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epi (Definition)

A morphism $ f : A\to B$ in a category $ \mathcal{C}$ is called epi if for any object $ C$ and any morphisms $ g_1,g_2 : B\to C$, if $ g_1 f = g_2 f$ then $ g_1 = g_2$. In other words, any diagram

$ \xymatrix{A \ar[r]^f & B \ar[r]^{g_1} & C}=\xymatrix{A \ar[r]^f & B \ar[r]^{g_2} & C}$
reduces to the diagram
$\displaystyle \xymatrix{B \ar[r]^{g_1} & C}=\xymatrix{B \ar[r]^{g_2} & C}.$

An epimorphism is just an epi morphism, and epi is also known as right cancellable, epimorphic, or simply epic.

Remarks.

  1. If $ \mathcal{C}$ is an abelian category, then an epi has the property that $ gf=0$ implies $ g=0$ (surely, since $ gf=0=0f$, and the result follows).
  2. Epi is the generalization of a function being onto. In some categories where surjections are well-defined (such as sets and groups), epi is the same as being onto.
  3. The dual notion of epi is that of monic.



"epi" is owned by CWoo. [ full author list (2) ]
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See Also: monic

Other names:  epimorphism, epimorphic
Also defines:  epic
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Cross-references: groups, well-defined, surjections, onto, function, implies, property, abelian category, right, object, category, morphism
There are 9 references to this entry.

This is version 11 of epi, born on 2004-11-21, modified 2007-06-16.
Object id is 6507, canonical name is Epi.
Accessed 2475 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms)
Pending Errata and Addenda
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