# Kerodon

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### 8.4.3 Cocompletion via the Yoneda Embedding

Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty$-category. Our goal in this section is to prove Theorem 8.4.0.3, which asserts that the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as a cocompletion of $\operatorname{\mathcal{C}}$. We begin by formulating a slightly more general assertion.

Notation 8.4.3.1. Let $\kappa$ be an uncountable regular cardinal. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\kappa$-cocomplete $\infty$-categories, we let $\operatorname{Fun}^{\kappa }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which preserve $\kappa$-small colimits.

Definition 8.4.3.2. Let $\kappa$ be an uncountable regular cardinal. We say that a functor of $\infty$-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa$-cocompletion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:

$(1)$

The $\infty$-category $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa$-cocomplete.

$(2)$

For every $\kappa$-cocomplete $\infty$-category $\operatorname{\mathcal{D}}$, precomposition with $h$ induces an equivalence of $\infty$-categories $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Following the convention of Remark 4.7.0.5, we can regard Theorem 8.4.0.3 as a special case of the following more general assertion:

Theorem 8.4.3.3. Let $\kappa$ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is essentially $\kappa$-small, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Then $h_{\bullet }$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ as a $\kappa$-cocompletion of $\operatorname{\mathcal{C}}$.

Warning 8.4.3.4. The conclusion of Theorem 8.4.3.3 is not necessarily satisfied if we assume only that $\operatorname{\mathcal{C}}$ is locally $\kappa$-small. For example, suppose that $\operatorname{\mathcal{C}}= S$ is a set of cardinality $\kappa$ (regarded as a discrete simplicial set), and let $\operatorname{\mathcal{D}}$ be (the nerve of) the partially ordered set $\{ 0 < 1 \}$. Then we can identify objects of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ with collections of $\kappa$-small Kan complexes $\{ X_ s \} _{s \in S}$. Define a functor $\lambda : \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ by the formula

$\lambda ( \{ X_ s \} _{s \in S} ) = \begin{cases} 0 & \text{ if } | \{ s \in S: X_{s} \neq \emptyset \} | < \kappa \\ 1 & \text{ otherwise, } \end{cases}$

and let $\lambda _0: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ be the constant functor taking the value $0$. The functors $\lambda$ and $\lambda _{0}$ both preserve $\kappa$-small colimits and coincide on the image of the Yoneda embedding $h_{\bullet }$, but do not coincide in general.

The proof of Theorem 8.4.3.3 will require some preliminaries. Let $\kappa$ be an uncountable cardinal, and let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is locally $\kappa$-small. In what follows, we let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ spanned by the representable functors $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. We will need the following elementary observation:

Lemma 8.4.3.5. Let $\kappa$ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is essentially $\kappa$-small, and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor. Then the $\infty$-category

$\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}} = \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}$

is essentially $\kappa$-small.

Proof. The $\infty$-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is equivalent to $\operatorname{\mathcal{C}}$ (Theorem 8.3.3.13), and is therefore essentially $\kappa$-small. Since $\kappa$ is regular, it will suffice to show that each fiber of the right fibration $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}} \rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an essentially $\kappa$-small Kan complex (Corollary 5.6.7.7). Equivalently, we must show that for each object $\mathscr {G} \in \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$, the mapping space $X = \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })}( \mathscr {G}, \mathscr {F} )$ is essentially $\kappa$-small. This follows from Proposition 8.3.1.3: if $\mathscr {G}$ is representable by the object $C \in \operatorname{\mathcal{C}}$, then $X$ is homotopy equivalent to the $\kappa$-small Kan complex $\mathscr {F}(C)$. $\square$

We will deduce Theorem 8.4.3.3 from the following more precise assertion:

Theorem 8.4.3.6. Let $\kappa$ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is essentially $\kappa$-small, and let $T: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $T$ preserves $\kappa$-small colimits.

$(2)$

The functor $T$ is left Kan extended from the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \kappa } ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.

Proof. We first show that $(1)$ implies $(2)$. Assume that the functor $T$ preserves $\kappa$-small colimits and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor; we wish to show that the composite functor

$\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {T} \operatorname{\mathcal{D}}$

is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$. Lemma 8.4.3.5 guarantees that the $\infty$-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}$ is essentially $\kappa$-small. Since $T$ preserves $\kappa$-small colimits, it will suffice to show that the map $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is a colimit diagram (Remark 7.6.7.6), which follows from Corollary 8.4.2.2.

