Kerodon

Definition 4.7.9.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\kappa $ be an uncountable regular cardinal. We say that $U$ is essentially $\kappa $-small if, for every simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small. We say that $U$ is locally $\kappa $-small if, for every simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-category $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small.