PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Very low
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) ]
(view preamble)

View style:

See Also: monic

Other names:  epimorphism, epimorphic
Also defines:  epic
Log in to rate this entry.
(view current ratings)

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 2466 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
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)