Variant 8.4.3.6 (03WE)

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

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

Proof of Variant 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 $\operatorname{\mathcal{D}}$ 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 8.3.3.13 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

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

is an equivalence of $\infty $-categories. By virtue of Theorem 8.4.3.9, $\operatorname{Fun}'( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }), \operatorname{\mathcal{D}})$ 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 7.3.6.16, 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 8.4.3.8. 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 7.6.7.5), which follows from Lemma 8.4.3.8. $\square$