Kerodon

Proof. By virtue of Proposition 8.5.4.9, an $\infty $-category $\operatorname{\mathcal{D}}$ is idempotent complete if and only if it admits $\operatorname{N}_{\bullet }( \operatorname{Idem})$-indexed colimits: that is, if and only if it is $\mathbb {K}$-cocomplete, where $\mathbb {K} = \{ \operatorname{N}_{\bullet }( \operatorname{Idem}) \} $ (see Definition 8.4.5.1). Moreover, every functor of $\infty $-categories $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ automatically preserves $\operatorname{N}_{\bullet }(\operatorname{Idem})$-indexed colimits (Corollary 8.5.3.12). We can therefore restate $(b)$ as follows:

$(b')$

The functor $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$, in the sense of Definition 8.4.5.1.

Using Variant 8.4.6.9, we see that this condition is satisfied if and only if $H$ satisfies conditions $(1)$ and $(2)$ of Definition 8.5.5.1, together with the following variant of $(3)$:

$(3')$

The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is generated by the essential image of $H$ under the formation of $\operatorname{N}_{\bullet }( \operatorname{Idem})$-indexed colimits. That is, if $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ is a replete full subcategory which contains the essential image of $H$ and is closed under retracts, then $\widehat{\operatorname{\mathcal{C}}}' = \widehat{\operatorname{\mathcal{C}}}$.

The implication $(3) \Rightarrow (3')$ is immediate. To prove the converse, let $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ be the full subcategory spanned by those objects $Y$ which are retracts of $H(X)$, for some $X \in \operatorname{\mathcal{C}}$. Condition $(3)$ of Definition 8.5.5.1 asserts that $\widehat{\operatorname{\mathcal{C}}}' = \widehat{\operatorname{\mathcal{C}}}$. This is a special case of $(3')$, since $\widehat{\operatorname{\mathcal{C}}}'$ is closed under the formation of retracts (Remark 8.5.1.6). $\square$

Proof. By virtue of Proposition 8.5.5.3 (and its proof), this is a special case of Proposition 8.4.5.3. $\square$

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 7.1.7.2. 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 8.5.4.9. It follows from Corollary 8.5.1.14 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 8.5.4.8).

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 8.4.3.9, 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 $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ 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 $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$. $\square$

Proof. Combine Corollary 8.5.5.8 with Variant 7.1.4.25. $\square$

Proof. By virtue of Remark 8.5.3.9, we can identify $\operatorname{\mathcal{D}}$ 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 $\operatorname{\mathcal{C}}$. The desired result is now a special case of Corollary 8.5.5.9. $\square$