8.4.7 Slices of Cocompletions
Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a functor between essentially small $\infty $-categories. Using Example 8.4.4.5, we see that $U$ admits an essentially unique extension $\operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ which preserves small colimits. Our goal in this section is to show that, up to equivalence, this construction carries right fibrations to right fibrations. More precisely, if $U$ is a right fibration, we show that $\operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is equivalent to the slice $\infty $-category $\operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} }$, where $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is a covariant transport representation for the left fibration $U^{\operatorname{op}}$ (Corollary 8.4.7.2). This is a consequence of the following:
Proposition 8.4.7.1. Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$, let $\mathscr {F} \in \widehat{\operatorname{\mathcal{C}}}$ be an object, and let
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [r]^-{\widetilde{h}} \ar [d] & \widehat{\operatorname{\mathcal{C}}}_{ / \mathscr {F} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \widehat{\operatorname{\mathcal{C}}} } \]
be a categorical pullback square of $\infty $-categories. Then $\widetilde{h}$ exhibits $\widehat{\operatorname{\mathcal{C}}}_{ / \mathscr {F} }$ as a cocompletion of $\widetilde{\operatorname{\mathcal{C}}}$.
Corollary 8.4.7.2. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ 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 $T: \operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} }$ for which the diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [dd]^{U} \ar [r]^-{ h_{\bullet }^{\widetilde{\operatorname{\mathcal{C}}}} }& \operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \ar [d]^{T} \\ & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}}} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) } \]
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 $\widetilde{\operatorname{\mathcal{C}}}$, respectively.
Proof.
Using Corollary 8.4.2.7, we can choose categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [r]^-{ \widetilde{h} } \ar [d]^{U} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}} } & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}). } \]
It follows from Theorem 8.4.0.3 that the functor $\widetilde{h}$ factors (up to isomorphism) as a composition
\[ \widetilde{\operatorname{\mathcal{C}}} \xrightarrow { h_{\bullet }^{ \widetilde{\operatorname{\mathcal{C}}} } } \operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {T} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} }, \]
where the functor $T$ preserves small colimits. To complete the proof, it will suffice to show that $T$ is an equivalence of $\infty $-categories. This is equivalent to the assertion that $\widetilde{h}$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} }$ as a cocompletion of $\widetilde{\operatorname{\mathcal{C}}}$, which is a special case of Proposition 8.4.7.1.
$\square$
Proposition 8.4.7.1 is a special case of the following more general assertion:
Proposition 8.4.7.3. Let $\mathbb {K}$ be a collection of simplicial sets, 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 $\operatorname{\mathcal{C}}$, let $f: L \rightarrow \widehat{\operatorname{\mathcal{C}}}$ be any morphism of simplicial sets, and let
8.60
\begin{equation} \begin{gathered}\label{equation:slice-cocompletion} \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [r]^-{\widetilde{h}} \ar [d] & \widehat{\operatorname{\mathcal{C}}}_{ / f } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \widehat{\operatorname{\mathcal{C}}} } \end{gathered} \end{equation}
be a categorical pullback square of $\infty $-categories. Then $\widetilde{h}$ exhibits $\widehat{\operatorname{\mathcal{C}}}_{ / f }$ as a $\mathbb {K}$-cocompletion of $\widetilde{\operatorname{\mathcal{C}}}$.
Proof.
We will show that $\widetilde{h}$ satisfies the hypotheses of Variant 8.4.6.9:
- $(0)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{ / f }$ is $\mathbb {K}$-cocomplete: that is, it admits $K$-indexed colimits, for each $K \in \mathbb {K}$. This follows from Corollary 7.1.4.21, since the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete.
- $(1)$
The functor $\widetilde{h}$ is fully faithful. This is a special case of Remark 4.6.2.7, since the diagram (8.60) 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 $\widehat{\operatorname{\mathcal{C}}}_{/f}$ are locally $\kappa $-small, and each simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. Choose an object $\widetilde{C} \in \widetilde{\operatorname{\mathcal{C}}}$ having image $C \in \operatorname{\mathcal{C}}$, and let
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad \widetilde{\mathscr {F}}: \widehat{\operatorname{\mathcal{C}}}_{/f} \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \]
be functors corepresented by the objects $X = h(C)$ and $\widetilde{X} = \widetilde{h}( \widetilde{C} )$, respectively. For every simplicial set $K \in \mathbb {K}$, the functor $\mathscr {F}$ preserves $K$-indexed colimits, and we must show that $\widetilde{ \mathscr {F} }$ has the same property. Choose a colimit diagram $\widetilde{g}: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{C}}}_{/f}$; we wish to show that $\widetilde{\mathscr {F}} \circ \widetilde{g}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$. Let $g: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{C}}}$ denote the composition of $\widetilde{g}$ with the projection map. Then $g$ is a colimit diagram in $\widehat{\operatorname{\mathcal{C}}}$ (Corollary 7.1.4.21), so $\mathscr {F} \circ g$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$. Define
\[ \operatorname{\mathcal{E}}= K^{\triangleright } \times _{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{X/} \quad \quad \widetilde{\operatorname{\mathcal{E}}} = K^{\triangleright } \times _{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{X/}. \]
Using Proposition 5.6.6.21, we see that $\mathscr {F} \circ g$ and $\widetilde{\mathscr {F}} \circ \widetilde{g}$ are covariant transport representations for the left fibrations $\operatorname{\mathcal{E}}\rightarrow K^{\triangleleft }$ and $\overline{\operatorname{\mathcal{E}}} \rightarrow K^{\triangleleft }$, respectively. Our assumption that $\mathscr {F} \circ g$ is a colimit diagram guarantees that the inclusion map $K \times _{ K^{\triangleleft } } \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}$ is left cofinal, and we wish to show that the inclusion map $K \times _{ K^{\triangleleft } } \widetilde{\operatorname{\mathcal{E}}} \hookrightarrow \widetilde{\operatorname{\mathcal{E}}}$ is also left cofinal (Corollary 7.4.3.14). This follows from Proposition 7.2.3.12, since the tautological map $\widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}$ is a pullback of the projection map $(\widehat{\operatorname{\mathcal{C}}}_{X/})_{/f} \rightarrow \widehat{\operatorname{\mathcal{C}}}_{X/}$, and therefore a right fibration (Proposition 4.3.6.1).
- $(3)$
Let $\widehat{\operatorname{\mathcal{C}}}'_{/f}$ denote the smallest replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}_{/f}$ 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 $\widehat{\operatorname{\mathcal{C}}}'_{/f} = \widehat{\operatorname{\mathcal{C}}}_{/f}$. 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 $\widetilde{X} \in \widehat{\operatorname{\mathcal{C}}}_{/f}$ lying over $X$ belongs to $\widehat{\operatorname{\mathcal{C}}}_{/f}^{0}$. We will complete the proof by showing that $\widehat{\operatorname{\mathcal{C}}}' = \widehat{\operatorname{\mathcal{C}}}$. Since the diagram (8.60) 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 $\widehat{\operatorname{\mathcal{C}}}_{/f}$ 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 $\widetilde{g}: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{C}}}_{/f}$ 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 $\widehat{\operatorname{\mathcal{C}}}_{/f}^{0}$. Since $g$ is a colimit diagram, $\widetilde{g}$ is also a colimit diagram (Corollary 7.1.4.21). It follows that $\widetilde{X}$ belongs to $\widehat{\operatorname{\mathcal{C}}}'_{/f}$. Allowing the object $\widetilde{X}$ to vary, we conclude that $X$ belongs to $\widehat{\operatorname{\mathcal{C}}}'$, as desired.
$\square$