Corollary 8.5.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The idempotent $F$ is split: that is, it can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$.
- $(2)$
The diagram $F$ admits a limit in $\operatorname{\mathcal{C}}$.
- $(3)$
The diagram $F$ admits a colimit in $\operatorname{\mathcal{C}}$.