Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ${\displaystyle X\hookrightarrow Y}$.

In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for 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.

The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category C^op. Every section is a monomorphism, and every retraction is an epimorphism.

Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ${\displaystyle l\circ f=\operatorname {id} _{X}}$), then f is monic, as

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