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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 $$\xymatrix{B \ar[r]^{g_1} & C}=\xymatrix{B \ar[r]^{g_2} & C}.$$
An epimorphism is just an epi morphism, and epi is also known as right cancellable, epimorphic, or simply epic.
Remarks.
- 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).
- 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.
- The dual notion of epi is that of monic.
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"epi" is owned by CWoo. [ full author list (2) ]
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See Also: monic
| Other names: |
epimorphism, epimorphic |
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Cross-references: groups, well-defined, surjections, onto, function, implies, property, abelian category, right, diagram, object, category, morphism
There are 17 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 4100 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) |
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Pending Errata and Addenda
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