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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 $Z$)?

Thanks for any answer.

(retitled the question as per the comment below)

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It's certainly true if C is abelian, but I guess you're looking for something a bit more interesting. –  Hugh Thomas 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
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2 Answers

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).

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This is true in boolean categories (extensive + terminal object + (T : 1 → 1 + 1) is subobject classifier), but for any monic, not just split monics. There's quite a nice and easy read on extensive categories from 93:

Carboni et al. Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra (1993) vol. 84 pp. 145-158

However, modulo axiom of choice, all monics in Set split, and I suspect this could be true for any boolean category.

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Even in Set, not all monics split: consider any function from an empty set to a nonempty one. (-: –  Mike Shulman May 21 '10 at 15:01
    
Also, I think the more usual definition of "Boolean category" includes regularity but not extensivity: ncatlab.org/nlab/show/Boolean%20category –  Mike Shulman May 21 '10 at 18:48
    
That sneaky empty set. :) –  Aleks Kissinger Jul 16 '10 at 15:38
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