9.4.1 Uniqueness of Fiberwise Cocompletions
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Our goal in this section is to show that if there exists a diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr] \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise cocompletion of $\operatorname{\mathcal{E}}$ (in the sense of Definition 9.4.0.1), then the inner fibration $\widetilde{U}$ is uniquely determined up to equivalence. To prove this, we will show that the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}$ can be characterized by a universal mapping property (Theorem 9.4.1.20). To formulate it, we will need some terminology.
Definition 9.4.1.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\mathbb {K}$ be a collection of simplicial sets. We say that $U$ is $\mathbb {K}$-cocomplete if it satisfies the following conditions:
- $(1)$
For every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete (Definition 8.4.5.1). That is, it admits $K$-indexed colimits, for every simplicial set $K \in \mathbb {K}$.
- $(2)$
Let $C$ be an object of $\operatorname{\mathcal{C}}$ and let $K \in \mathbb {K}$. Then every colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is a $U$-colimit diagram in $\operatorname{\mathcal{E}}$.
If $\kappa $ is a regular cardinal, we say that the inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it $\mathbb {K}$-cocomplete, where $\mathbb {K}$ denotes the collection of all $\kappa $-small simplicial sets.
Example 9.4.1.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\mathbb {K}$ be a collection of simplicial sets. Then $\operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete (in the sense of Definition 8.4.5.1) if and only if the inner fibration $\operatorname{\mathcal{E}}\rightarrow \Delta ^0$ is $\mathbb {K}$-cocomplete (in the sense of Definition 9.4.1.1).
Proposition 9.4.1.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\mathbb {K}$ be a collection of simplicial sets. Then $U$ is $\mathbb {K}$-cocomplete if and only if, for every morphism $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the inner fibration $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is $\mathbb {K}$-cocomplete.
Proof.
This follows immediately from the characterization of $U$-colimit diagrams supplied by Proposition 7.1.6.24.
$\square$
Using Proposition 9.4.1.3, we can formulate a slightly more general version of Definition 9.4.1.1:
Variant 9.4.1.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\mathbb {K}$ be a collection of simplicial sets. We say that $U$ is $\mathbb {K}$-cocomplete if, for every edge $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the projection map $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is a $\mathbb {K}$-cocomplete inner fibration of $\infty $-categories, in the sense of Definition 9.4.1.1. By virtue of Proposition 9.4.1.3, this agrees with Definition 9.4.1.1 in the special case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.
Example 9.4.1.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of simplicial sets, and let $\mathbb {K}$ be a collection of simplicial sets. Then $U$ is $\mathbb {K}$-cocomplete if and only if, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$, so that $U$ is a cartesian fibration. In this case, the desired result follows from Corollary 7.1.6.22.
Example 9.4.1.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. We say that $U$ is idempotent complete if it satisfies either of the following equivalent conditions:
- $(a)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is idempotent complete.
- $(b)$
The inner fibration $U$ is $\{ \operatorname{N}_{\bullet }( \operatorname{Idem}) \} $-cocomplete, where $\operatorname{Idem}$ is the category introduced in Construction 8.5.2.7.
The implication $(b) \Rightarrow (a)$ is immediate (see the proof of Proposition 8.5.5.2). To prove the converse, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$; in this case, the result follows from the observation that the inclusion map $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}$ preserves $\operatorname{N}_{\bullet }(\operatorname{Idem})$-indexed colimiits (Corollary 8.5.3.12).
We will soon show that if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an arbitrary inner fibration, then there is a universal way to enlarge $\operatorname{\mathcal{E}}$ to obtain a $\mathbb {K}$-cocomplete inner fibration $\widetilde{U}: \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$. We begin by characterizing this enlargement in the special case $\operatorname{\mathcal{C}}= \Delta ^1$.
Definition 9.4.1.8. Let $\mathbb {K}$ be a collection of simplicial sets and suppose that we are given a commutative diagram of $\infty $-categories
9.28
\begin{equation} \begin{gathered}\label{equation:relative-cocompletion-over-1-simplex} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1. & } \end{gathered} \end{equation}
We say that the diagram (9.28) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ if the following conditions are satisfied:
- $(1)$
For $i \in \{ 0,1\} $, the map of fibers $H_{i}: \operatorname{\mathcal{E}}_{i} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{i}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{i}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{i}$ (Definition 8.4.5.1).
