Kerodon

Proof. To simplify the notation, set $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. For each object $C \in \operatorname{\mathcal{C}}$, the representable functor $h_{C} \in \operatorname{\mathcal{D}}$ corepresents the functor

given by evaluation at $C$, which preserves $\kappa $-small colimits by virtue of Proposition 7.1.6.1. It follows that the covariant Yoneda embedding $h_{\bullet }$ factors through the subcategory $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{\mathcal{D}}$. Moreover, the functor $h_{\bullet }$ is fully faithful (Theorem 8.3.3.13).

The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits, and is therefore idempotent complete by virtue of Proposition 8.5.4.7. It follows from Corollary 8.5.1.11 that the full category $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{\mathcal{D}}$ is closed under the formation of retracts, and is therefore also idempotent complete (Proposition 8.5.4.6).

To complete the proof, it will suffice to show that every object $\mathscr {F} \in \widehat{\operatorname{\mathcal{C}}}$ is a retract of $h_{C}$ for some object $C \in \operatorname{\mathcal{C}}$. Applying Corollary 8.4.3.12, we deduce that $\mathscr {F}$ can be realized as the colimit of a diagram

where $\operatorname{\mathcal{K}}$ is an essentially $\kappa $-small $\infty $-category. Since the functor $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( \mathscr {F}, \bullet )$ preserves $\kappa $-small colimits, it follows that the identity map $\operatorname{id}_{\mathscr {F}} \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \mathscr {F}, \mathscr {F} )$ factors (up to homotopy) through $h_{ T(B) }$ for some $B$. In particular, $\mathscr {F}$ is a retract of the representable functor $h_{ T(B) }$. $\square$

Proof. Choose a functor $F: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ 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 $F(X)$ for some object $X \in \operatorname{\mathcal{C}}$. It follows that $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(Y, Y' )$ is a retract of $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( F(X), Y' )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. By virtue of Corollary 8.5.1.14, it will suffice to show that the Kan complex $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( F(X), Y' )$ is essentially $\kappa $-small. Applying the same argument to $Y'$, we are reduced to showing that the mapping space $\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( F(X), F(X') )$ 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 $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, X' ) \rightarrow \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( F(X), F(X') )$ 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 5.4.6.14). By virtue of the criterion of Proposition 5.4.8.8, 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 $F(X)$. 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 $F(X)$ 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 $(\operatorname{id}, e): F(X) \rightarrow F(X)$. It follows that the cardinality of the set of isomorphism classes $S_{X}$ is bounded above by the cardinality of the set $\operatorname{Hom}_{\mathrm{h} \mathit{ \widehat{\operatorname{\mathcal{C}}}} }( F(X), F(X) )$ of morphisms $e: F(X) \rightarrow F(X)$ in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which we can identify with the $\kappa $-small set $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$. $\square$

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is a full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ and that $H$ is the inclusion functor. Let $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admit an extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$. It follows from Proposition 8.5.1.7 that, in this case, the functor $\widehat{G}$ is automatically left (and right) Kan extended from $\operatorname{\mathcal{C}}$. Applying Corollary 7.3.6.16, we deduce that the restriction functor $\theta : \operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is a trivial Kan extension. Note that any functor $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ carries split idempotents in $\widehat{\operatorname{\mathcal{C}}}$ to split idempotents in $\operatorname{\mathcal{D}}$, so that $G = \widehat{G}|_{\operatorname{\mathcal{C}}}$ satisfies condition $(\ast )$. To complete the proof, it will suffice to prove the reverse implication. Fix a functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfies condition $(\ast )$; we wish to show that $G$ admits an extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$.

Choose an uncountable regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Using Proposition 8.5.4.7, we can choose a fully faithful functor $H': \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$, where the $\infty $-category $\widehat{\operatorname{\mathcal{D}}}$ admits $\kappa $-small colimits. Replacing $\operatorname{\mathcal{D}}$ by the essential image of $H'$, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{D}}}$. Invoking Proposition 7.6.7.12, we deduce that the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ admits a left Kan extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{D}}}$. We will complete the proof by showing that $\widehat{G}$ factors through $\operatorname{\mathcal{D}}$.

