Kerodon

Example 8.5.3.6. Let $\operatorname{\mathcal{C}}$ be a category and let $e: X \rightarrow X$ be an idempotent endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is split (in the sense of Example 8.5.2.3) if and only if the induced map $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a split idempotent in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 8.5.3.5).