Kerodon

Definition 8.5.6.3 (Idempotent Endomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. We will say that $e$ is idempotent if there exists a functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \widetilde{e} ) = e$; here $\widetilde{e}$ denotes the (unique) non-identity morphism in the category $\operatorname{Idem}$. We let $\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}}$ denote the full subcategory of $\operatorname{End}_{\operatorname{\mathcal{C}}}$ spanned by the idempotent endomorphisms.