# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 7.2.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. The following conditions are equivalent:

$(1)$

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

$(2)$

For every small filtered category $\operatorname{\mathcal{K}}$, the $\infty$-category $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$-indexed colimits.

$(3)$

For every directed partially ordered set $(A, \leq )$, the $\infty$-category $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(A)$-indexed colimits.

Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 7.2.5.8 and the implication $(2) \Rightarrow (3)$ follows from Exercise 7.2.4.2. The implication $(3) \Rightarrow (1)$ follows from Theorem 7.2.7.2 and Corollary 7.2.2.3. $\square$