Monomorphisms and epimorphisms
The classes of monomorphisms, epimorphisms, bimorphisms, split monomorphisms, and split epimorphisms are closed under composition and contain all isomorphisms. To show these classes are closed under composition, consider two morphisms and . If and are monomorphisms then for ,
and so is a monomorphism. If and are epimorphisms then for ,
and so is an epimorphism. If and are split monomorphisms then there exists and such that and . So is a split monomorphism since . Similarly, if and are split epimorphisms then and so is a split epimorphism.
Let be an isomorphism then, has inverse and so for every pair of morphisms and we have that
and
Hence, every isomorphism is a monomorphism and an epimorphism and so it is a bimorphism. Also, every isomorphism is a split monomorphism and a split epimorphism since it has a unique inverse.
Let and be two morphisms:
- If the composition is a monomorphism then for , and so is a monomorphism.
- If the composition is an epimorphism then for , and so is an epimorphism.
- If the composition is a split monomorphism then there exists a morphism such that , so and hence, is a split monomorphism.
- If the composition is a split epimorphism then there exists a morphism such that , so and hence, is a split epimorphism.
- If is a split monomorphism then there is a morphism with and so and since is an epimorphism, we have that . Hence, is an isomorphism.
The axiom of choice can be stated as follows.
For every surjective map of sets , there exists a map with 1.
Using this, and the fact that every epimorphism in is a surjective map of sets, it is easy to see that the axiom of choice can be formulated as
Every epimorphism in is split 1.
The inclusion map from the additive monoid of natural numbers to the additive group of integers (considered as a monoid) is a bimorphism in . Let denote the additive monoid of natural numbers and denote the additive group of integers considered as a monoid. Then for every object in and for every pair of morphisms we have that if then for every , and so . Hence, is a monomorphism. Additionally, for every pair of morphisms and , for all with . But this means and so since . Thus, we have that and so . Hence, is an epimorphism and so it is a bimorphism.