8.4.9 Cocompletion and Right Fibrations
The goal of this section is to show that the formation of cocompletions preserves right fibrations, at least up to equivalence. Our main result can be stated more precisely as follows:
Theorem 8.4.9.1. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories. Then there exists a categorical pullback diagram of $\infty $-categories
8.68
\begin{equation} \begin{gathered}\label{equation:slice-cocompletion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{h}} & \widehat{\operatorname{\mathcal{E}}} \ar [d]^{ \widehat{U} } \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \widehat{\operatorname{\mathcal{C}}} } \end{gathered} \end{equation}
satisfying the following conditions:
- $(a)$
The functor $\widehat{U}$ is a right fibration.
- $(b)$
The functor $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$.
- $(c)$
The $\infty $-category $\widehat{\operatorname{\mathcal{E}}}$ is $\mathbb {K}$-cocomplete and the functor $\widehat{U}$ is $\mathbb {K}$-cocontinuous.
Moreover, for any diagram satisfying these conditions, the functor $\widetilde{h}$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$.
We will give the proof of Theorem 8.4.9.1 at the end of this section.
Example 8.4.9.3. Let $\mathbb {K}$ be a collection of simplicial sets, let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$, and suppose we are given a diagram $Q: L \rightarrow \widehat{\operatorname{\mathcal{C}}}$. It follows from Corollary 7.1.4.27 that the slice $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{/Q}$ is $\mathbb {K}$-cocomplete and that the forgetful functor $\widehat{\operatorname{\mathcal{C}}}_{/Q} \rightarrow \widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocontinuous. Applying Theorem 8.4.9.1 to the (categorical) pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\times _{ \widehat{\operatorname{\mathcal{C}}}} \widehat{\operatorname{\mathcal{C}}}_{/Q} \ar [d] \ar [r] & \widehat{\operatorname{\mathcal{C}}}_{/Q} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \widehat{\operatorname{\mathcal{C}}}, } \]
we deduce that projection onto the second factor exhibits $\widehat{\operatorname{\mathcal{C}}}_{/Q}$ as a $\mathbb {K}$-cocompletion of the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{/Q}$.
Example 8.4.9.4 (Cocompletions of Slice $\infty $-Categories). In the situation of Example 8.4.9.3, suppose that the diagram $Q$ factors through $\operatorname{\mathcal{C}}$: that is, we have $Q = h \circ q$ for some diagram $q: L \rightarrow \operatorname{\mathcal{C}}$. Since the functor $h$ is fully faithful, the induced map $\operatorname{\mathcal{C}}_{/q} \rightarrow \operatorname{\mathcal{C}}\times _{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{/Q}$ is an equivalence of $\infty $-categories. It follows that the functor $h_{/q}: \operatorname{\mathcal{C}}_{/q} \rightarrow \widehat{\operatorname{\mathcal{C}}}_{ / Q}$ exhibits $\widehat{\operatorname{\mathcal{C}}}_{/Q}$ as a $\mathbb {K}$-cocompletion of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/q}$.
Example 8.4.9.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration between essentially small $\infty $-categories and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for the left fibration $U^{\operatorname{op}}$. It follows from Corollary 8.4.2.7 that there is a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{h} } \ar [d]^{U} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}} } & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), } \]
where $h_{\bullet }^{\operatorname{\mathcal{C}}}$ is the covariant Yoneda embedding of $\operatorname{\mathcal{C}}$. Since $h_{\bullet }^{\operatorname{\mathcal{C}}}$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as a cocompletion of $\operatorname{\mathcal{C}}$ (Theorem 8.4.0.3), Theorem 8.4.9.1 guarantees that $\widetilde{h}$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} }$ as a cocompletion of the $\infty $-category $\operatorname{\mathcal{E}}$.
