Proposition 9.1.8.10. Let $\kappa $ be an infinite cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-small. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-filtered.
- $(2)$
The idempotent completion of $\operatorname{\mathcal{C}}$ has a final object.
- $(3)$
There exists a right cofinal functor $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.
In particular, if $\operatorname{\mathcal{C}}$ is idempotent complete, then it is $\kappa $-filtered if and only if it has a final object.
Proof.
The equivalence $(2) \Leftrightarrow (3)$ follows from Remark 8.5.6.19 (and does not require the assumption that $\operatorname{\mathcal{C}}$ is $\kappa $-small). Since the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$ is $\kappa $-filtered (Example 9.1.1.8), the implication $(3) \Rightarrow (1)$ is a special case of Corollary 9.1.5.11. We will complete the proof by showing that $(1)$ implies $(2)$. If condition $(1)$ is satisfied, then there exists an object $X \in \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \underline{X}$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. Fix an uncountable cardinal $\lambda $ such that $\operatorname{\mathcal{C}}$ is locally $\lambda $-small, let $\widehat{\operatorname{\mathcal{C}}}$ denote the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, and let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Since $H$ is dense (Theorem 8.4.2.1), every object $\mathscr {F} \in \widehat{\operatorname{\mathcal{C}}}$ can be recovered as the colimit of the diagram
\[ \operatorname{\mathcal{C}}\times _{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{ / \mathscr {F} } \rightarrow \operatorname{\mathcal{C}}\xrightarrow { H } \widehat{\operatorname{\mathcal{C}}}. \]
In particular, if $\mathscr {F}$ is a final object of $\widehat{\operatorname{\mathcal{C}}}$, then it is a colimit of the diagram $H$. It follows that there exists a morphism $i: \mathscr {F} \rightarrow H(X)$ in the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ and a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & \underline{\mathscr {F}} \ar [dr]^{\underline{i}} & \\ H \ar [rr]^{ H( \alpha ) } \ar [ur] & & \underline{ H(X) } } \]
in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \widehat{\operatorname{\mathcal{C}}} )$. Since $\mathscr {F}$ is a final object of $\widehat{\operatorname{\mathcal{C}}}$, the morphism $i$ automatically has a left homotopy inverse (given by any morphism $r: H(X) \rightarrow \mathscr {F}$). In particular, $\mathscr {F}$ is a retract of $H(X)$, and can therefore be viewed as a final object in the idempotent completion of $\operatorname{\mathcal{C}}$ (see Proposition 8.5.5.5).
$\square$