8.7.2 Fiberwise Cocompletions of Cocartesian Fibrations
We now specialize our theory of fiberwise cocompletions to the setting of cocartesian fibrations. In this case, Definition 8.7.1.5 can be reformulated:
Theorem 8.7.2.1. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then a commutative diagram
8.99
\begin{equation} \begin{gathered}\label{equation:fiberwise-cocompletion-cocartesian-case} \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}}& } \end{gathered} \end{equation}
exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ if and only if the following conditions are satisfied:
- $(a)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the functor $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\mathbb {K}$-cocompletion of the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ (Definition 8.4.6.1).
- $(b)$
The morphism $\widehat{U}$ is a cocartesian fibration of simplicial sets.
- $(c)$
For every edge $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $f_{!}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C'}$ is $\mathbb {K}$-cocontinuous.
- $(d)$
The morphism $H$ carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to $\widehat{U}$-cocartesian edges of $\widehat{\operatorname{\mathcal{E}}}$.
Moreover, there exists a diagram (8.99) which satisfies these conditions.
Our proof of Theorem 8.7.2.1 will require some preliminaries.
Lemma 8.7.2.2. Let $\mathbb {K}$ be a collection of simplicial sets, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and suppose we are given a commutative diagram
8.100
\begin{equation} \begin{gathered}\label{equation:fiberwise-cocompletion-one-direction} \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}}& } \end{gathered} \end{equation}
which satisfies the conditions of Theorem 8.7.2.1. Then $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$. For $i \in \{ 0,1\} $, let $\operatorname{\mathcal{E}}_{i}$ denote the fiber $\{ i\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and define $\widehat{\operatorname{\mathcal{E}}}_{i}$ similarly, so that $H$ restricts to a functor $H_{i}: \operatorname{\mathcal{E}}_{i} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{i}$. By assumption, the functor $H_{i}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{i}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{i}$, and is therefore fully faithful (Proposition 8.4.6.3). Since $H$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ to $\widehat{U}$-cocartesian morphisms of $\widehat{\operatorname{\mathcal{E}}}$, it follows that $H$ is fully faithful (Proposition 5.1.6.7). It follows from Example 7.6.6.32 that the cocartesian fibration $\widehat{U}$ is $\mathbb {K}$-cocomplete.
Fix an uncountable regular cardinal $\kappa $ such that $\widehat{\operatorname{\mathcal{E}}}$ is locally $\kappa $-small and every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. To complete the proof, it will suffice to show that for every object $X \in \operatorname{\mathcal{E}}_{0}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{\mathcal{S}}_{< \kappa } \quad \quad Y \mapsto \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}} }( H(X), Y ) \]
commutes with $K$-indexed colimits, for each $K \in \mathbb {K}$. Since $U$ is a cocartesian fibration, we can choose a $U$-cocartesian morphism $u: X \rightarrow X'$ of $\operatorname{\mathcal{E}}$ with $X' \in \operatorname{\mathcal{E}}_{1}$. By assumption, $H(u): H(X) \rightarrow H(X')$ is a $\widehat{U}$-cocartesian morphism in the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}$, so the functor $\mathscr {F}$ is corepresented by the object $H(X')$. The desired result now follows from the characterization of $\mathbb {K}$-cocompletions given in Variant 8.4.8.9.
$\square$
Our next goal is to prove the existence assertion of Theorem 8.7.2.1. We begin by treating the case where $\mathbb {K}$ is the collection of $\kappa $-small simplicial sets for some uncountable regular cardinal $\kappa $, and the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is essentially $\kappa $-small. In the special case $\operatorname{\mathcal{C}}= \Delta ^0$, Theorem 8.7.2.1 reduces to the assertion that there exists a functor $H: \operatorname{\mathcal{E}}\rightarrow \widehat{\operatorname{\mathcal{E}}}$ which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$. In §8.4, we proved this using an explicit construction: we can take $\widehat{\operatorname{\mathcal{E}}}$ to be the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ and $H$ to be the covariant Yoneda embedding (Theorem 8.4.0.3). We now extend this construction to the relative setting, using the duality theory developed in §8.6 (see Exercise 
for an alternative approach).
Lemma 8.7.2.3. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be essentially $\kappa $-small cocartesian fibrations of simplicial sets, and let $\operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } )$ be the relative exponential of Construction 4.5.8.1. Suppose we are given a morphism
\[ \mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa }, \]
which exhibits $U^{\vee }$ as a cocartesian dual of $U$ (see Variant 8.6.4.13). Then $\mathscr {K}$ is classified by a commutative diagram with a diagram
8.101
\begin{equation} \begin{gathered}\label{equation:describe-relative-cocompletion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } ) \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}. & } \end{gathered} \end{equation}
which satisfies the conditions of Theorem 8.7.2.1.
Proof.
It follows from Proposition 8.6.5.12 that $\widehat{U}$ is both a cartesian and cocartesian fibration, so that condition $(b)$ of Theorem 8.7.2.1 is satisfied and condition $(c)$ is automatic (Corollary 7.1.4.28). Condition $(d)$ follows from Proposition 8.6.5.18, and condition $(a)$ from Theorem 8.4.3.2.
