Concrete categories: injective/surjective implies mono/epi; functor generalization

A category \mathrm C is by definition concrete if there exists a faithful functor \mathcal F from \mathrm C to the category of sets \mathrm {Set}. (A functor is faithful if it is one-to-one (injective) regarding morphisms.)

It happens to be the case that in the category \mathrm {Set}, monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections.

We will show that for any concrete category \mathrm C, with faithful functor \mathcal F from \mathrm C to \mathrm {Set}, a sufficient (though not always necessary) condition for a morphism \phi in \mathrm C to be mono is that its image by \mathcal F be itself mono, that is, one-to-one.

More generally, we will show that if categories \mathrm C and \mathrm D are such that there exists a faithful functor \mathcal F from \mathrm C to \mathrm D, a sufficient condition for a morphism \phi in \mathrm C to be mono is that its image in \mathrm D by \mathcal F be itself mono. The same will hold for epimorphisms.

Proof

I will first prove the general case last mentioned.

Assume categories \mathrm C and \mathrm D with a faithful functor \mathcal F from the former to the latter.

Let A and B be two objects in \mathrm C and \phi a morphism from A to B such that \mathcal F(\phi) is a monomorphism.

Let X be an object in \mathrm C and \chi_1 and \chi_2 two morphisms from X to A such that \phi \circ \chi_1 = \phi \circ \chi_2.

Since \mathcal F is a functor, we can write \mathcal F(\phi \circ \chi_1) = \mathcal F(\phi) \circ \mathcal F(\chi_1), and similarly for \chi_2; hence:

\mathcal F(\phi) \circ \mathcal F(\chi_1) = \mathcal F(\phi) \circ \mathcal F(\chi_2)

Since \mathcal F(\phi) is a monomorphism, it follows that \mathcal F(\chi_1) = \mathcal F(\chi_2).

Since \mathcal F is faithful, this implies \chi_1 = \chi_2.

This being the case for any X object in \mathrm C and \chi_1 and \chi_2 morphisms from X to A such that \phi \circ \chi_1 = \phi \circ \chi_2, it follows that \phi is a monomorphism.

We have thus shown that if \mathcal F(\phi) is mono, then \phi itself is mono.

A similar reasoning shows that if \mathcal F(\phi) is epi, then \phi itself is epi.

In the case of concrete categories, \mathrm D is the set category, in which injective maps are mono and surjective maps are epi. Typically, the \mathrm C-morphisms are particular mappings between the underlying sets, that is the sets obtained through the functor \mathcal F. In this case, one can say that injective implies mono and surjective implies epi.