Proposition 9.1.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be an infinite cardinal. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-filtered.
- $(2)$
For every $\kappa $-small simplicial set $K$ and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is $\kappa $-filtered.
- $(3)$
For every $\kappa $-small simplicial set $K$ and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is filtered.
- $(4)$
For every $\kappa $-small simplicial set and every morphism $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is weakly contractible.
- $(5)$
For every $\kappa $-small simplicial set $K$, the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal.