# Morphism=monomorphism•epimorphism?

Is it true that any morphism in any category can be written as a combination of monomorphism and epimorphism? In SET and categories where monomorphism is an injective function and epimorphism is a surjective function is it true. But I am interested about the issue in any category. How to prove that statement or there is a counterexample?

• – Arnaud D. Nov 6 '18 at 17:08

In general there is no reason for this to be true. A simple counterexample is the category with one object $$x$$ and exactly one non-identity arrow $$f:x\to x$$, where $$f\circ f=f$$. In this category $$f$$ is neither a monomorphism nor an epimorphism, so the only monomorphism/epimorphism is the identity, and $$f$$ is not a composition of the identity with itself. More generally, if $$\mathcal{C}$$ is a category where every epimorphism or monomorphism is an isomorphism, then clearly an arrow that is not an isomorphism cannot be decomposed like that (though I can't think of any other example of this kind).
• @Vladislav It's certainly not enough to be regular or coregular, and I'm pretty sure it is not enough to have epi-mono factorizations. I think one can make a variation of my counterexample (to a different question) in this answer by adding a collection of endomaps $\alpha_n:A\to A$ and a kernel of $f$, in such a way that $f$ would not have any factorization. – Arnaud D. Oct 3 '19 at 9:52