- $(2)$
The functor $H$ is fully faithful.
- $(3)$
The inner fibration $\widehat{U}$ is $\mathbb {K}$-cocomplete (Definition 9.4.1.1).
- $(4)$
For every object $X \in \operatorname{\mathcal{E}}_0$, the functor
\[ \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{\mathcal{S}}\quad \quad Y \mapsto \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }( H(X), Y) \]
preserves $\mathbb {K}$-indexed colimits.
If $\kappa $ is a regular cardinal, we say that (9.28) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ if it exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets.
Proposition 9.4.1.11. Let $\kappa $ be an uncountable regular cardinal and suppose we are given a commutative diagram of $\infty $-categories
9.29
\begin{equation} \begin{gathered}\label{equation:compare-notions-of-cocompletion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1, & } \end{gathered} \end{equation}
where $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Then (9.29) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ if and only if it satisfies conditions $(1)$ and $(2)$ of Definition 9.4.1.8, together with the following:
- $(3')$
The functor $\widehat{U}$ is a cartesian fibration.
- $(4')$
The contravariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \widehat{\operatorname{\mathcal{E}}}_0$ preserves $\kappa $-small colimits.
Proof.
Without loss of genearlity, we may assume that $H$ satisfies conditions $(1)$ and $(2)$ of Definition 9.4.1.8. The implication $(3') \Rightarrow (3)$ is a special case of Example 9.4.1.6. Moreover, if $(3')$ is satisfied, then we can use condition $(1)$ (together with Proposition 8.4.2.5) to identify the contravariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \widehat{\operatorname{\mathcal{E}}}_0$ with the functor
\[ \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \quad \quad \widehat{Y} \mapsto \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }( H(-), \widehat{Y} ). \]
In this case, the equivalence $(4) \Leftrightarrow (4')$ follows from the observation that $\kappa $-small colimits in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ are computed levelwise (Proposition 7.1.7.2).
We will complete the proof by showing that if $H$ satisfies conditions $(3)$ and $(4)$, then $U$ is a cartesian fibration. Fix a regular cardinal $\lambda $ having exponential cofinality $\geq \kappa $ such that $\widehat{\operatorname{\mathcal{E}}}$ is locally $\lambda $-small. By virtue of Corollary 6.2.3.2, it will suffice to show that for each object $\widehat{Y} \in \widehat{\operatorname{\mathcal{E}}}_{1}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{E}}}_0^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda } \quad \quad \widehat{X} \mapsto \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }(\widehat{X}, \widehat{Y}) \]
is representable by an object of $\widehat{\operatorname{\mathcal{E}}}_{0}$. Condition $(3)$ guarantees that $\mathscr {F}$ preserves $\kappa $-small limits. Using condition $(1)$ and Corollary 8.4.3.10, we are reduced to showing that for each object $X \in \operatorname{\mathcal{E}}_{0}$, the Kan complex $\mathscr {F}( H(X) ) = \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} )$ is $\kappa $-small.
Let now us regard the object $X$ as fixed. Condition $(4)$ guarantees that the functor
\[ \widehat{\operatorname{\mathcal{E}}}_{0} \rightarrow \operatorname{\mathcal{S}}^{< \lambda } \quad \quad \widehat{Y} \mapsto \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} ) \]
preserves $\kappa $-small colimits. In particular, the collection of objects $\widehat{Y}$ for which $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} )$ is essentially $\kappa $-small is closed under $\kappa $-small colimits (Example 7.6.6.8). Since $\widehat{\operatorname{\mathcal{E}}}_{1}$ is generated under $\kappa $-small colimits by the image of $H$, we can assume that $\widehat{Y} = H(Y)$ for some object $Y \in \operatorname{\mathcal{E}}_1$. In this case, condition $(2)$ supplies a homotopy equivalence
\[ \operatorname{Hom}_{ \operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} ), \]
so the desired result follows from our assumption that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small.
$\square$
We now extend the scope of Definition 9.4.1.8.
