Does there exists an example of a category that comes equipped with an $(Epi,Mono)$-factorization structure and such that epimorphisms are not split?
The choice of the factorization structure is too restrictive. So far, I think to the category of sets and functions, where any epimorphism is split. I hope that an $(Epi,Mono)$-category whose epimorphisms are split is not isomorphic to the category of sets, otherwise the assumptions I'm considering are very restrictive and useless.