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In the context of
abstract algebra or
universal algebra, an '''epimorphism''' is simply a
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
:''g''
1 O ''f'' = ''g''
2 O ''f'' implies ''g''
1 = ''g''
2
for all morphisms ''g''
1, ''g''
2 : ''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 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 epimorhpism.
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.
== See also ==
*
Monomorphism
*
Isomorphism
