Example 8.5.3.13 (Idempotent Automorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an idempotent $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ and let $e: X \rightarrow X$ be the morphism in $\operatorname{\mathcal{C}}$ given by the image of the nondegenerate edge of $\operatorname{N}_{\bullet }(\operatorname{Idem})$. The following conditions are equivalent:
- $(1)$
The morphism $e$ is an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
The functor $F$ factors through the core $\operatorname{\mathcal{C}}^{\simeq }$.
- $(3)$
There is an isomorphism from $F$ to the constant functor $\underline{X}: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.
In particular, if $e$ is an isomorphism, then $F$ is a split idempotent. The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definition of the core $\operatorname{\mathcal{C}}^{\simeq }$, and the equivalence $(2) \Leftrightarrow (3)$ follows from the weak contractibility of $\operatorname{N}_{\bullet }(\operatorname{Idem})$ (Remark 8.5.3.4).