9.4.2 Fiberwise Cocompletions of Cocartesian Fibrations
We now specialize our theory of fiberwise cocompletions to the setting of cocartesian fibrations. In this case, Definition 9.4.1.12 can be reformulated:
Theorem 9.4.2.1. 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
9.35
\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 8.4.5.1).
- $(b)$
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 $f_{!}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C'}$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$.
- $(d)$
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 (9.35) which satisfies these conditions.
Our proof of Theorem 9.4.2.1 will require some preliminaries.
Lemma 9.4.2.2. Let $\mathbb {K}$ be a collection of simplicial sets and let $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration. Then $\widehat{U}$ is $\mathbb {K}$-cocomplete (in the sense of Variant 9.4.1.4) if and only if it satisfies the following conditions:
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 $\mathbb {K}$-cocomplete.
For each edge $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category (or even that $\operatorname{\mathcal{C}}= \Delta ^1$). In this case, the desired result follows from the characterization of $\widehat{U}$-colimit diagrams supplied by Proposition 7.3.9.2.
$\square$
Lemma 9.4.2.3. 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
9.36
\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 9.4.2.1. 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 8.4.5.3). 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 Lemma 9.4.2.2 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 8.4.6.9.
$\square$
Our next goal is to prove the existence assertion of Theorem 9.4.2.1. 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 9.4.2.1 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 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.
Lemma 9.4.2.4. 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 4.5.9.1. 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
9.37
\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 9.4.2.1.
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 9.4.2.1 is satisfied and condition $(c)$ is automatic (Corollary 7.1.4.22). Condition $(d)$ follows from Proposition 8.6.5.18, and condition $(a)$ from Theorem 8.4.3.3.
$\square$
Example 9.4.2.5. 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 9.4.2.4 (with the roles of $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\vee }$ switched) and Lemma 9.4.2.3, 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 9.4.2.6. 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
9.38
\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 9.4.2.1.
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 9.4.2.4 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 9.4.2.1 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 9.4.2.1. Moreover, for each edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the associated covariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{D}$ is 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 8.4.5.7.
$\square$
Proof of Theorem 9.4.2.1.
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 9.4.2.6, we can choose a commutative diagram
9.39
\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 9.4.2.1. Lemma 9.4.2.3 guarantees any such diagram (9.39) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Conversely, if we are given another diagram
9.40
\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 9.4.1.21 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 (9.40) also satisfies conditions $(a)$, $(b)$, $(c)$, and $(d)$ of Theorem 9.4.2.1.
$\square$