Concrete categories: injective/surjective implies mono/epi; functor generalization
A category is by definition concrete if there exists a faithful functor from to the category of sets . (A functor is faithful if it is one-to-one (injective) regarding morphisms.)
It happens to be the case that in the category , monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections.
We will show that for any concrete category , with faithful functor from to , a sufficient (though not always necessary) condition for a morphism in to be mono is that its image by be itself mono, that is, one-to-one.
More generally, we will show that if categories and are such that there exists a faithful functor from to , a sufficient condition for a morphism in to be mono is that its image in by be itself mono. The same will hold for epimorphisms.
I will first prove the general case last mentioned.
Assume categories and with a faithful functor from the former to the latter.
Let and be two objects in and a morphism from to such that is a monomorphism.
Let be an object in and and two morphisms from to such that .
Since is a functor, we can write , and similarly for ; hence:
Since is a monomorphism, it follows that .
Since is faithful, this implies .
This being the case for any object in and and morphisms from to such that , it follows that is a monomorphism.
We have thus shown that if is mono, then itself is mono.
A similar reasoning shows that if is epi, then itself is epi.
In the case of concrete categories, is the set category, in which injective maps are mono and surjective maps are epi. Typically, the -morphisms are particular mappings between the underlying sets, that is the sets obtained through the functor . In this case, one can say that injective implies mono and surjective implies epi.