Example 9.1.1.8. Let $(A, \leq )$ be a nonempty linearly ordered set. Then the $\infty $-category $\operatorname{N}_{\bullet }(A)$ is automatically a filtered $\infty $-category. If $A$ has a largest element, then it is $\kappa $-filtered for every infinite cardinal $\kappa $. Otherwise, $A$ is $\kappa $-filtered if and only if it has cofinality $\geq \kappa $ (see Definition 4.7.1.28).
In particular, if $\kappa $ is a regular cardinal, then the $\infty $-category $\operatorname{N}_{\bullet }(\mathrm{Ord}_{< \kappa })$ is $\kappa $-filtered (but not $\lambda $-filtered for $\lambda > \kappa $).