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Are there some examples of epimorphisms which are not split epimorphisms? Thank you very much.

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This question isn't really focussed. There are lots lots lots of examples, in various categories, for various objects, ... – Martin Brandenburg May 6 '13 at 9:19
In fact,it is harder to find examples where all epis are split, in a sense! :-) – Mariano Suárez-Alvarez May 6 '13 at 9:31
up vote 5 down vote accepted

The projection from $\mathbb{Z}$ to $\mathbb{Z}/2\mathbb{Z}$ (in, for example, the category of abelian groups). It has no splitting, because there are no non-trivial homomorphism from $\mathbb{Z}/2\mathbb{Z}$ to $\mathbb{Z}$ due to lack of elements of order two in the latter group.

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"In the category of groups, the projection ..." – Martin Brandenburg May 6 '13 at 9:19
Thanks, @Martin. It is, indeed, essential to specify the category. I guess it was implicit in my answer :-) – Jyrki Lahtonen May 6 '13 at 9:21

Every split epimorphism is an coequalizer; every coequalizer is an epimorphism. So every non-coequalizer is a non-split epimorphism.

Addition: Thanking Zhen Lin, I changed "equalizer" by "coequalizer".

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You mean coequaliser. – Zhen Lin May 6 '13 at 10:04
@Zhen Lin:Yes, thank you. I will correct the answer. – Boris Novikov May 6 '13 at 10:35
This is not an example in the usual sense of the word "example". – Martin Brandenburg May 6 '13 at 13:40
@Martin Brandenburg: Of course. It's a way to get examples. – Boris Novikov May 6 '13 at 14:06

See this blog post and this blog post for examples. (Any fake isomorphism, in the terminology of the second post, is mono and epi but cannot be either split mono or split epi.) Being split is a very strong condition, and in fact epimorphisms just don't split in general. $\text{Set}$ is a bad place to look for intuition about the different kinds of epimorphisms.

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