= Idempotent splitting through a coequalizer
{title2=$e^2=e,\quad e\text{ splits}\iff\operatorname{coeq}(e,1_E)\text{ exists}$}
If $e=ir$ and $ri=1$, then $r$ is the <coequalizer> of $(e,1)$: an arrow $h$ with $he=h$ factors uniquely as $(hi)r$. Conversely a <coequalizer> $q$ makes $e$ factor as $iq=e$; epimorphic cancellation gives $qi=1$. Thus the <splitting of an idempotent morphism> is equivalent to this particular <coequalizer>, without assuming arbitrary <coequalizers> exist.
Back to article page
