Proposition 8.5.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is idempotent complete.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits limits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits colimits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.