Kerodon

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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 $\operatorname{\mathcal{C}}$.

Proof. The implications $(1) \Rightarrow (2)$ and $(1) \Rightarrow (3)$ follow from Remark 8.5.3.9, and the reverse implications follow from Corollary 7.3.5.8. $\square$