$\square$
Example 8.7.2.4. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is essentially $\kappa $-small, and let $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } )$ be as in Construction 8.6.5.6. It follows from Proposition 8.6.5.8 that the evaluation functor
\[ \operatorname{ev}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa }) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa } \quad \quad ((C, \mathscr {F}_{C}), X) \mapsto \mathscr {F}_{C}(X) \]
exhibits $U$ as a cocartesian dual of the projection map $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$. Applying Lemma 8.7.2.3 (with the roles of $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\vee }$ switched) and Lemma 8.7.2.2, we conclude that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } ) \ar [rr] \ar [dr] & & \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } ) \ar [dl] \\ & \operatorname{\mathcal{C}}& } \]
exhibits $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } )$ as a fiberwise $\kappa $-cocompletion of $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } )$.
Lemma 8.7.2.5. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then there exists a commutative diagram
8.102
\begin{equation} \begin{gathered}\label{equation:describe-relative-cocompletion2} \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}}. & } \end{gathered} \end{equation}
which satisfies the conditions of Theorem 8.7.2.1.
Proof.
Choose an uncountable regular cardinal $\kappa $ such that $U$ is essentially $\kappa $-small and each $K \in \mathbb {K}$ is $\kappa $-small. Using Theorem 8.6.5.1 (and Proposition 8.6.4.15), we can choose a cocartesian fibration $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ and a diagram $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{<\kappa }$ which exhibits $U^{\vee }$ as a cocartesian dual of $U$. Let $\mathbb {K}'$ be the collection of all $\kappa $-small simplicial sets and set $\widehat{\operatorname{\mathcal{E}}'} = \operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \kappa } )$, so that Lemma 8.7.2.3 supplies 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 satisfies the conditions of Theorem 8.7.2.1 with respect to $\mathbb {K}'$.
For each object $C \in \operatorname{\mathcal{C}}$, let $\widehat{\operatorname{\mathcal{E}}}_{C}$ denote smallest full subcategory of $\widehat{\operatorname{\mathcal{E}}}$ which contains the essential image of the functor $H'_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}'_{C}$ and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$. For each edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, Remark 5.2.2.14 guarantees that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r] \ar [d]^{ H'_ C } & \operatorname{\mathcal{E}}_{D} \ar [d]^{ H'_ D } \\ \widehat{\operatorname{\mathcal{E}}}'_{C} \ar [r]^-{f_{!}} & \widehat{\operatorname{\mathcal{E}}}'_{D} } \]
commutes up to isomorphism, where the horizontal maps are given by covariant transport along $f$. In particular, the covariant transport functor $f_{!}: \widehat{\operatorname{\mathcal{E}}}'_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}'_{D}$ carries the essential image of $H'_{C}$ into the essential image of $H'_{D}$. Since $f_{!}$ preserves $\kappa $-small colimits, it carries $\widehat{\operatorname{\mathcal{E}}}_{C}$ into $\widehat{\operatorname{\mathcal{E}}}_{D}$.
Let $\widehat{\operatorname{\mathcal{E}}}$ denote the full simplicial subset of $\widehat{\operatorname{\mathcal{E}}}'$ spanned by those vertices $X$ which belong to $\widehat{\operatorname{\mathcal{E}}}_{C}$, where $C = \widehat{U}'(X)$. Applying Proposition 5.1.4.17, we see that $\widehat{U} = \widehat{U}'|_{ \widehat{\operatorname{\mathcal{E}}} }$ is a cocartesian fibration from $\widehat{\operatorname{\mathcal{E}}}$ to $\operatorname{\mathcal{C}}$, and that an edge of $\widehat{\operatorname{\mathcal{E}}}$ is $\widehat{U}$-cocartesian if and only if it is $\widehat{U}'$-cocartesian when regarded as an edge of $\widehat{\operatorname{\mathcal{E}}}'$. It follows that the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{\widehat{U}} \\ & \operatorname{\mathcal{C}}& } \]
automatically satisfies conditions $(b)$ and $(d)$ of Theorem 8.7.2.1. Moreover, for each edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the associated covariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{D}$ can be identified with the restriction of $f_{!}$ to $\widehat{\operatorname{\mathcal{E}}}_{C}$, and therefore preserves $K$-indexed colimits for each $K \in \mathbb {K}$. To complete the proof, it will suffice to show that for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{C}$. This follows from Proposition 8.4.6.8.
$\square$
Proof of Theorem 8.7.2.1.
Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Using Lemma 8.7.2.5, we can choose a commutative diagram
8.103
\begin{equation} \begin{gathered}\label{equation:proof-of-fiberwise-cocompletion} \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}}& } \end{gathered} \end{equation}
which satisfies conditions $(a)$, $(b)$, $(c)$, and $(d)$ of Theorem 8.7.2.1. Lemma 8.7.2.2 guarantees any such diagram (8.103) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Conversely, if we are given another diagram
8.104
\begin{equation} \begin{gathered}\label{equation:proof-of-fiberwise-cocompletion2} \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}}& } \end{gathered} \end{equation}
which exhibits $\widehat{\operatorname{\mathcal{E}}}'$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, then Remark 8.7.1.15 guarantees that there is an equivalence $F: \widehat{\operatorname{\mathcal{E}}} \rightarrow \widehat{\operatorname{\mathcal{E}}}'$ of inner fibrations over $\operatorname{\mathcal{C}}$ such that $H'$ is isomorphic to $F \circ H$ as an object of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \widehat{\operatorname{\mathcal{E}}}' )$. It then follows that (8.104) also satisfies conditions $(a)$, $(b)$, $(c)$, and $(d)$ of Theorem 8.7.2.1.
$\square$