Proposition 8.5.6.10 (Lifting Split Idempotents). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is split idempotent endomorphism if and only if it extends to a split idempotent $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$, in the sense of Definition 8.5.3.5. In particular, every split idempotent endomorphism is an idempotent endomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume that the endomorphism $e$ is split idempotent; we will show that $e$ can be extended to a split idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ (the reverse implication follows immediately from the definitions). Choose a retraction diagram
8.66
\begin{equation} \begin{gathered}\label{equation:lifting-split-idempotents} \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_{Y} } & & Y } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $[e] = [i] \circ [r]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Using Corollary 8.5.1.28, we can extend the diagram (8.66) to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$. By construction, $\overline{F}$ carries the unique non-identity morphism of $\operatorname{Idem}$ to a morphism $e': X \rightarrow X$ of $\operatorname{\mathcal{C}}$ which is homotopic to $e$. Replacing $\overline{F}$ by an isomorphic functor if necessary, we may assume that $e' = e$ (see Corollary 4.4.5.3). Then $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem})}$ is a split idempotent in $\operatorname{\mathcal{C}}$ extending $e$. $\square$