Proposition 9.1.5.18 (02QC)

Proposition 9.1.5.18. Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category and let $\kappa $ be a small infinite cardinal. The following conditions are equivalent:

$(1)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-filtered.
$(2)$: The $\infty $-category $\operatorname{\mathcal{C}}$ can be realized as the colimit of a small diagram $\mathscr {F}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{QC}}$, where $\operatorname{\mathcal{K}}$ is $\kappa $-filtered and each of the $\infty $-categories $\mathscr {F}(k)$ is $\kappa $-filtered.
$(3)$: The $\infty $-category $\operatorname{\mathcal{C}}$ can be realized as the colimit of a small diagram $\mathscr {F}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{QC}}$, where the $\infty $-category $\operatorname{\mathcal{K}}$ is $\kappa $-filtered and each of the $\infty $-categories $\mathscr {F}(k)$ has a final object.

Proof. For any (small) $\infty $-category $\operatorname{\mathcal{C}}$, the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration, whose covariant transport representation

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}\quad \quad C \mapsto \operatorname{\mathcal{C}}\vec{\times }_{\operatorname{\mathcal{C}}} \{ C\} \]

carries each $C \in \operatorname{\mathcal{C}}$ to an $\infty $-category containing a final object (Proposition 4.7.3.20). Moreover, the $\infty $-category $\operatorname{\mathcal{C}}$ is a colimit of this diagram (Example 7.4.5.7). This proves the implication $(1) \Rightarrow (3)$. The implication $(2) \Rightarrow (1)$ from Corollary 9.1.5.16 and the implication $(3) \Rightarrow (2)$ from Example 9.1.1.6. $\square$