We now show that $(2)$ implies $(1)$. Assume that $T$ is left Kan extended from the $\infty$-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$; we wish to show that it preserves $\kappa$-small colimits. Choose a cardinal $\lambda$ such that $\operatorname{\mathcal{D}}$ is locally $\lambda$-small. Enlarging $\lambda$ if necessary, we may assume that it has exponential cofinality $\geq \kappa$ (Remark 4.7.3.19). By virtue of Proposition 7.4.5.17 (and Remark 7.4.5.19), it will suffice to show that for every representable functor $H: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the composition $H^{\operatorname{op}} \circ T$ preserves $\kappa$-small colimits. Since $H^{\operatorname{op}}$ preserves $\kappa$-small colimits (Proposition 7.4.5.17 and Remark 7.4.5.19), the functor $H^{\operatorname{op}} \circ T$ is left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Consequently, to show that $(2)$ implies $(1)$, we may replace $T$ by $H^{\operatorname{op}} \circ T$ and thereby reduce to the case where $\operatorname{\mathcal{D}}= (\operatorname{\mathcal{S}}^{< \lambda })^{\operatorname{op}}$, for some cardinal $\lambda$ of exponential cofinality $\geq \kappa$.

Let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$, and let $\mathscr {F}$ denote the composite functor

$\operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { h_{\bullet }^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\operatorname{op}} \xrightarrow { T^{\operatorname{op}} } \operatorname{\mathcal{S}}^{< \lambda }.$

Using Remark 4.7.3.19 again, we can choose a cardinal $\lambda ' \geq \lambda$ of exponential cofinality $\geq \kappa$ such that $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$ is locally $\lambda '$-small. In what follows, we abuse notation by identifying $T$ with the composite functor $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {T} ( \operatorname{\mathcal{S}}^{< \lambda } )^{\operatorname{op}} \hookrightarrow (\operatorname{\mathcal{S}}^{< \lambda '} )^{\operatorname{op}}$. Note that, since the inclusion $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \lambda '}$ preserves $\kappa$-small limits (see Variant 7.4.5.8), this composite functor is also left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.

Let $H': \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda '}$ be a functor represented by $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$, and let $U$ denote the composite functor

$\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \xrightarrow {H'^{\operatorname{op}} } (\operatorname{\mathcal{S}}^{< \lambda ' })^{\operatorname{op}}.$

Applying Proposition 8.4.2.5, we see that the composition $U \circ h_{\bullet }$ is isomorphic to the functor $\mathscr {F} = T \circ h_{\bullet }$. Since the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an equivalence of $\infty$-categories (Theorem 8.3.3.13), it follows that the functors $U$ and $T$ are isomorphic when restricted to $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Proposition 7.4.5.17 and Remark 7.4.5.19 guarantee that the functor $U$ preserves $\kappa$-small colimits. Invoking the implication $(1) \Rightarrow (2)$, we see that $U$ is left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Applying the universal property of Kan extensions (Corollary 7.3.6.13), we deduce that the functor $T$ is isomorphic to $U$, and therefore also preserves $\kappa$-small colimits. $\square$

Remark 8.4.3.7. In the statement of Theorem 8.4.3.6, it is not necessary to assume that the $\infty$-category $\operatorname{\mathcal{D}}$ admits $\kappa$-small colimits (though we will primarily be interested in cases where this condition is satisfied).

Example 8.4.3.8. Let $\operatorname{\mathcal{C}}$ be a small $\infty$-category and let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors. Then a functor of $\infty$-categories $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ preserves small colimits if and only if it is left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.

Proof of Theorem 8.4.3.3. Let $\kappa$ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is essentially $\kappa$-small. It follows from Example 7.6.7.8 that the $\infty$-category $\operatorname{\mathcal{S}}^{< \kappa }$ is $\kappa$-cocomplete, so that the functor $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is also $\kappa$-cocomplete (Remark 7.6.7.5). Let $\operatorname{\mathcal{D}}$ be an $\infty$-category which admits $\kappa$-small colimits. We wish to show that that the composite functor

$\operatorname{Fun}^{\kappa }( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ), \operatorname{\mathcal{D}}) \xrightarrow { \circ h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$

is an equivalence of $\infty$-categories. Theorem 8.3.3.13 guarantees that the covariant Yoneda embedding $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an equivalence of $\infty$-categories. We are therefore reduced to showing that the restriction functor

$U: \operatorname{Fun}^{\kappa }( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ), \operatorname{\mathcal{D}}) \quad \quad F \mapsto F|_{ \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )}$

is an equivalence of $\infty$-categories. By virtue of Theorem 8.4.3.6, $\operatorname{Fun}^{\kappa }( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}})$ is the full subcategory of $\operatorname{Fun}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}})$ spanned by those functors which are left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Applying Corollary 7.3.6.15, we see that $U$ restricts to a trivial Kan fibration

$\operatorname{Fun}^{\kappa }( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'( \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}}),$

where $\operatorname{Fun}'( \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}})$ spanned by those functors which admit a left Kan extension to $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.

