# When are all split monomorphisms complemented?

In a category $\mathcal C$, let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$. Is there a general criterion on $\mathcal C$ such that the following holds: there is an object $Z$ such that $Y = X \sqcup Z$ (the coproduct of $X$ and $Z$)?

(retitled the question as per the comment below)

• It's certainly true if C is abelian, but I guess you're looking for something a bit more interesting. – Hugh Thomas supports Monica May 21 '10 at 3:07
• As a terminological point, it is usual in category theory to say that a "split monomorphism" is one with a retract, whereas one which satisfies the conclusion of your question is "complemented." So your question would be less confusing if it were titled "when are all split monomorphisms complemented?" – Mike Shulman May 21 '10 at 18:30
• I retitled it accordingly. – Jakob May 22 '10 at 7:42

In the setting of the question, the composite $f\circ g$ is idempotent. Thus one hypothesis on $\mathcal C$ that will guarantee the existence of the desired splittings is that $\mathcal C$ be Karoubian, or (same concept, alternative name) pseudo-abelian, which is to say: $\mathcal C$ is pre-additive (the Hom-sets are abelian groups) and all idempotents have kernels (and hence cokernels).