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# Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism.

In the more general (and abstract) setting of category theory, a monomorphism (also called a monic morphism) is a morphism f : XY such that

f o g1 = f o g2 implies g1 = g2

for all morphisms g1, g2 : ZX.

The dual of a monomorphism is an epimorphism (i.e. a monomorphism in a category C is an epimorphism in the dual category Cop).

In the category of sets the monomorphisms are exactly the injective morphisms. Thus the algebraic and categorical notions are the same. The same is true in many other concrete categories such as those of groups, rings, and vector spaces.

An example of a monomorphism that is not injective arises in the category Div of divisible abelian groups and group homomorphisms between them. Consider the quotient q: QQ/Z. This is clearly not an injective map; nevertheless, it is a monomorphism in this category. To see this, note that if q o f = q o g for some morphisms f,g: G → Q where G is some divisible abelian group then q o h = 0 where h = f - g (this makes sense as this is an additive category). This implies that h(x) is an integer if xG. If h(x) is not 0 then, for instance,

$h\left(\frac{x}{4h(x)}\right) = \frac{1}{4}$

so that

$(q \circ h)\left(\frac{x}{4h(x)}\right) \neq 0$,

contradicting q o h = 0, so h(x) = 0 and q is therefore a monomorphism [1].

There are also useful concepts of regular monomorphism and extremal monomorphism. A regular monomorphism equalizes some parallel pair of morphisms. An extremal monomorphism is a monomorphism that has no epimorphism as a first factor, unless that epimorphism is an isomorphism.