Kerodon

Proof. Choose a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$. We first prove $(1)$. Assume that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small; we wish to show that $\widehat{\operatorname{\mathcal{C}}}$ is locally $\kappa $-small (the reverse implication follows immediately from the definition). Fix a pair of objects $Y,Y' \in \widehat{\operatorname{\mathcal{C}}}$; we wish to show that the morphism space $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(Y, Y' )$ is essentially $\kappa $-small. By assumption, the object $Y \in \widehat{\operatorname{\mathcal{C}}}$ is a retract of $H(X)$ for some object $X \in \operatorname{\mathcal{C}}$. It follows that $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(Y, Y' )$ is a retract of $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( H(X), Y' )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. By virtue of Corollary 8.5.1.17, it will suffice to show that the Kan complex $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( H(X), Y' )$ is essentially $\kappa $-small. Applying the same argument to $Y'$, we are reduced to showing that the mapping space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( H(X), H(X') )$ is essentially $\kappa $-small for every pair of objects $X,X' \in \operatorname{\mathcal{C}}$. Since the functor $F$ is fully faithful, the canonical map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, X' ) \rightarrow \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( H(X), H(X') )$ is a homotopy equivalence. The desired result now follows from our assumption that the $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small.

We now prove $(2)$. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small; we wish to show that $\widehat{\operatorname{\mathcal{C}}}$ is also essentially $\kappa $-small (again, the reverse implication follows immediately from the definitions). Without loss of generality, we may assume that $\kappa $ is the smallest cardinal for which $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, and is therefore regular (Corollary 4.7.6.17). By virtue of the criterion of Proposition 4.7.8.7, it will suffice to show that the set of isomorphism classes $S = \pi _0( \widehat{\operatorname{\mathcal{C}}}^{\simeq } )$ is $\kappa $-small. For each object $X \in \operatorname{\mathcal{C}}$, let $S_{X} \subseteq S$ be the collection of isomorphism classes of objects $Y \in \widehat{\operatorname{\mathcal{C}}}$ which can be realized as a retract of $H(X)$. Note that we can write $S$ as a union of the subsets $S_{X}$, where the $X$ ranges over a set of representatives for the isomorphism classes in $\operatorname{\mathcal{C}}$. Since $\kappa $ is regular, and the set $\pi _0( \operatorname{\mathcal{C}}^{\simeq } )$ is $\kappa $-small, it will suffice to show that each of the sets $S_{X}$ is $\kappa $-small. Let us henceforth regard the object $X \in \operatorname{\mathcal{C}}$ as fixed, and let $Y$ be any retract of $H(X)$ in the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$. It follows from Proposition 8.5.2.4 that, as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, $Y$ can be identified with the equalizer of a pair of morphisms $(\operatorname{id}, e): H(X) \rightarrow H(X)$. It follows that the cardinality of the set of isomorphism classes $S_{X}$ is bounded above by the cardinality of the set $\operatorname{Hom}_{\mathrm{h} \mathit{ \widehat{\operatorname{\mathcal{C}}}} }( H(X), H(X) )$ of morphisms $e: H(X) \rightarrow H(X)$ in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which we can identify with the $\kappa $-small set $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$. $\square$