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 : X → Y such that

f O g₁ = f O g₂ implies g₁ = g₂

for all morphisms g₁, g₂ : Z → X.

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

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. (Are there any counterexamples?)

There are also useful concepts of regular monomorphism and extremal monomorhpism. 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.

See also:

• epimorphism
• isomorphism
• subobject