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.

File:Monomorphism-01.png

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,

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

so that

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

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.

Note on usage

The use of the words monomorphism and epimorphism is somewhat unsettled. They were originally introduced by Bourbaki to mean injective and surjective, respectively. Early category theorists argued that the correct category-theoretic generalization of injective (one-to-one) was the definition of monomorphism given above, and simply gave the word this new, somewhat different, meaning; this made the term ambiguous. Mac Lane attempted to restore its original meanings by using the terms monic morphism or mono to refer to the category-theoretic concept, but this distinction has not caught on. However, whereas the difference is more notable in the case of epimorphisms, in "most" naturally occurring categories of algebras the categorical and algebraic meaning coincide because in any concrete category with a free object on a one element set the categorical monomorphisms are all one-to-one.