Definition 9.4.1.12. Let $\mathbb {K}$ be a collection of simplicial sets and suppose we are given a commutative diagram of simplicial sets
9.30
\begin{equation} \begin{gathered}\label{equation:relative-cocompletion-general} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \end{gathered} \end{equation}
where $U$ and $\widehat{U}$ are inner fibrations. We will say that the diagram (9.30) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ if, for every edge $e: \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$, the induced map
\[ H_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \]
exhibits $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (in the sense of Definition 9.4.1.8).
If $\kappa $ is a regular cardinal, we say that the diagram (9.30) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ if it exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets.
We now show that, in the special case where $\mathbb {K}$ is the collection of small simplicial sets and $U$ is an essentially small inner fibration of $\infty $-categories, Definition 9.4.1.12 reduces to Definition 9.4.0.1. Following the convention of Remark 4.7.0.5, this can be regarded as a special case of the following:
Proposition 9.4.1.16. Let $\kappa $ be an uncountable regular cardinal and suppose we are given a commutative diagram of $\infty $-categories
9.31
\begin{equation} \begin{gathered}\label{equation:compare-notions-of-cocompletion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \end{gathered} \end{equation}
where $U$ and $\widehat{U}$ are inner fibrations and $U$ is essentially $\kappa $-small. Then (9.31) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ (in the sense of Definition 9.4.1.12) if and only if the following conditions are satisfied:
- $(1)$
For each object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{E}}_{C}$.
- $(2)$
The functor $H$ is fully faithful.
- $(3)$
The inner fibration $\widehat{U}$ is locally cartesian.
- $(4)$
For each morphism $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $f^{\ast }: \widehat{\operatorname{\mathcal{E}}}_{C'} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ preserves $\kappa $-small colimits.
Proof.
Using Remark 9.4.1.14, we can reduce to the special case $\operatorname{\mathcal{C}}= \Delta ^1$, which follows from Proposition 9.4.1.11.
$\square$
Example 9.4.1.17 (Fiberwise Idempotent Completion). Suppose we are given a commutative diagram of simplicial sets
9.32
\begin{equation} \begin{gathered}\label{equation:fiberwise-idempotent-completion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \end{gathered} \end{equation}
where $U$ and $\widehat{U}$ are inner fibrations. We say that (9.32) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$ if it exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a $\{ \operatorname{N}_{\bullet }( \operatorname{Idem}) \} $-cocompletion of $\operatorname{\mathcal{E}}$, in the sense of Definition 9.4.1.12; here $\operatorname{Idem}$ is the category introduced in Construction 8.5.2.7.
Proposition 9.4.1.18. Suppose we are given a commutative diagram of $\infty $-categories
9.33
\begin{equation} \begin{gathered}\label{equation:fiberwise-idempotent-completion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \end{gathered} \end{equation}
where $U$ and $\widehat{U}$ are inner fibrations. Then (9.33) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$ if and only if it satisfies the following pair of conditions:
- $(1)$
For each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}_{C}$ is idempotent complete.
- $(2)$
The functor $H$ is fully faithful.
Proof.
As in Proposition 9.4.1.16, we can use Remark 9.4.1.14 to reduce to the case $\operatorname{\mathcal{C}}= \Delta ^1$; in this case, the result follows from Example 9.4.1.7.
$\square$
We can now formulate our main result.
Notation 9.4.1.19. Let $\mathbb {K}$ be a collection of simplicial sets, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be $\mathbb {K}$-cocomplete inner fibrations of simplicial sets. We let $\operatorname{Fun}^{\mathbb {K}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' )$ denote the full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}')$ spanned by those commutative diagrams
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ F } \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}& } \]
having the property that, for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
Theorem 9.4.1.20. Let $\mathbb {K}$ be a collection of simplicial sets. Suppose we are given a diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Then, for every $\mathbb {K}$-cocomplete inner fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$, precomposition with $H$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}^{\mathbb {K}}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}'). \]
Example 9.4.1.22. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$. Then, for any idempotent complete inner fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$, precomposition with $H$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ). \]
This follows by combining Theorem 9.4.1.20 with Corollary 8.5.3.12.
