Epimorphism

In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all morphisms g₁, g₂ : Y → Z,

Epimorphisms are categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts; for example, the inclusion ${\displaystyle \mathbb {Z} \to \mathbb {Q} }$ is a ring-epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category C^op).

Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see the section on Terminology below.

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