Definition 9.4.1.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\mathbb {K}$ be a collection of simplicial sets. We say that $U$ is $\mathbb {K}$-cocomplete if it satisfies the following conditions:
- $(1)$
For every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\mathbb {K}$-cocomplete (Definition 8.4.5.1). That is, it admits $K$-indexed colimits, for every simplicial set $K \in \mathbb {K}$.
- $(2)$
Let $C$ be an object of $\operatorname{\mathcal{C}}$ and let $K \in \mathbb {K}$. Then every colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ is a $U$-colimit diagram in $\operatorname{\mathcal{E}}$.
If $\kappa $ is a regular cardinal, we say that the inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it $\mathbb {K}$-cocomplete, where $\mathbb {K}$ denotes the collection of all $\kappa $-small simplicial sets.