$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 8.5.5.2. 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 following conditions are equivalent:
- $(a)$
The functor $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$, in the sense of Definition 8.5.5.1.
- $(b)$
For every idempotent complete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $H$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Proof.
By virtue of Proposition 8.5.4.7, 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).
$\square$