Our proof of Theorem 9.4.1.20 will require some preliminaries.
Lemma 9.4.1.23. Let $\widehat{\operatorname{\mathcal{D}}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ be a full subcategory. Suppose that the inclusion functor $\operatorname{\mathcal{D}}\hookrightarrow \widehat{\operatorname{\mathcal{D}}}$ exhibits $\widehat{\operatorname{\mathcal{D}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{D}}$, for some collection of simplicial sets $\mathbb {K}$. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocomplete inner fibration of $\infty $-categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $F: \widehat{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{E}}_{C}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
- $(2)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{D}}$.
- $(3)$
When regarded as a functor from $\widehat{\operatorname{\mathcal{D}}}$ to $\operatorname{\mathcal{E}}$, the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{D}}$.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Lemma 8.4.5.9, and the implication $(3) \Rightarrow (2)$ is a special case of Corollary 7.3.3.23. We will complete the proof by showing that $(1)$ implies $(3)$. By virtue of Remark 7.3.3.24, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $C$ is the initial vertex; in this case, we wish to show that $F$ is left Kan extended from $\operatorname{\mathcal{D}}$ (when regarded as a functor from $\widehat{\operatorname{\mathcal{D}}}$ to $\operatorname{\mathcal{E}}$). Fix an uncountable regular cardinal $\kappa $ such that every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. Let $\lambda $ be a cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{E}}$ is locally $\lambda $-small. By virtue of Proposition 7.4.1.18, it will suffice to show that for every object $E \in \operatorname{\mathcal{E}}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{D}}} \rightarrow (\operatorname{\mathcal{S}}^{< \lambda })^{\operatorname{op}} \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{E}}}( F(D), E ) \]
is left Kan extended from $\operatorname{\mathcal{D}}$. By virtue of Lemma 8.4.5.9, this is equivalent to the requirement that $\mathscr {F}$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. This follows from $(1)$ together with our assumption that $U$ is $\mathbb {K}$-cocomplete.
$\square$
Lemma 9.4.1.24. Let $\mathbb {K}$ be a collection of simplicial sets and suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Let $X$ be an object of $\widehat{\operatorname{\mathcal{E}}}$ having image $C = \widehat{U}( X )$ and set $\operatorname{\mathcal{E}}_{ / X} = \operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/ X}$. If $U$ is an isofibration, then the inclusion map
\[ \{ \operatorname{id}_{C} \} \times _{ \operatorname{\mathcal{C}}_{/C} } \operatorname{\mathcal{E}}_{ / X } \hookrightarrow \operatorname{\mathcal{E}}_{ /X } \]
is right cofinal.
Proof.
Fix an object $\widetilde{Y} \in \operatorname{\mathcal{E}}_{ / X }$, which we identify with a pair $(Y,v)$ where $Y$ is an object of $\operatorname{\mathcal{E}}$ and $v: H(Y) \rightarrow X$ is a morphism in $\widehat{\operatorname{\mathcal{E}}}$. By virtue of Theorem 7.2.3.1, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{A}}= \{ \operatorname{id}_{C} \} \times _{ \operatorname{\mathcal{C}}_{/C} } ( \operatorname{\mathcal{E}}_{ / C} )_{\widetilde{Y}/}$ is weakly contractible. Since $\widehat{U}$ is an inner fibration, the map
\[ \operatorname{\mathcal{E}}_{ / X } = \operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/ X } \rightarrow \operatorname{\mathcal{E}}\times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \]
is a right fibration (Proposition 4.3.6.8). Using our assumption that $U$ is an isofibration, we deduce that the map $\operatorname{\mathcal{E}}_{ / X } \rightarrow \operatorname{\mathcal{C}}_{/C}$ is also an isofibration. Let $\widetilde{D}$ denote the image of $\widetilde{Y}$ under this isofibration, which we identify with a morphism $f: D \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (so that $D = U(Y)$ and $f = \widehat{U}(v)$). Set $\operatorname{\mathcal{B}}= \{ \operatorname{id}_{C} \} \times _{ \operatorname{\mathcal{C}}_{/C} } ( \operatorname{\mathcal{C}}_{/C} )_{\widetilde{D} /}$, so that $U$ induces an isofibration
