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Example 8.5.2.3 (Split Idempotents). Let $\operatorname{\mathcal{C}}$ be a category containing a retraction diagram

8.49
\begin{equation} \begin{gathered}\label{equation:retracts-and-idempotents} \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_{Y} } & & Y } \end{gathered} \end{equation}

(see Definition 8.5.1.16). Then $e = i \circ r$ is an idempotent endomorphism of $X$. This follows from the calculation

\[ e \circ e = (i \circ r) \circ (i \circ r) = i \circ \operatorname{id}_{Y} \circ r = i \circ r = e. \]

We will say that an idempotent endomorphism $e: X \rightarrow X$ is split if it can be obtained in this way (that is, if $e = i \circ r$, for some pair of morphisms $i: Y \rightarrow X$ an $r: X \rightarrow Y$ satisfying $r \circ i = \operatorname{id}_{Y}$.