# Kerodon

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Definition 9.3.1.8. Let $\mathbb {K}$ be a collection of simplicial sets and suppose that we are given a commutative diagram of $\infty$-categories

9.26
$$\begin{gathered}\label{equation:relative-cocompletion-over-1-simplex} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1. & } \end{gathered}$$

We say that the diagram (9.26) 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 9.3.1.1).

$(4)$

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 (9.26) 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.