# Kerodon

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## 9.3 Fiberwise Cocompletions

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty$-categories. For each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C}$ is an $\infty$-category. In §8.4, we proved that $\operatorname{\mathcal{E}}_{C}$ admits a cocompletion: that is, an $\infty$-category $\widehat{\operatorname{\mathcal{E}}}_{C}$ which is universal among cocomplete $\infty$-categories with a functor from $\operatorname{\mathcal{E}}_{C}$ (Proposition 8.4.5.3). Our goal in this section is to show that the collection of $\infty$-categories $\{ \widehat{\operatorname{\mathcal{E}}}_{C} \} _{C \in \operatorname{\mathcal{C}}}$ can be regarded as the fibers of another inner fibration $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$, where $\widehat{\operatorname{\mathcal{E}}}$ is an $\infty$-category we refer to as the fiberwise cocompletion of $\operatorname{\mathcal{E}}$. We begin by articulating the relationship between the fibrations $U$ and $\widehat{U}$. To simplify the discussion, let us assume that the inner fibration $U$ is essentially small (see Definition 9.3.1.12 for a more general discussion).

Definition 9.3.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small inner fibration of $\infty$-categories. We say that a commutative diagram of $\infty$-categories

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

exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$ if the following conditions are satisfied:

$(1)$

For every object $C \in \operatorname{\mathcal{C}}$, the map of fibers $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a cocompletion of $\operatorname{\mathcal{E}}_{C}$ (see Definition 8.4.0.1).

$(2)$

The functor $H$ is fully faithful.

$(3)$

The functor $\widehat{U}$ is a locally cartesian fibration.

$(4)$

For every morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $f^{\ast }: \widehat{\operatorname{\mathcal{E}}}_{D} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ preserves small colimits.

One of our principal goals in this section is to prove the following:

Theorem 9.3.0.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small inner fibration of $\infty$-categories. Then there exists a commutative diagram

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

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$. Moreover, the inner fibration $\widetilde{U}$ is unique up to equivalence.

We begin by establishing the uniqueness assertion of Theorem 9.3.0.2. In the situation of Definition 9.3.0.1, the inner fibration $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ satisfies the following condition:

$(\ast )$

For each object $C \in \operatorname{\mathcal{C}}$, the $\infty$-category $\widehat{\operatorname{\mathcal{E}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ is cocomplete and the inclusion functor $\widehat{\operatorname{\mathcal{E}}}_{C} \hookrightarrow \widehat{\operatorname{\mathcal{E}}}$ carries (small) colimit diagrams in $\operatorname{\mathcal{E}}_{C}$ to $U$-colimit diagrams in $\operatorname{\mathcal{E}}$.

More generally, we say that an inner fibration of $\infty$-categories $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ is cocomplete if it satisfies condition $(\ast )$ (Definition 9.3.1.1). In §9.3.1, we show that if a commutative diagram

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

exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$, then it is universal in the following sense: every functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ satisfying $U' \circ F = U$ admits an essentially unique extension $\widehat{F}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}'$ having the property that, for each object $C \in \operatorname{\mathcal{C}}$, the map of fibers $\widehat{F}_{C}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ preserves small colimits (Theorem 9.3.1.20). It follows that $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ is determined (up to equivalence) by the inner fibration $U$ (Remark 9.3.1.21).

To complete the proof of Theorem 9.3.0.2, we must show that every (essentially small) inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ admits a fiberwise cocompletion $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$. We begin by treating some special cases:

• Assume that $U$ is a cocartesian fibration. In §9.3.2 we show that there exists a diagram

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

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$. Moreover, $\widehat{U}$ is also a cocartesian fibration and the functor $H$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ to $\widehat{U}$-cocartesian morphisms of $\widehat{\operatorname{\mathcal{E}}}$ (Theorem 9.3.2.1). Our proof uses the theory of dual fibrations developed in §8.6. More precisely, we show that we can take $\widehat{\operatorname{\mathcal{E}}}$ to be the relative exponential $\operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$, where $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian dual of $U$ (Lemma 9.3.2.4).

• Assume that $U$ is a cartesian fibration. In §9.3.3 we show that there exists a diagram

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

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$. Moreover, $\widehat{U}$ is also a cartesian fibration and the functor $H$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{E}}$ to $\widehat{U}$-cartesian morphisms of $\widehat{\operatorname{\mathcal{E}}}$ (Theorem 9.3.3.1). Our strategy is to reduce to the cocartesian case, using the compatibility of fiberwise cocompletion with the formation of conjugate fibrations (Proposition 9.3.3.3).

In §9.3.4, we complete the proof of Theorem 9.3.0.2 by proving the existence of a fiberwise cocompletion for a general inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Our strategy is to embed $\operatorname{\mathcal{E}}$ into the oriented fiber product $\operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, and thereby reduce to the case where $U$ is a cocartesian fibration (using the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, we could instead reduce to the case where $U$ is a cartesian fibration).

Recall that an inner fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is exponentiable if the pullback functor

$(\operatorname{Set_{\Delta }})_{ / \operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

preserves categorical equivalences (Definition 4.5.9.10). In §9.3.6, we introduce a closely related condition which has somewhat better formal properties. We say that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is flat if, for every $2$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$, the inclusion map

$\Lambda ^{2}_{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \Delta ^{2} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$

is a categorical equivalence of simplicial sets (Definition 9.3.6.1). It follows immediately from the definitions that if $U$ is an exponentiable inner fibration, then it is flat. In §9.3.6, we show that the converse holds if $U$ is an isofibration (Corollary 9.3.6.30). The key observation is that that an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is flat if and only if the fiberwise cocompletion $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration (Lemma 9.3.6.13). In §9.3.8 we show that, under this assumption, we can give a concrete description of $\widehat{\operatorname{\mathcal{E}}}$ in terms of the relative Yoneda embedding of $U$ (Theorem 9.3.8.4).

## Structure

• Subsection 9.3.1: Uniqueness of Fiberwise Cocompletions
• Subsection 9.3.2: Fiberwise Cocompletions of Cocartesian Fibrations
• Subsection 9.3.3: Fiberwise Cocompletion of Cartesian Fibrations
• Subsection 9.3.4: Existence of Fiberwise Cocompletions
• Subsection 9.3.5: Digression: Morita Equivalence
• Subsection 9.3.6: Application: Flat Inner Fibrations
• Subsection 9.3.7: Flatness and Morphism Spaces
• Subsection 9.3.8: Fiberwise Cocompletion via the Yoneda Embedding