Idempotent morphism

An endomorphism $e:A\to A$ is idempotent when $e^2=e$. It splits when $e=ir$ for maps $r:A\to B$, $i:B\to A$ satisfying $ri=1_B$.

An idempotent morphism $e:E\to E$ splits if there are $r:E\to H$ and $s:H\to E$ with $sr=e$ and $rs=1_H$. The object $H$ is a retract of $E$. In a functor category, the splitting maps must be natural transformations; objectwise splittings without compatible functorial choices are not enough.

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.

A category is Cauchy-complete when every idempotent morphism splits: for $e:A\to A$ with $e^2=e$, there are $r:A\to B$ and $i:B\to A$ with $ir=e$ and $ri=1_B$. Its Karoubi envelope freely supplies these splittings. For a small category, this condition makes every retract of a representable functor representable.

A collection of arrows is a two-sided ideal when arbitrary precomposition and postcomposition preserve it. If a finite category has split idempotent morphisms, every ideal $I$ satisfying $I^2=I$ is generated by identities belonging to $I$. Factor an arrow $f\in I$ repeatedly as a longer product of arrows of $I$. Two prefixes coincide, since there are only finitely many arrows. The intervening endomorphism $t\in I$ fixes that prefix; an idempotent power $e$ of $t$ still fixes it. Write $e=ir$, $ri=1_B$. Then $1_B=rei\in I$, and $f$ factors through $B$. Applied to the least covering sieves of a finite site, this proves rigid coverage.

The Karoubi envelope freely splits idempotents. Its objects are pairs $(A,e)$ and a morphism $(A,e)\to (B,f)$ is a map $u:A\to B$ satisfying $u=fue$.