Epimorphism

From Exampleproblems

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 : X → Y such that

g₁ o f = g₂ o f implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z.

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).

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 N → Z 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 Z → Q 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.

Notes on Usage

The use of the words epimorphism and monomorphism is somewhat unsettled. They were originally introduced by Bourbaki to mean surjective and injective, respectively. Early category theorists argued that the correct category-theoretic generalization of surjective was the definition of epimorphism 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 epic morphism and epi to refer to the category-theoretic concept, but this distinction has not caught on. There are many areas where the category-theoretic meaning is well established or at least used very often, e.g. in semigroups or modules.

See also

• Monomorphism
• Isomorphism

References

• George M. Bergman, An invitation to General Algebra and Universal Constructions Harry Helson Publisher, 1998, Berkeley. ISBN 0-9655211-4-1.de:Epimorphismus