Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

8.3.3 The Universal Property of Presheaf $\infty $-Categories

When studying a small $\infty $-category $\operatorname{\mathcal{C}}$, it is often useful to embed $\operatorname{\mathcal{C}}$ into a larger $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ with better closure properties. For example, we can take $\widehat{\operatorname{\mathcal{C}}}$ to be the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. By virtue of Theorem 8.2.5.5, the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is a fully faithful functor. Moreover, since the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ admits small limits and colimits (Corollary 7.4.5.6), the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ also admits small limits and colimits, which can be computed levelwise (Proposition 7.1.6.1). Our goal in this section is to show that the Yoneda embedding $h_{\bullet }$ can be characterized by a universal mapping property:

Theorem 8.3.3.1. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ (Definition 8.2.5.1). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small colimits, and let $\operatorname{Fun}'( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}})$ spanned by those functors $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ which preserve small colimits. Then precomposition with $h_{\bullet }$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}'( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \quad \quad F \mapsto F \circ h_{\bullet }. \]

Remark 8.3.3.2. Stated more informally, Theorem 8.3.3.1 asserts that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ can be obtained from $\operatorname{\mathcal{C}}$ by “freely” adjoining small colimits.

Remark 8.3.3.3. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially small and $\operatorname{\mathcal{D}}$ admits small colimits. Fix a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Theorem 8.3.3.1 implies that $f$ is isomorphic to the composition $F \circ h_{\bullet }$, for some functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ which preserves small colimits. Moreover, the functor $F$ is uniquely determined up to isomorphism.

Example 8.3.3.4. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small colimits, and let $\operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve small colimits. Then the evaluation functor

\[ \operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}\quad \quad F \mapsto F( \Delta ^0 ) \]

is an equivalence of $\infty $-categories (this follows by applying Theorem 8.3.3.1 in the special case $\operatorname{\mathcal{C}}= \Delta ^0$). Note that this property characterizes the $\infty $-category $\operatorname{\mathcal{S}}$ up to equivalence: it is “freely generated” under small colimits by the object $\Delta ^0$.

Remark 8.3.3.5. In the situation of Theorem 8.3.3.1, let $\operatorname{LFun}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}})$ spanned by those functors $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ which admit a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (see Notation 6.2.1.3). By virtue of Corollary 7.1.3.21, this condition implies that $F$ preserves small colimits: that is, we have an inclusion

\[ \operatorname{LFun}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}'( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \operatorname{\mathcal{D}}). \]

We will see later that equality holds if the $\infty $-category $\operatorname{\mathcal{D}}$ is locally small (see Proposition )

Following the convention of Remark 5.4.0.5, we can regard Theorem 8.3.3.1 as a special case of the following more general assertion, which we will prove at the end of this section:

Variant 8.3.3.6. 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 $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\kappa $-small colimits and let $\operatorname{Fun}^{\kappa }( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ), \operatorname{\mathcal{D}})$ denote 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 preserve $\kappa $-small colimits. Then precomposition with $h_{\bullet }$ determines an equivalence of $\infty $-categories

\[ \operatorname{Fun}^{\kappa }( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \quad \quad F \mapsto F \circ h_{\bullet }. \]

Warning 8.3.3.7. The conclusion of Variant 8.3.3.6 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.

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 8.3.3.8. 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 8.2.5.5), and is therefore essentially $\kappa $-small. Since $\kappa $ is regular, it will suffice to show that each fiber of the right fibration $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {G}} \rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an essentially $\kappa $-small Kan complex (Corollary 5.4.8.11). 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 8.2.1.3: 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 Variant 8.3.3.6 from the following more precise assertion:

Theorem 8.3.3.9. 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 8.3.3.8 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 7.6.7.5), which follows from Corollary 8.3.2.2.

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 5.4.3.19). By virtue of Proposition 7.4.5.13 (and Remark 7.4.5.15), 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 7.4.5.13 and Remark 7.4.5.15), 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 5.4.3.19 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 8.3.2.5, 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 8.2.5.5), 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 7.4.5.13 and Remark 7.4.5.15 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 7.3.6.11), we deduce that the functor $T$ is isomorphic to $U$, and therefore also preserves $\kappa $-small colimits. $\square$

Remark 8.3.3.10. In the statement of Theorem 8.3.3.9, 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 8.3.3.11. 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 Variant 8.3.3.6. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\kappa $-small colimits. We wish to show that the composite functor

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

is an equivalence of $\infty $-categories. Theorem 8.2.5.5 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

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

is an equivalence of $\infty $-categories. By virtue of Theorem 8.3.3.9, $\operatorname{Fun}'( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}})$ 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 7.3.6.13, we see that $U$ restricts to a trivial Kan fibration

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

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 8.3.3.8. By virtue of Corollary 7.3.5.6, 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 7.6.7.5), which follows from Lemma 8.3.3.8. $\square$

Corollary 8.3.3.12. 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 8.3.3.8 with Theorem 8.2.5.5, 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 Variant 8.3.3.6, we deduce that $\mathscr {F}$ is a colimit of the composite functor

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