Fix an object $Y \in \widehat{\operatorname{\mathcal{C}}}$; we wish to show that $\widehat{G}(Y)$ belongs to $\operatorname{\mathcal{D}}$. By assumption, there exists a retraction diagram

in $\widehat{\operatorname{\mathcal{C}}}$, where the object $X$ belongs to $\operatorname{\mathcal{C}}$. Using Corollary 8.5.1.24, we can extend (8.50) to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \overline{\operatorname{\mathcal{C}}}$. Then $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) }$ is an idempotent in $\operatorname{\mathcal{C}}$ which splits in $\widehat{\operatorname{\mathcal{C}}}$. Invoking assumption $(\ast )$, we deduce that $G \circ F$ is a split idempotent in $\operatorname{\mathcal{D}}$. That is, there exists a functor $\overline{F}': \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{D}}$ satisfying $\overline{F}' |_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } = G \circ F$. Applying Corollary 8.5.3.10, we deduce that $\widehat{G} \circ \overline{F}$ is isomorphic to $\overline{F}'$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \widehat{\operatorname{\mathcal{D}}} )$. Evaluating on the final object of $\operatorname{Ret}$, we deduce that $\widehat{G}(Y)$ is isomorphic to an object of $\operatorname{\mathcal{D}}$ and therefore belongs to $\operatorname{\mathcal{D}}$ (since the full subcategory $\operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ was assumed to be replete). $\square$

Proof of Proposition 8.5.5.8. Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories, where $\widehat{\operatorname{\mathcal{C}}}$ is idempotent complete. The implication $(a) \Rightarrow (b)$ follows from Lemma 8.5.5.9, the implication $(b) \Rightarrow (c)$ from Remark 4.5.1.19, and the implication $(c) \Rightarrow (d)$ from Remark 3.1.6.5. We will complete the proof by showing that $(d)$ implies $(a)$.

Using Corollary 8.5.5.6, we can choose an $\infty $-category $\widehat{\operatorname{\mathcal{C}}}'$ and a functor $H': \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}'$ which exhibits $\widehat{\operatorname{\mathcal{C}}'}$ as an idempotent completion of $\operatorname{\mathcal{C}}$. Since the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}'$ is idempotent complete, assumption $(d)$ guarantees that there exists a functor $G: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{C}}}'$ such that $H'$ is isomorphic to $G \circ H$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \widehat{\operatorname{\mathcal{C}}}' )$). By virtue of Remark 8.5.5.3, the functor $G \circ H$ also exhibits $\widehat{\operatorname{\mathcal{C}}}'$ as an idempotent completion of $\operatorname{\mathcal{C}}$. To show that $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$, it will suffice to show that the functor $G$ is an equivalence of $\infty $-categories (Remark 8.5.5.2). This is equivalent to the assertion that the homotopy class $[G]$ is an isomorphism when regarded as a morphism in the homotopy category of idempotent complete $\infty $-categories. Fix an idempotent complete $\infty $-category $\operatorname{\mathcal{D}}$; we wish to show that precomposition with $[G]$ induces a bijection

Invoking assumption $(d)$, this is equivalent to the bijectivity of the composite map

which follows from Lemma 8.5.5.9. $\square$

Proof. Combine Corollaries 8.5.5.10 and 7.1.3.22. $\square$

Proof. By virtue of Remark 8.5.3.9, we can identify $\operatorname{\mathcal{Y}}$ with the limit of a diagram $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{QC}}$ carrying the unique object of $\operatorname{Idem}$ to the idempotent complete $\infty $-category $\operatorname{\mathcal{X}}$. The desired result is now a special case of Corollary 8.5.5.11. $\square$