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Suppose that $f: A \to B$ is an epimorphism and $x: X \to B$ is a morphism. Is it true that there exists a morphism $y: X \to A$ such that $f \circ y = x$? Is it necessary that the category is abelian?

I can't see how I might prove this, but also I can't think of an example of an abelian category where it is not true.

Harry Partridge
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  • Yes that is very helpful. My question then becomes, is there a type of category in which every object is projective? Obviously Set is one such category, but is there a more general class of such categories? – Harry Partridge Aug 03 '23 at 07:12
  • Certainly, if $f$ is a split epimorphism then such a $y$ exists. And conversely, if such a morphism $y$ exists in the case that $x$ is the identity on $B$, then that says exactly that $f$ is a split epimorphism. – Daniel Schepler Aug 03 '23 at 17:32

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