# Epimorphisms of groups

In my previous post, we saw that injections (surjections) in concrete categories are always monomorphisms (epimorphisms), and in some cases the converse holds.

We now wish to classify all epimorphisms of groups. To show that all epimorphisms are surjective, for any strict subgroup we want to construct maps to some group that differ on but agree on . In the case of abelian groups this is relatively easy, because we can take to be the cokernel, the quotient map, and the zero map. But in general the cokernel only exists if the image is normal, so a different argument is needed.

Lemma. Let be a group homomorphism. Then is an epimorphism if and only if is surjective.

Proof. We already saw that surjections are epimorphisms. Conversely, let be an epimorphism of groups. We may replace by its image in , since the map is still an epimorphism. Let be the coset space, viewed as a pointed set with distinguished element . Let be the set “ with the distinguished point doubled”, and write and for these distinguished points.

Let be the symmetric group on , and define homomorphisms by letting act naturally on the copy of in (for ). Since the action of on fixes the trivial coset , we see that the maps agree. Since is an epimorphism, this forces . But then showing that is surjective (and a fortiori ). Note however that the result is not true in every algebraic category. For example, the map is an epimorphism of (commutative) rings that is not surjective. More generally, every localisation is an epimorphism, by the universal property of localisation; these maps are rarely surjective.

# Concrete categories and monomorphisms

This post serves to collect some background on concrete categories for my next post.

Concrete categories are categories in which objects have an underlying set:

Definition. A concrete category is a pair of a category with a faithful functor . In cases where is understood, we will simply say is a concrete category.

Example. The categories of groups, of topological spaces, of rings, and of -modules are concrete in an obvious way. The category of sheaves on a site with enough points is concrete by mapping a sheaf to the disjoint union of its stalks (the same holds for any Grothendieck topos, but a different argument is needed). Similarly, the category of schemes can be concretised by sending to , where is the contravariant power set functor.

Today we will study the relationship between monomorphisms and injections in :

Lemma. Let be a concrete category, and let be a morphism in . If is a monomorphism (resp. epimorphism), then so is .

Proof. A morphism in is a monomorphism if and only if the induced map is injective. Faithfulness implies that the vertical maps in the commutative diagram are injective, hence if the bottom map is injective so is the top. The statement about epimorphisms follows dually. For example, this says that any injection of groups is a monomorphism, and any surjection of rings is an epimorphism, since the monomorphisms (epimorphisms) in are exactly the injections (surjections).

In some concrete categories, these are the only monomorphisms and epimorphisms. For example:

Lemma. Let be a concrete category such that the forgetful functor admits a left (right) adjoint. Then every monomorphism (epimorphism) in is injective (surjective).

Proof. If is a right adjoint, it preserves limits. But is a monomorphism if and only if the square is a pullback. Thus, preserves monomorphisms if it preserves limits. The statement about epimorphisms is dual. For example, the forgetful functors on algebraic categories like , , and have left adjoints (a free functor), so all monomorphisms are injective.

The forgetful functor has adjoints on both sides: the left adjoint is given by the discrete topology, and the right adjoint by the indiscrete topology. Thus, monomorphisms and epimorphisms in are exactly injections and surjections, respectively.

On the other hand, in the category of Hausdorff topological spaces, the inclusion is an epimorphism that is not surjective. Indeed, a map to a Hausdorff space is determined by its values on .