Monomorphism

That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.

), then f is monic, as A left-invertible morphism is called a split mono or a section.

In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms.

The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator.

It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense.

If h : G → Q, where G is some divisible group, and q ∘ h = 0, then h(x) ∈ Z, ∀ x ∈ G. Now fix some x ∈ G. Without loss of generality, we may assume that h(x) ≥ 0 (otherwise, choose −x instead).

To go from that implication to the fact that q is a monomorphism, assume that q ∘ f = q ∘ g for some morphisms f, g : G → Q, where G is some divisible group.

While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms.