# Epimorphism

In the context of abstract algebra or universal algebra, an epimorphism is simply an onto or surjective homomorphism.

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

g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : YZ.
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Epimorphism-01.png
Image:Epimorphism-01.png

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

In the category of sets the epimorphisms are exactly the surjective morphisms. Thus the algebraic and categorical notions are the same. This, however, does not always hold in other concrete categories. For example:

• In the category of monoids, Mon, the inclusion function NZ is a non-surjective monoid homomorphism, and hence not an algebraic epimorphism. It is, however, a epimorphism in the categorical sense.
• In the category of rings, Ring, the inclusion map ZQ is a categorical epimorphism but not an algebraic one. (To see this note that any ring homomorphism on Q is determined entirely by its action on Z).

In general, algebraic epimorphisms are always categorical ones but not vice-versa.

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

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