $\infty $-category. Since $F$ is a categorical equivalence of simplicial sets (Corollary 4.5.2.11), Proposition 5.6.6.2 guarantees that $U_0$ is the pullback of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 5.6.6.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then $U$ is an isofibration.

Proof. By virtue of Corollary 5.6.6.3, we may assume without loss of generality that $U$ is a cocartesian fibration of $\infty $-categories, in which case the desired result follows from Proposition 5.1.4.8. $\square$

Corollary 5.6.6.5. Suppose we are given a pullback diagram of simplicial sets

5.43
\begin{equation} \begin{gathered}\label{equation:pullback-cocartesian-fibration} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{F} } \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{C}}', } \end{gathered} \end{equation}

where $U'$ is a cocartesian fibration. If $F$ is a categorical equivalence of simplicial sets, then $\widetilde{F}$ is also a categorical equivalence of simplicial sets.

Proof. Using Corollary 5.6.6.3, we can extend (5.43) to a diagram of pullback squares

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

where $U''$ is a cocartesian fibration, $G$ is inner anodyne, and $\operatorname{\mathcal{C}}''$ is an $\infty $-category. Since $G$ and $G \circ F$ are categorical equivalences, it follows from Proposition 5.2.6.24 that $\widetilde{G}$ and $\widetilde{G} \circ \widetilde{F}$ are categorical equivalences. Applying the two-out-of-three property (Remark 4.5.2.5), we conclude that $\widetilde{F}$ is also a categorical equivalence. $\square$

Corollary 5.6.6.6. Suppose we are given a commutative diagram of simplicial sets

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

where $U$ and $V$ are cocartesian fibrations. Then $F$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (Definition 5.6.2.1) if and only if it is a categorical equivalence of simplicial sets.

Proof. Combine Proposition 5.6.2.5 with Corollary 5.6.6.4. $\square$