Kerodon

Remark 9.2.1.11. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose that $\operatorname{\mathcal{C}}$ is locally $\mu $-small, for some uncountable cardinal $\mu $ of exponential cofinality $\geq \lambda $. Then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is also locally $\mu $-small (Proposition 8.7.3.11). In particular, if $\operatorname{\mathcal{C}}$ is a locally small $\infty $-category, then $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is also locally small.

Remark 9.2.1.14 ($\operatorname{Ind}$-Extensions as Kan Extensions). In the situation of Definition 9.2.1.13, suppose that we are given functors $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\alpha : f \rightarrow F \circ H$. The following conditions are equivalent:

$(1)$

The natural transformation $\alpha $ exhibits $F$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$, in the sense of Definition 9.2.1.13. That is, the functor $F$ is $(\kappa ,\lambda )$-finitary and $\alpha $ is an isomorphism.

$(2)$

The natural transformation $\alpha $ exhibits $F$ as a left Kan extension of $f$ along $H$, in the sense of Variant 7.3.1.5.

See Corollary 8.4.5.11.

Remark 9.2.1.15. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits.

$(2)$

The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ admits a left adjoint $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.

Moreover, if these conditions are satisfied, then the functor $F$ is the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. See Proposition 8.4.5.13.

Remark 9.2.1.17. Let $\kappa \leq \lambda $ be regular cardinals, and let $\mathbb {K}$ be the collection of all $\lambda $-small $\kappa $-filtered $\infty $-categories. Then $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor (in the sense of Definition 9.2.1.16) if and only if it is a $\mathbb {K}$-cocompletion functor (in the sense of Definition 8.7.3.3).

Remark 9.2.1.20. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Applying the functor $\operatorname{Ind}_{\kappa }^{\lambda }(-)$ of Notation 9.2.1.19 (for any sufficiently large $\mu $), we obtain a functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, which is characterized (up to isomorphism) by the requirements that it is $(\kappa ,\lambda )$-finitary and that the diagram

commutes up to isomorphism. In other words, $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ can be identified with the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ (see Definition 9.2.1.13).

Remark 9.2.1.21. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Choose a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\mu $-small, and let

be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. By virtue of Example 8.4.4.5, left Kan extension along $F$ determines a $\mu $-cocontinuous functor $F_{!}: \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{ < \mu } )$ for which the diagram

commutes up to (canonical) isomorphism. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ with the smallest full subcategories of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ and $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ which are closed under $\lambda $-small $\kappa $-filtered colimits and which contain the essential images of $h_{\bullet }^{\operatorname{\mathcal{C}}}$ and $h_{\bullet }^{\operatorname{\mathcal{D}}}$, respectively. It follows from the commutativity of the diagram (9.3) that $F_{!}$ restricts to a functor $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, which we can identify with the functor $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ of Remark 9.2.1.20.