We will complete the proof by showing that every functor $f: \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ admits a left Kan extension to $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Fix an object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ and let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}$ be as in the statement of Lemma 8.4.3.5. By virtue of Corollary 7.3.5.8, it will suffice to show the diagram

$\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}} \rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {f} \operatorname{\mathcal{D}}$

admits a colimit in the $\infty$-category $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ admits $\kappa$-small colimits, we are reduced to showing that the $\infty$-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}$ is essentially $\kappa$-small (see Remark 7.6.7.6), which follows from Lemma 8.4.3.5. $\square$

Corollary 8.4.3.9. Let $\kappa$ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is essentially $\kappa$-small, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Then every object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ can be realized as the colimit of a diagram

$\operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}\xrightarrow {h_{\bullet }} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ),$

where $\operatorname{\mathcal{K}}$ is a $\kappa$-small $\infty$-category.

Proof. Let $\operatorname{\mathcal{K}}'$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F}}$. Combining Lemma 8.4.3.5 with Theorem 8.3.3.13, we deduce that $\operatorname{\mathcal{K}}'$ is essentially $\kappa$-small. We can therefore choose an equivalence of $\infty$-categories $e: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}'$, where $\operatorname{\mathcal{K}}$ is $\kappa$-small. Applying Theorem 8.4.3.3, we deduce that $\mathscr {F}$ is a colimit of the composite functor

$\operatorname{\mathcal{K}}\xrightarrow {e} \operatorname{\mathcal{K}}' = \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F}} \rightarrow \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ).$
$\square$

Corollary 8.4.3.10. Let $\kappa$ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is essentially $\kappa$-small, and let $h^{\operatorname{\mathcal{C}}}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding. Suppose we are given a functor

$T: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda },$

where $\lambda$ is a cardinal of exponential cofinality $< \kappa$. The following conditions are equivalent:

$(1)$

For each object $C \in \operatorname{\mathcal{C}}$, the Kan complex $T( h^{\operatorname{\mathcal{C}}}_{C} )$ is essentially $\kappa$-small.

$(2)$

The functor $T$ preserves $\kappa$-small limits.

Moreover, if these conditions are satisfied, then the functor $T$ is representable by the object $\mathscr {F} = T \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$.

Proof. Assume first that $T$ is representable by an object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. In this case, Corollary 7.4.5.18 guarantees that $T$ satisfies condition $(2)$. To verify $(1)$, we note that for each object $C \in \operatorname{\mathcal{C}}$, Proposition 8.3.1.3 supplies a homotopy equivalence

$T( h^{\operatorname{\mathcal{C}}}_{C} ) \simeq \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h^{\operatorname{\mathcal{C}}}_{C}, \mathscr {G} ) \xrightarrow {\sim } \mathscr {G}(C);$

the desired result now follows from the observation that $\mathscr {G}(C)$ is essentially $\kappa$-small.

We now prove the converse. Assume that $T$ satisfies conditions $(1)$ and $(2)$ and set $\mathscr {F} = T \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$. Condition $(1)$ guarantees that we can view $\mathscr {F}$ as an object of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Since $\lambda$ has exponential cofinality $\geq \kappa$, the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda$-small (Corollary 4.7.8.8). We can therefore choose a functor $T': \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ which is representable by $\mathscr {F}$ (Theorem 5.6.6.13). It follows from Proposition 8.4.2.5 that the composition $T' \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$ is isomorphic to the functor $\mathscr {F} = T \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$. Using condition $(2)$ (and Corollary 7.4.5.18), we see that the functors $T$ and $T'$ both preserve $\kappa$-small limits. Applying Theorem 8.4.3.3, we conclude that $T'$ is isomorphic to $T$, so that the functor $T$ is also representable by the object $\mathscr {F}$. $\square$