Editing Epimorphism (section)

== Related concepts ==
Among other useful concepts are ''regular epimorphism'', ''extremal epimorphism'', ''immediate epimorphism'', ''strong epimorphism'', and ''split epimorphism''. 

* An epimorphism is said to be '''regular''' if it is a [[coequalizer]] of some pair of parallel morphisms. 
* An epimorphism <math>\varepsilon</math> is said to be '''extremal'''{{sfn|Borceux|1994}} if in each representation <math>\varepsilon=\mu\circ\varphi</math>, where <math>\mu</math> is a [[monomorphism]], the morphism <math>\mu</math> is automatically an [[isomorphism]]. 
* An epimorphism <math>\varepsilon</math> is said to be '''immediate''' if in each representation <math>\varepsilon=\mu\circ\varepsilon'</math>, where <math>\mu</math> is a [[monomorphism]] and <math>\varepsilon'</math> is an epimorphism, the morphism <math>\mu</math> is automatically an [[isomorphism]].
* [[File:Diagram-orthogonality-2.jpg|thumb]] An epimorphism <math>\varepsilon:A\to B</math> is said to be '''strong'''{{sfn|Borceux|1994}}{{sfn|Tsalenko|Shulgeifer|1974}} if for any [[monomorphism]] <math>\mu:C\to D</math> and any morphisms <math>\alpha:A\to C</math> and <math>\beta:B\to D</math> such that <math>\beta\circ\varepsilon=\mu\circ\alpha</math>, there exists a morphism <math>\delta:B\to C</math> such that <math>\delta\circ\varepsilon=\alpha</math> and <math>\mu\circ\delta=\beta</math>.
* An epimorphism <math>\varepsilon</math> is said to be '''split''' if there exists a morphism <math>\mu</math> such that <math>\varepsilon\circ\mu=1</math> (in this case <math>\mu</math> is called a right-sided inverse for <math>\varepsilon</math>).

There is also the notion of '''homological epimorphism''' in ring theory. A morphism  ''f'': ''A'' → ''B'' of rings is a homological epimorphism if it is an epimorphism and it induces a [[full and faithful functor]] on [[derived categories]]:
D(''f'') : D(''B'') → D(''A'').

A morphism that is both a monomorphism and an epimorphism is called a [[bimorphism]]. Every isomorphism is a bimorphism but the converse is not true in general. For example, the map from the [[half-open interval]] [0,1) to the [[unit circle]] S<sup>1</sup> (thought of as a [[topological subspace|subspace]] of the [[complex plane]]) that sends ''x'' to exp(2πi''x'') (see [[Euler's formula]]) is continuous and bijective but not a [[homeomorphism]] since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the category '''Top'''. Another example is the embedding {{math|'''Q''' → '''R'''}} in the category '''Haus'''; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism. Similarly, in the category of [[ring (algebra)|ring]]s, the map {{math|'''Z''' → '''Q'''}} is a bimorphism but not an isomorphism.

Epimorphisms are used to define abstract [[quotient object]]s in general categories: two epimorphisms ''f''<sub>1</sub> : ''X'' → ''Y''<sub>1</sub> and ''f''<sub>2</sub> : ''X'' → ''Y''<sub>2</sub> are said to be ''equivalent'' if there exists an isomorphism ''j'' : ''Y''<sub>1</sub> → ''Y''<sub>2</sub> with {{math|1=''j'' ''f''<sub>1</sub> = ''f''<sub>2</sub>.}} This is an [[equivalence relation]], and the equivalence classes are defined to be the quotient objects of ''X''.