# Kerodon

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Proposition 7.2.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. The following conditions are equivalent:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is filtered.

$(2)$

For every finite simplicial set $K$ and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{C}}_{f/}$ is filtered.

$(3)$

For every finite simplicial set $K$ and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty$-category $\operatorname{\mathcal{C}}_{f/}$ is weakly contractible.

$(4)$

For every finite simplicial set $K$, the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.2.4.8, the implication $(2) \Rightarrow (3)$ from Proposition 7.2.4.9, and the implication $(3) \Rightarrow (1)$ is immediate from the definitions (Remark 7.2.4.7). The equivalence $(3) \Leftrightarrow (4)$ is a special case of Corollary 7.2.3.9. $\square$