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Is it true that all epimorphisms have sections? Or does this depend on the category we are in?

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That property is called the (internal) axiom of choice: ncatlab.org/nlab/show/axiom+of+choice –  Colin McQuillan Feb 11 '12 at 21:36
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It is surprising that you have not found examples to answer this yourself... What exactly did you try doin before asking? –  Mariano Suárez-Alvarez Feb 11 '12 at 23:50

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up vote 5 down vote accepted

Very much depends on the category.

For example, in the category of all abelian groups, epimorphisms are all surjective on the underlying set. An surjection $A\to B$ has a section if and only if we can write $A$ as $A\cong B\oplus C$ in such a way that the surjection corresponds to projection onto the first coordinate. But, for example, the epimorphism $\mathbb{Z}/4\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}$ does not have a section, since $\mathbb{Z}/4\mathbb{Z}$ is not the direct sum of two groups of order $2$.

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Do all surjective morphisms $B\oplus C\to B$ have a section? –  Mariano Suárez-Alvarez Feb 11 '12 at 23:51
    
@Mariano: Ah, good point. No; take $C_2\oplus C_4$, and the surjection to $C_2$ that maps the first coordinate to $0$ and the second coordinate to the generator. Thanks for the prompt. –  Arturo Magidin Feb 12 '12 at 3:21

In the category of Rings, consider the epimorphism $\mathbb{Z}\to\mathbb{Q}$.

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Every epi out of $Z$ which is not an iso does not have a section! –  Mariano Suárez-Alvarez Feb 11 '12 at 23:53

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