\[ \operatorname{\mathcal{A}}= \{ \operatorname{id}_{C} \} \times _{ \operatorname{\mathcal{C}}_{/C} } (\operatorname{\mathcal{E}}_{ / X})_{\widetilde{Y}/} ) \rightarrow \{ \operatorname{id}_ C \} \times _{ \operatorname{\mathcal{C}}_{/C} } ( \operatorname{\mathcal{C}}_{/C} )_{\widetilde{D}/} = \operatorname{\mathcal{B}}. \]
Since $\operatorname{id}_{C}$ is final when viewed as an object of $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.7.22), the simplicial set $\operatorname{\mathcal{B}}$ is a contractible Kan complex. Let $B \in \operatorname{\mathcal{B}}$ be the vertex which corresponds to the degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & C \ar [dr]_{ \operatorname{id}_{C} } & \\ D \ar [ur]^{f} \ar [rr]^-{f} & & C, } \]
so that the inclusion map $\{ B\} \hookrightarrow \operatorname{\mathcal{B}}$ is an equivalence of $\infty $-categories. Applying Corollary 4.5.2.30, we deduce that the inclusion map $\{ B\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{A}}\hookrightarrow \operatorname{\mathcal{A}}$ is an equivalence of $\infty $-categories. In particular, it is a weak homotopy equivalence. Consequently, to show that $\operatorname{\mathcal{A}}$ is weakly contractible, it will suffice to show that the fiber $\{ B\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{A}}$ is weakly contractible. Replacing $\operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}$ by the fiber products $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$, respectively, we are reduced to proving that $\operatorname{\mathcal{A}}$ is weakly contractible under the additional assumption that $\operatorname{\mathcal{C}}= \Delta ^1$, where $U(Y) = 0$ and $\widehat{U}(X) = 1$.
For $i \in \{ 0,1\} $, set $\operatorname{\mathcal{E}}_{i} = \{ i\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}_{i} = \{ i\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$. 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. Then the projection map $\widehat{\operatorname{\mathcal{E}}}_{ H(Y) / } \rightarrow \widehat{\operatorname{\mathcal{E}}}$ admits a covariant transport representation $\mathscr {F}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, given informally by the formula $\mathscr {F}(E) = \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }( H(Y), E )$. Our assumption that $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ guarantees that the restriction $\mathscr {F}|_{ \widehat{\operatorname{\mathcal{E}}}_{1} }$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$. Applying Corollary 8.4.5.10 (and our assumption that $H$ is fully faithful), we conclude that $H$ induces a left cofinal functor
\[ \operatorname{\mathcal{E}}_1 \times _{ \operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{Y/} \xrightarrow {\sim } \operatorname{\mathcal{E}}_1 \times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{ H(Y) / } \rightarrow \widehat{\operatorname{\mathcal{E}}}_{1} \times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{H(Y) / }. \]
By virtue of Theorem 7.2.3.1, this is a reformulation of the assertion that the $\infty $-category $\operatorname{\mathcal{A}}$ is weakly contractible (for every choice of object $X \in \widehat{\operatorname{\mathcal{E}}}_{1}$).
$\square$
Proposition 9.4.1.25. Let $\mathbb {K}$ be a collection of simplicial sets. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \widehat{\operatorname{\mathcal{E}}} \ar [rr]^-{F} \ar [dr]_{ \widehat{U} } & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}& } \]
and let $\operatorname{\mathcal{E}}\subseteq \widehat{\operatorname{\mathcal{E}}}$ be a full subcategory. Assume that $U = \widehat{U}|_{\operatorname{\mathcal{E}}}$ is an isofibration, that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \widehat{\operatorname{\mathcal{E}}}$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, and that $U'$ is a $\mathbb {K}$-cocomplete inner fibration. The following conditions are equivalent:
- $(1)$
The functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$.
- $(2)$
For each object $C \in \operatorname{\mathcal{C}}$ and each $K \in \mathbb {K}$, the functor $F_{C}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ preserves $K$-indexed colimits.
