Proposition 9.4.6.2 (06K7)

Proposition 9.4.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be a small regular cardinal. The following conditions are equivalent:

$(1)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-accessible: that is, it is an $\operatorname{Ind}_{\kappa }$-completion of a small $\infty $-category $\operatorname{\mathcal{C}}_0$.
$(2)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$.
$(3)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated and the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects is essentially small.

Proof. The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definitions and the equivalence $(2) \Leftrightarrow (3)$ follows from the description of $\operatorname{Ind}_{\kappa }^{\lambda }$-completions supplied by Theorem 9.3.4.14. The implication $(3) \Rightarrow (2)$ follows Proposition 9.4.1.11. Conversely, if $\operatorname{\mathcal{C}}$ is the $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$, then $\operatorname{\mathcal{C}}_{< \kappa }$ is an idempotent-completion of $\operatorname{\mathcal{C}}_0$ (Proposition 9.4.1.19) and is therefore essentially small (Proposition 8.5.5.7). $\square$