Corollary 8.4.9.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration between essentially small $\infty $-categories and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for the left fibration $U^{\operatorname{op}}$. Then there exists an equivalence of $\infty $-categories $T: \operatorname{Fun}( \operatorname{\mathcal{E}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} }$ for which the diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dd]^{U} \ar [r]^-{ h_{\bullet }^{\widetilde{\operatorname{\mathcal{C}}}} }& \operatorname{Fun}( \operatorname{\mathcal{E}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \ar [d]^{T} \\ & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}}} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) } \]
commutes up to isomorphism. Here $h_{\bullet }^{\operatorname{\mathcal{C}}}$ and $h_{\bullet }^{ \widetilde{\operatorname{\mathcal{C}}} }$ denote covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$, respectively.
Proof.
By virtue of Theorem 8.4.0.3, this is a reformulation of Example 8.4.9.5.
$\square$
We now turn to the proof of Theorem 8.4.9.1.
Lemma 8.4.9.7. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ be $\infty $-categories which are $\mathbb {K}$-cocomplete, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration which is $\mathbb {K}$-cocontinuous. Let $X \in \operatorname{\mathcal{E}}$ be an object having image $C = U(X)$. Fix an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ are locally $\kappa $-small and each $K \in \mathbb {K}$ is $\kappa $-small, so that $C$ and $X$ determine corepresentable functors
\[ h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa } \quad \quad h^{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa }. \]
If $h^{C}$ is $\mathbb {K}$-cocontinuous, then $h^{X}$ is $\mathbb {K}$-cocontinuous.
Proof.
Suppose we are given a colimit diagram $\mathscr {F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ for some $K \in \mathbb {K}$; we wish to show that $h^{X} \circ \mathscr {F}$ is a colimit diagram in $\operatorname{\mathcal{S}}_{< \kappa }$. By virtue of Corollary 7.4.3.14, it will suffice to show that the upper horizontal map in the pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{X/} \times _{\operatorname{\mathcal{E}}} K \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{X/} \times _{\operatorname{\mathcal{E}}} K^{\triangleright } \ar [d] \\ \operatorname{\mathcal{C}}_{C / } \times _{\operatorname{\mathcal{C}}} K \ar [r] & \operatorname{\mathcal{C}}_{C / } \times _{\operatorname{\mathcal{C}}} K^{\triangleright } } \]
is a weak homotopy equivalence of simplicial sets. Since $U$ is a right fibration, the vertical maps appearing in the diagram are Kan fibrations (Corollary 4.3.7.3). It will therefore suffice to show that the bottom horizontal map is a weak homotopy equivalence (Corollary 3.3.7.4). This follows from Corollary 7.4.3.14, together with our assumption that $h^{C}$ preserves the colimit diagram $U \circ \mathscr {F}$.
$\square$
Proof of Theorem 8.4.9.1.
Let $\mathbb {K}$ be a collection of simplicial sets, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Fix a regular cardinal $\kappa $ having the property that each $K \in \mathbb {K}$ is $\kappa $-small. Let $\lambda $ be an uncountable regular cardinal of exponential cofinality $\geq \kappa $, so that the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ admits $\kappa $-small limits (Variant 7.4.1.15). Enlarging $\lambda $ if necessary, we may assume that $\widehat{\operatorname{\mathcal{C}}}$ is locally $\lambda $-small and that the right fibration $U$ is essentially $\lambda $-small, so that $U^{\operatorname{op}}$ admits a covariant transport representation $F: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$. Since $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$, we may assume without loss of generality that $F^{\operatorname{op}}$ factors as a composition $\operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}} \xrightarrow { \widehat{F}^{\operatorname{op}}} \operatorname{\mathcal{S}}_{< \lambda }^{\operatorname{op}}$, where the functor $\widehat{F}^{\operatorname{op}}$ is $\mathbb {K}$-cocontinuous. Then we can identify $\widehat{F}$ with the covariant transport representation of $\widehat{U}^{\operatorname{op}}$, where $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \widehat{\operatorname{\mathcal{C}}}$ is a right fibration of $\infty $-categories. Since $\widehat{F}$ is $\mathbb {K}$-cocontinuous, the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}$ is $\mathbb {K}$-cocomplete and the right fibration $\widehat{U}$ is $\mathbb {K}$-cocontinuous (Corollary 7.4.1.21). By construction, there exists a categorical pullback square
8.69
\begin{equation} \begin{gathered}\label{equation:slice-cocompletion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{h}} & \widehat{\operatorname{\mathcal{E}}} \ar [d]^{ \widehat{U} } \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \widehat{\operatorname{\mathcal{C}}}. } \end{gathered} \end{equation}
We will complete the proof by showing that $\widetilde{h}$ satisfies the hypotheses of Variant 8.4.8.9, and therefore exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$:
- $(1)$
The functor $\widetilde{h}$ is fully faithful. This is a special case of Remark 4.8.3.31, since the diagram (8.69) is a categorical pullback square and the functor $h$ is fully faithful.