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$, and let $X$ be an object of $\widehat{\operatorname{\mathcal{E}}}$ satisfying $\widehat{U}(X) = C$. Combining Lemma 9.4.1.24 with Corollary 7.2.2.2, we see that the following conditions are equivalent:
- $(1_ X)$
The functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$ at the object $X \in \widehat{\operatorname{\mathcal{E}}}$.
- $(2_ X)$
The functor $F_ C$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}_{C}$ at the object $X \in \widehat{\operatorname{\mathcal{E}}}_{C}$.
Allowing the object $X$ to vary (with $C$ fixed) and applying Lemma 9.4.1.23, we see that the following are equivalent:
- $(1_ C)$
For each object $X \in \widehat{\operatorname{\mathcal{E}}}$ satisfying $\widehat{U}(X) = C$, the functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$ at the object $X$.
- $(2_ C)$
For each $K \in \mathbb {K}$, the functor $F_{C}$ preserves $K$-indexed colimits.
The equivalence $(1) \Leftrightarrow (2)$ now follows by allowing the object $C \in \operatorname{\mathcal{C}}$ to vary.
$\square$
Variant 9.4.1.26. Let $\mathbb {K}$ be a collection of simplicial sets, let $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories, and let $\operatorname{\mathcal{E}}\subseteq \widehat{\operatorname{\mathcal{E}}}$ be a full subcategory. Assume that $U = \widehat{U}|_{\operatorname{\mathcal{E}}}$ is an isofibration and that the inclusion functor $\operatorname{\mathcal{E}}\hookrightarrow \widehat{\operatorname{\mathcal{E}}}$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocomplete isofibration. Then every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ F^0 } \ar [d] & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \widehat{\operatorname{\mathcal{E}}} \ar@ {-->}[ur]^{F} \ar [r] & \operatorname{\mathcal{C}}} \]
admits a solution, where the functor $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}$.
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$, so that $F^0$ restricts to a functor $F^{0}_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$. Since the inclusion functor $\operatorname{\mathcal{E}}_{C} \hookrightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{C}$, the functor $F^0_{C}$ admits an (essentially unique) extension $F_{C}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ which preserves $K$-indexed colimits for each $K \in \mathbb {K}$. It follows from Lemma 9.4.1.23 that $F_{C}$ is $U'$-left Kan extended from $\operatorname{\mathcal{E}}_{C}$. In particular, for each object $X \in \widehat{\operatorname{\mathcal{E}}}_{C}$, the composition
\[ (\operatorname{\mathcal{E}}_{C} \times _{ \widehat{\operatorname{\mathcal{E}}}_{C} } (\widehat{\operatorname{\mathcal{E}}}_{C})_{/X})^{\triangleright } \widehat{\operatorname{\mathcal{E}}}_{C} \xrightarrow { F_{C} } \operatorname{\mathcal{E}}' \]
is a $U'$-colimit diagram. Since the inclusion map
\[ \operatorname{\mathcal{E}}_{C} \times _{ \widehat{\operatorname{\mathcal{E}}}_{C} } (\widehat{\operatorname{\mathcal{E}}}_{C})_{/X} \hookrightarrow \operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/X} \]
is right cofinal (Lemma 9.4.1.24), Proposition 7.2.2.9 guarantees that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/X} \ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [r]^-{F} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ (\operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{E}}} } \widehat{\operatorname{\mathcal{E}}}_{/X})^{\triangleright } \ar [r] \ar@ {-->}[urr] & \widehat{\operatorname{\mathcal{E}}} \ar [r]^-{ \widehat{U} } & \operatorname{\mathcal{C}}} \]
admits a solution, where the dotted arrow is a $U'$-colimit diagram. The desired result now follows by allowing the object $X$ to vary and applying the criterion of Proposition 7.3.5.5.
$\square$
We now prove a special case of Theorem 9.4.1.20 (which is sufficient for most applications).
Lemma 9.4.1.27. Let $\mathbb {K}$ be a collection of simplicial sets. Suppose we are given a diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \]
where $U$ and $\widehat{U}$ are isofibrations and $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Then, for every $\mathbb {K}$-cocomplete isofibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$, precomposition with $H$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}^{\mathbb {K}}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}'). \]
Proof.
