Editing Epimorphism (section)

==Properties==
Every [[isomorphism]] is an epimorphism; indeed only a right-sided inverse is needed: suppose there exists a morphism {{math|''j'' : ''Y'' → ''X''}} such that ''fj'' = id<sub>''Y''</sub>. For any morphisms <math>h_1, h_2: Y \to Z</math> where <math>h_1f = h_2f</math>, you have that <math>h_1 = h_1 id_Y = h_1fj = h_2fj = h_2</math>. A map with such a right-sided inverse is called a '''[[Section (category theory)|split epi]]'''.  In a [[topos]], a map that is both a [[monic morphism]] and an epimorphism is an isomorphism.

The composition of two epimorphisms is again an epimorphism. If the composition ''fg'' of two morphisms is an epimorphism, then ''f'' must be an epimorphism.

As some of the above examples show, the property of being an epimorphism is not determined by its behavior as a function, but also by the category of context. If ''D'' is a  [[subcategory]] of ''C'', then every morphism in ''D'' that is an epimorphism when considered as a morphism in  ''C'' is also an epimorphism in ''D''. However the converse need not hold; the smaller category can (and often will) have more epimorphisms.

As for most concepts in category theory, epimorphisms are preserved under [[equivalence of categories|equivalences of categories]]: given an equivalence ''F'' : ''C'' → ''D'', a morphism ''f'' is an epimorphism in the category ''C'' if and only if ''F''(''f'') is an epimorphism in ''D''. A [[Duality (category theory)|duality]] between two categories turns epimorphisms into monomorphisms, and vice versa.

The definition of epimorphism may be reformulated to state that ''f'' : ''X'' → ''Y'' is an epimorphism if and only if the induced maps
:<math>\begin{matrix}\operatorname{Hom}(Y,Z) &\rightarrow& \operatorname{Hom}(X,Z)\\
g &\mapsto& gf\end{matrix}</math>
are [[injective]] for every choice of ''Z''. This in turn is equivalent to the induced [[natural transformation]]
:<math>\begin{matrix}\operatorname{Hom}(Y,-) &\rightarrow& \operatorname{Hom}(X,-)\end{matrix}</math>
being a monomorphism in the [[functor category]] '''Set'''<sup>''C''</sup>.

Every [[coequalizer]] is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in particular that every [[cokernel]] is an epimorphism. The converse, namely that every epimorphism be a coequalizer, is not true in all categories.

In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphism ''f'' : ''G'' → ''H'', we can define the group ''K'' = im(''f'') and then write ''f'' as the composition of the surjective homomorphism ''G'' → ''K'' that is defined like ''f'', followed by the injective homomorphism ''K'' → ''H'' that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all [[Abelian category|abelian categories]] and also in all the concrete categories mentioned above in {{section link||Examples}} (though not in all concrete categories).