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

olivierd

Jan 23, 2023 3 min read

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.