Definition 9.4.1.8. Let $\mathbb {K}$ be a collection of simplicial sets and suppose that we are given a commutative diagram of $\infty $-categories
We say that the diagram (9.28) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ if the following conditions are satisfied:
- $(1)$
For $i \in \{ 0,1\} $, the map of fibers $H_{i}: \operatorname{\mathcal{E}}_{i} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{i}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{i}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{i}$ (Definition 8.4.5.1).
- $(2)$
The functor $H$ is fully faithful.
- $(3)$
The inner fibration $\widehat{U}$ is $\mathbb {K}$-cocomplete (Definition 9.4.1.1).
- $(4)$
For every object $X \in \operatorname{\mathcal{E}}_0$, the functor
\[ \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{\mathcal{S}}\quad \quad Y \mapsto \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }( H(X), Y) \]preserves $\mathbb {K}$-indexed colimits.
If $\kappa $ is a regular cardinal, we say that (9.28) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ if it exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets.