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9.3.4 Existence of Fiberwise Cocompletions

Our goal in this section is to prove the following existence result for fiberwise cocompletions:

Theorem 9.3.4.1. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. 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 $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ (in the sense of Definition 9.3.1.12).

Our strategy for proving Theorem 9.3.4.1 is to reduce to the special case where $U$ is a cocartesian fibration, which was treated in ยง9.3.2 (see Theorem 9.3.2.1). To carry out the reduction, we will need the following relative version of Corollary 8.4.6.10:

Lemma 9.3.4.2. Let $\mathbb {K}$ be a collection of simplicial sets, let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, and and suppose we are given a commutative diagram

9.44
$$\begin{gathered}\label{equation:cocompletion-of-full-subcategory-revisited} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [rr]^-{H'} \ar [dr]_{U'} & & \widehat{\operatorname{\mathcal{E}}}' \ar [dl]^{ \widehat{U}' } \\ & \operatorname{\mathcal{C}}& } \end{gathered}$$

which exhibits $\widehat{\operatorname{\mathcal{E}}}'$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}'$. Let $\operatorname{\mathcal{E}}$ be a full simplicial subset of $\operatorname{\mathcal{E}}'$ and let $\widehat{\operatorname{\mathcal{E}}} \subseteq \widehat{\operatorname{\mathcal{E}}}'$ be the full simplicial subset spanned by those vertices $Y$ with the following property:

• If $C = \widehat{U}'(Y)$, then $Y$ belongs to the smallest full subcategory of $\widehat{\operatorname{\mathcal{E}}}'_{C}$ which contains the essential image of $H|_{ \operatorname{\mathcal{E}}_{C} }$ and is closed under $K$-indexed colimits, for each $K \in \mathbb {K}$.

Set $H = H'|_{\operatorname{\mathcal{E}}}$, $U = U'|_{\operatorname{\mathcal{E}}}$, and $\widehat{U} = \widehat{U}'|_{\widehat{\operatorname{\mathcal{E}}}}$. Then the diagram

9.45
$$\begin{gathered}\label{equation:cocompletion-of-full-subcategory-revisited2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U}} \\ & \operatorname{\mathcal{C}}& } \end{gathered}$$

exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$; in this case, we must show that the diagram (9.45) satisfies conditions $(1)$ through $(4)$ of Definition 9.3.1.8. Condition $(1)$ follows from Corollary 8.4.6.10, and the remaining conditions follow immediately from the corresponding conditions on the diagram (9.44) $\square$

We now prove a special case of Theorem 9.3.4.1, which is already sufficient for most applications:

Lemma 9.3.4.3. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an isofibration of $\infty$-categories Then there exists a commutative diagram of $\infty$-categories

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

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ (in the sense of Definition 9.3.1.12). Moreover, $\widehat{U}$ is also an isofibration.

Proof. Choose a factorization of $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {F} \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $U'$ is a cocartesian fibration of $\infty$-categories and $F$ is fully faithful. For example, we can take $\operatorname{\mathcal{E}}'$ to be the oriented fiber product $\operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be the projection onto the second factor (which is a cocartesian fibration by virtue of Corollary 5.3.7.3), and $F$ to be the inclusion map

$\operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$

(which is fully faithful by virtue of Corollary 4.5.2.22). Let $\operatorname{\mathcal{E}}'_0 \subseteq \operatorname{\mathcal{E}}'$ be the essential image of $F$. Since $U'$ is an isofibration (Proposition 5.1.4.8), it restricts to an isofibration $\operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{C}}$. Applying Proposition 5.1.7.5, we see that the functor $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'_0$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$. We may therefore replace $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{E}}'_0$ and thereby reduce to the special case where $\operatorname{\mathcal{E}}$ is a replete full subcategory of $\operatorname{\mathcal{E}}'$.