- $(2)$
Choose an uncountable regular cardinal $\kappa $ such that $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{E}}}$ are locally $\kappa $-small, and each simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. Let $X$ be an object of $\operatorname{\mathcal{E}}$ having image $C \in \operatorname{\mathcal{C}}$ and let $\mathscr {F}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{S}}_{< \kappa }$ be the functor corepresented by $\widetilde{h}(X)$; we wish to show that $\mathscr {F}$ is $\mathbb {K}$-cocontinuous. This is a special case of Lemma 8.4.9.7, since $\widehat{U}( \widetilde{h}(X) ) = h( C )$ corepresents a $\mathbb {K}$-cocontinuous functor $\widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}_{< \kappa }$.
- $(3)$
Let $\widehat{\operatorname{\mathcal{E}}}'$ denote the smallest replete full subcategory of $\widehat{\operatorname{\mathcal{E}}}$ which contains the essential image of $\widetilde{h}$ and is closed under the formation of $K$-indexed colimits for $K \in \mathbb {K}$. We wish to show that $\widehat{\operatorname{\mathcal{E}}}' = \widehat{\operatorname{\mathcal{E}}}$. Let $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ denote the full subcategory spanned by those objects $X \in \widehat{\operatorname{\mathcal{C}}}$ having the property that every object $\widetilde{X} \in \widehat{\operatorname{\mathcal{E}}}$ lying over $X$ belongs to $\widehat{\operatorname{\mathcal{E}}}'$. We will complete the proof by showing that $\widehat{\operatorname{\mathcal{C}}}' = \widehat{\operatorname{\mathcal{C}}}$. Since the diagram (8.68) is a categorical pullback square, $\widehat{\operatorname{\mathcal{C}}}'$ contains the essential image of the functor $h$. It will therefore suffice to show that $\widehat{\operatorname{\mathcal{C}}}'$ is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$. Fix a colimit diagram $g: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{C}}}$ carrying the cone point of $K^{\triangleright }$ to an object $X \in \widehat{\operatorname{\mathcal{C}}}$. Assume that $g|_{K}$ factors through $\widehat{\operatorname{\mathcal{C}}}'$; we wish to show that $X$ also belongs to $\widehat{\operatorname{\mathcal{C}}}'$. Let $\widetilde{X}$ be an object of $\widehat{\operatorname{\mathcal{E}}}$ lying over $X$. Since the inclusion of the cone point into $K^{\triangleright }$ is right anodyne (Example 4.3.7.11), we can lift $g$ to a diagram $\widetilde{g}: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{E}}}$ carrying the cone point to $\widetilde{X}$ (Proposition 4.2.4.5). The assumption that $g|_{K}$ factors through $\widehat{\operatorname{\mathcal{C}}}'$ guarantees that $\widetilde{g}|_{K}$ factors through $\widehat{\operatorname{\mathcal{E}}}'$. Since $g$ is a colimit diagram, the $\mathbb {K}$-cocontinuity of $\widehat{U}$ guarantees that $\widetilde{g}$ is also a colimit diagram. It follows that $\widetilde{X}$ belongs to $\widehat{\operatorname{\mathcal{E}}}'$. Allowing the object $\widetilde{X}$ to vary, we conclude that $X$ belongs to $\widehat{\operatorname{\mathcal{C}}}'$, as desired.
$\square$