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.