Proposition 9.4.1.11. Let $\kappa $ be an uncountable regular cardinal and suppose we are given a commutative diagram of $\infty $-categories
9.29
\begin{equation} \begin{gathered}\label{equation:compare-notions-of-cocompletion} \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} \end{equation}
where $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Then (9.29) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ if and only if it satisfies conditions $(1)$ and $(2)$ of Definition 9.4.1.8, together with the following:
- $(3')$
The functor $\widehat{U}$ is a cartesian fibration.
- $(4')$
The contravariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \widehat{\operatorname{\mathcal{E}}}_0$ preserves $\kappa $-small colimits.
Proof.
Without loss of genearlity, we may assume that $H$ satisfies conditions $(1)$ and $(2)$ of Definition 9.4.1.8. The implication $(3') \Rightarrow (3)$ is a special case of Example 9.4.1.6. Moreover, if $(3')$ is satisfied, then we can use condition $(1)$ (together with Proposition 8.4.2.5) to identify the contravariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \widehat{\operatorname{\mathcal{E}}}_0$ with the functor
\[ \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \quad \quad \widehat{Y} \mapsto \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }( H(-), \widehat{Y} ). \]
In this case, the equivalence $(4) \Leftrightarrow (4')$ follows from the observation that $\kappa $-small colimits in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ are computed levelwise (Proposition 7.1.7.2).
We will complete the proof by showing that if $H$ satisfies conditions $(3)$ and $(4)$, then $U$ is a cartesian fibration. Fix a regular cardinal $\lambda $ having exponential cofinality $\geq \kappa $ such that $\widehat{\operatorname{\mathcal{E}}}$ is locally $\lambda $-small. By virtue of Corollary 6.2.3.2, it will suffice to show that for each object $\widehat{Y} \in \widehat{\operatorname{\mathcal{E}}}_{1}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{E}}}_0^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda } \quad \quad \widehat{X} \mapsto \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }(\widehat{X}, \widehat{Y}) \]
is representable by an object of $\widehat{\operatorname{\mathcal{E}}}_{0}$. Condition $(3)$ guarantees that $\mathscr {F}$ preserves $\kappa $-small limits. Using condition $(1)$ and Corollary 8.4.3.10, we are reduced to showing that for each object $X \in \operatorname{\mathcal{E}}_{0}$, the Kan complex $\mathscr {F}( H(X) ) = \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} )$ is $\kappa $-small.
Let now us regard the object $X$ as fixed. Condition $(4)$ guarantees that the functor
\[ \widehat{\operatorname{\mathcal{E}}}_{0} \rightarrow \operatorname{\mathcal{S}}^{< \lambda } \quad \quad \widehat{Y} \mapsto \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} ) \]
preserves $\kappa $-small colimits. In particular, the collection of objects $\widehat{Y}$ for which $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} )$ is essentially $\kappa $-small is closed under $\kappa $-small colimits (Example 7.6.6.8). Since $\widehat{\operatorname{\mathcal{E}}}_{1}$ is generated under $\kappa $-small colimits by the image of $H$, we can assume that $\widehat{Y} = H(Y)$ for some object $Y \in \operatorname{\mathcal{E}}_1$. In this case, condition $(2)$ supplies a homotopy equivalence
\[ \operatorname{Hom}_{ \operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}}}( H(X), \widehat{Y} ), \]
so the desired result follows from our assumption that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small.
$\square$