Kerodon

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 $\mathscr {F}(C)$. $\square$

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

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 7.6.6.6), 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 7.4.1.18, 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 7.4.1.18), 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

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 7.4.1.15), 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

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 7.4.1.18 guarantees 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. $\square$

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 7.6.6.8 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 7.6.6.5). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\kappa $-small colimits. We wish to show that that the composite functor

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

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

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

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 7.6.6.6), which follows from Lemma 8.4.3.5. $\square$

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

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 7.4.1.19 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

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 7.4.1.19), 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 $\mathscr {F}$. $\square$