$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.1.4.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a left fibration of $\infty $-categories, where the $\infty $-category $\operatorname{\mathcal{D}}$ is filtered. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered.
- $(2)$
The functor $F$ is right cofinal.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is weakly contractible.
Proof.
The equivalence $(1) \Leftrightarrow (3)$ is Theorem 9.1.3.2 and the implication $(1) \Rightarrow (2)$ follows from Corollary 9.1.4.13. The implication $(2) \Rightarrow (3)$ follows from the weak contractibility of the filtered $\infty $-category $\operatorname{\mathcal{D}}$ (Proposition 9.1.1.18), since every right cofinal functor is a weak homotopy equivalence (Proposition 7.2.1.5).
$\square$