# Epimorphism and Monomorphism = Isomorphism?

It seems to be that if a map is both an epimorphism and a monomorphism, it is not necessarily the case that it is an isomorphism.

However, in the category of sets, if a map is both an epimorphism and a monomorphism it is an isomorphism.

Are these accurate statements?

• Some notes: the converse holds, i.e. any isomorphism is both an epimorphism and a monomorphism. Also in a topos (a nice kind of category), any morphism that is both epic (epimorphism) and monic (monomorphism) is iso (isomorphism). en.wikipedia.org/wiki/Monomorphism Jul 10 '16 at 4:33
• In abelian categories this statements holds, too. Mar 12 '19 at 16:32
• Apparently categories where this holds are known as balanced categories Mar 19 at 9:56

Yes, those are accurate statements.

The inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is both a monomorphism and epimorphism in the category $\mathsf{Ring}$ (rings and ring homomorphisms), but not an isomorphism. The inclusion $\mathbb{Q}\hookrightarrow\mathbb{R}$ is both a monomorphism and epimorphism in the category $\mathsf{Haus}$ (Hausdorff topological spaces and continuous maps), but it is not an isomorphism.

In the category $\mathsf{Set}$, monomorphisms and epimorphisms are precisely the injective and surjective maps, respectively, so that a map of sets that is both a monomorphism and epimorphism is a bijection, i.e. an isomorphism of sets.

More examples and information can be found in Wikipedia and in Mac Lane's Categories for the Working Mathematician.

A simpler example: take a category with two objects and three morphisms, one of which goes from one object to the other. This last morphism is monic and epic, but not an iso.

Yes, that's correct. For example, every morphism in a poset is a monomorphism and an epimorphism, but only the identity morphisms are isomorphisms. (Mariano's answer describes the smallest poset with a non-identity morphism.)

However, there are various salvages of this statement available:

1. A morphism which is both a monomorphism and a split epimorphism (resp. both an epimorphism and a split monomorphism) is an isomorphism. See this blog post. In the category of sets, every monomorphism (edit: with non-empty domain) is split, and the statement that every epimorphism is split is equivalent to the axiom of choice.

2. Better: a morphism which is both a monomorphism and a regular epimorphism (resp. both an epimorphism and a regular monomorphism) is an isomorphism. See this blog post. In the category of sets, every epimorphism and monomorphism is regular (with no need to invoke the axiom of choice).

For more on the subject of morphisms which are both epimorphisms and monomorphisms but which are not isomorphisms see this blog post.

• Every monomorphism with non-empty domain is split. Dec 29 '12 at 9:03
• Whoops. I think I've been corrected on this in the past too. Dec 29 '12 at 9:19
• Great blog. Will be following it. Hope you keep posting. Dec 29 '12 at 19:48