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## Regular and effective monomorphisms and epimorphisms

Previously we observed that although monomorphisms tended to give expected generalizations of injective function in many categories, epimorphisms sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective morphisms in familiar concrete categories, the definition of a split epimorphism is too restrictive.

In this post we will discuss two other intermediate notions of epimorphism. (These all give dual notions of monomorphisms, but their epimorphic variants are more interesting as a possible solution to the above problem.) There are yet others, but these two appear to be the most relevant in the context of abelian categories.

## Split epimorphisms and split monomorphisms

• What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
• What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
• A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
• If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?

The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.

In this post, for convenience all categories will be locally small (that is, $\text{Set}$-enriched).