0
$\begingroup$

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

  • 2
    That property is called the (internal) axiom of choice: http://ncatlab.org/nlab/show/axiom+of+choice2012-02-11
  • 1
    It is surprising that you have not found examples to answer this yourself... What exactly did you try doin before asking?2012-02-11

2 Answers 2

5

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 $2$.

  • 0
    Do all surjective morphisms $B\oplus C\to B$ have a section?2012-02-11
  • 0
    @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.2012-02-12
4

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

  • 1
    Every epi out of $Z$ which is not an iso does not have a section!2012-02-11