Since the functor $H$ is fully faithful (see Variant 4.8.6.19), we can assume without loss of generality that $\operatorname{\mathcal{E}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{E}}}$ (and that $H$ is the inclusion functor). In this case, Proposition 9.4.1.25 identifies $\operatorname{Fun}^{\mathbb {K}}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' )$ with the full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' )$ spanned by those functors $F: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}'$ which are $U'$-left Kan extended from $\operatorname{\mathcal{E}}$. Combining this observation with Theorem 7.3.6.14 and Variant 9.4.1.26, we see that the restriction functor $\operatorname{Fun}^{\mathbb {K}}_{ / \operatorname{\mathcal{C}}}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}')$ is a trivial Kan fibration.
$\square$
Proof of Theorem 9.4.1.20.
For every morphism of simplicial sets $S \rightarrow \operatorname{\mathcal{C}}$, we let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( S \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' )$ denote the (replete) full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( S \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' )$ spanned by those morphisms $F: S \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}'$ having the property that, for each vertex $s \in S$, the functor $F_{s}: \{ s\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \rightarrow \{ s\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. We will prove the following:
- $(\ast )$
For every morphism of simplicial sets $S \rightarrow \operatorname{\mathcal{C}}$, precomposition with $H$ induces an equivalence of $\infty $-categories
\[ \theta _{S}: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( S \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( S \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ). \]
Writing $S$ as the union of its skeleta $\operatorname{sk}_{n}(S)$, we can realize $\theta _{S}$ as the limit of a tower of comparison maps
\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( \operatorname{sk}_1(S) \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \ar [d]^{ \theta _{ \operatorname{sk}_1(S) }} \ar [r] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( \operatorname{sk}_0(S) \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \ar [d]^{ \theta _{ \operatorname{sk}_0(S) }} \\ \cdots \ar [r] & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{sk}_1(S) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) \ar [r] & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{sk}_0(S) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ), } \]
where the horizontal maps are isofibrations (Variant 4.4.5.11). Consequently, to show that $\theta _{S}$ is an equivalence of $\infty $-categories, it will suffice to show that $\theta _{ \operatorname{sk}_ n(S) }$ is an equivalence of $\infty $-categories for each $n \geq -1$. We may therefore assume without loss of generality that $S$ has dimension $\leq n$, for some integer $n \geq -1$.
We now proceed by induction on $n$. Assume that $n \geq 0$ (otherwise, $S$ is empty and there is nothing to prove). Let $S'$ be the $(n-1)$-skeleton of $S$, so that Proposition 1.1.4.12 supplies a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ T' \ar [r] \ar [d] & T \ar [d] \\ S' \ar [r] & S } \]
where $T$ is a coproduct of standard $n$-simplices (indexed by the nondegenerate $n$-simplices of $S$) and $T'$ is the coproduct of their boundaries. It follows that $\theta _{S}$ can be realized as the horizontal limit of a diagram of comparison maps
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( S' \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \ar [d]^{\theta _{S'}} \ar [r] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( T' \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \ar [d]^{ \theta _{T'} } \ar [r] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( T \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{E}}' ) \ar [d]^{\theta _{T}} \\ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( S' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) \ar [r] & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( T' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ) & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( T \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ), \ar [l] } \]
where the horizontal maps on the right are isofibrations (Variant 4.4.5.11). Since the simplicial sets $S'$ and $T'$ have dimension $< n$, our inductive hypothesis guarantees that $\theta _{S'}$ and $\theta _{T'}$ are equivalences of $\infty $-categories. Consequently, to show that $\theta _{S}$ is an equivalence of $\infty $-categories, it will suffice to show that $\theta _{T}$ is an equivalence of $\infty $-categories (Corollary 4.5.2.30). Replacing $\operatorname{\mathcal{C}}$ by $T$, we are reduced to proving Theorem 9.4.1.20 in the special case where $\operatorname{\mathcal{C}}$ is a coproduct of standard simplices. In this case, $\operatorname{\mathcal{C}}$ is an $\infty $-category and the inner fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$, and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ are automatically isofibrations (Example 4.4.1.6), so the desired result follows from Lemma 9.4.1.27.
$\square$