Applying Theorem 9.3.2.1, we deduce that there is 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 $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}'$. Moreover, the morphism $\widehat{U}'$ is a cocartesian fibration (and therefore an isofibration; see Proposition 5.1.4.8). Define $\widehat{\operatorname{\mathcal{E}}} \subseteq \widehat{\operatorname{\mathcal{E}}}'$ as in the statement of Lemma 9.3.4.2. Then $\widehat{\operatorname{\mathcal{E}}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{E}}}'$, so the restriction map $\widehat{U} = \widehat{U}'|_{\widehat{\operatorname{\mathcal{E}}} }$ is an isofibration. Moreover, Lemma 9.3.4.2 guarantees that the diagram

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

exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. $\square$

We now prove a relative version of Theorem 9.3.4.1.

Lemma 9.3.4.4. Let $\mathbb {K}$ be a collection of simplicial sets, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset. Set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be the projection onto the first factor, and suppose we are given a diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [rr]^-{H_0} \ar [dr]_{ U_0 } & & \widehat{\operatorname{\mathcal{E}}}_0 \ar [dl]^{ \widehat{U}_0 } \\ & \operatorname{\mathcal{C}}_0 & }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}_0$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_0$. Then 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 $\operatorname{\mathcal{E}}$ and an isomorphism $\widehat{\operatorname{\mathcal{E}}}_0 \simeq \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ which carries $H_0$ to the map $(U_0, H): \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$.

Proof. Using Proposition 1.1.4.12, we can reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex and $\operatorname{\mathcal{C}}_0 = \operatorname{\partial \Delta }^ n$ is its boundary. In this case, $U$ is an isofibration of $\infty$-categories (Example 4.4.1.6). Applying Lemma 9.3.4.3, we deduce 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 $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Set $\widehat{\operatorname{\mathcal{E}}}'_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$, so that $H'_0 = H'|_{\operatorname{\mathcal{E}}_0}$ exhibits $\widehat{\operatorname{\mathcal{E}}}'_0$ as a fiberwise $\mathbb {K}$-completion of $\operatorname{\mathcal{E}}_0$. Applying Remark 9.3.1.21, we deduce that there exists an isomorphism $H'_0 \simeq F_0 \circ H_0$ in the $\infty$-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{E}}_0, \widehat{\operatorname{\mathcal{E}}}'_0 )$, where $F_0: \widehat{\operatorname{\mathcal{E}}}_0 \rightarrow \widehat{\operatorname{\mathcal{E}}}'_0$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}_0$. Using Lemma 5.6.7.1, we can choose an inner fibration $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ and an equivalence $F: \widehat{\operatorname{\mathcal{E}}} \rightarrow \widehat{\operatorname{\mathcal{E}}}'$ satisfying $F_0 = F|_{\widehat{\operatorname{\mathcal{E}}}_0}$. Composition with $F$ induces an equivalence of $\infty$-categories $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \widehat{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \widehat{\operatorname{\mathcal{E}}}' )$. We can therefore choose a functor $H: \operatorname{\mathcal{E}}\rightarrow \widehat{\operatorname{\mathcal{E}}}$ such that $F \circ H$ is isomorphic to $H'$ in the $\infty$-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \widehat{\operatorname{\mathcal{E}}}' )$. It follows that $F_0 \circ H|_{ \operatorname{\mathcal{E}}_0 }$ is isomorphic to $F_0 \circ H_0$ as objects of the $\infty$-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{E}}_0, \widehat{\operatorname{\mathcal{E}}}'_0 )$. Since $F_0$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}_0$, it follows that $H_0$ and $H|_{\operatorname{\mathcal{E}}_0}$ are isomorphic as objects of the $\infty$-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{E}}_0, \widehat{\operatorname{\mathcal{E}}}_0 )$. Since the restriction functor $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \widehat{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_0 }( \operatorname{\mathcal{E}}_0, \widehat{\operatorname{\mathcal{E}}}_0 )$ is an isofibration (Proposition 4.1.4.1), we can arrange (after replacing $H$ by an isomorphic functor if necessary) that $H|_{\operatorname{\mathcal{E}}_0} = H_0$. We conclude by observing that the 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 $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. $\square$

Proof of Theorem 9.3.4.1. Apply Lemma 9.3.4.4 in the special case $\operatorname{\mathcal{C}}_0 = \emptyset$. $\square$