Editing Epimorphism (section)

==Examples==
Every morphism in a [[concrete category]] whose underlying [[function (mathematics)|function]] is [[surjective]] is an epimorphism.  In many concrete categories of interest the converse is also true. For example, in the following categories, the epimorphisms are exactly those morphisms that are surjective on the underlying sets:
*'''[[category of sets|Set]]''': [[Set (mathematics)|sets]] and functions. To prove that every epimorphism ''f'': ''X'' → ''Y'' in '''Set''' is surjective, we compose it with both the [[indicator function|characteristic function]] {{math|''g''<sub>1</sub>: ''Y'' → {0,1} }} of the image ''f''(''X'') and the map ''g''<sub>2</sub>: ''Y'' → {0,1} that is constant 1.
*'''Rel''': sets with [[binary relation]]s and relation-preserving functions. Here we can use the same proof as for '''Set''', equipping {0,1} with the full relation {0,1}&times;{0,1}.
*'''Pos''': [[partially ordered set]]s and [[monotone function]]s. If {{math|''f'' : (''X'', ≤) → (''Y'', ≤)}} is not surjective, pick ''y''<sub>0</sub> in {{math|''Y'' \ ''f''(''X'')}} and let ''g''<sub>1</sub> : ''Y'' → {0,1} be the characteristic function of {''y'' | ''y''<sub>0</sub> ≤ ''y''} and ''g''<sub>2</sub> : ''Y'' → {0,1} the characteristic function of {''y'' | ''y''<sub>0</sub> < ''y''}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
*'''[[category of groups|Grp]]''': [[group (mathematics)|groups]] and [[group homomorphism]]s. The result that every epimorphism in '''Grp''' is surjective is due to [[Otto Schreier]] (he actually proved more, showing that every [[subgroup]] is an [[equaliser (mathematics)|equalizer]] using the [[free product]] with one amalgamated subgroup); an [[elementary proof]] can be found in (Linderholm 1970).
*'''FinGrp''': [[finite groups]] and group homomorphisms. Also due to Schreier; the proof given in (Linderholm 1970) establishes this case as well.
*'''[[category of abelian groups|Ab]]''': [[abelian group]]s and group homomorphisms.
*'''[[Category of vector spaces|''K''-Vect]]''': [[vector space]]s over a [[field (mathematics)|field]] ''K'' and [[linear transformation|''K''-linear transformations]].
*'''Mod'''-''R'': [[module (mathematics)|right module]]s over a [[ring (mathematics)|ring]] ''R'' and [[module homomorphism]]s. This generalizes the two previous examples; to prove that every epimorphism ''f'': ''X'' → ''Y'' in '''Mod'''-''R'' is surjective, we compose it with both the canonical [[quotient module|quotient map]] {{math|''g'' <sub>1</sub>: ''Y'' → ''Y''/''f''(''X'')}} and the [[zero map]] {{math|''g''<sub>2</sub>: ''Y'' → ''Y''/''f''(''X'').}}
*'''[[Category of topological spaces|Top]]''': [[topological spaces]] and [[continuous function]]s. To prove that every epimorphism in '''Top''' is surjective, we proceed exactly as in '''Set''', giving {0,1} the [[trivial topology|indiscrete topology]], which ensures that all considered maps are continuous.
*'''HComp''': [[compact space|compact]] [[Hausdorff space]]s and continuous functions. If ''f'': ''X'' → ''Y'' is not surjective, let {{math|''y'' &isin; ''Y'' − ''fX''.}} Since ''fX'' is closed, by [[Urysohn's Lemma]] there is a continuous function {{math|''g''<sub>1</sub>:''Y'' → [0,1]}} such that ''g''<sub>1</sub> is 0 on ''fX'' and 1 on ''y''. We compose ''f'' with both ''g''<sub>1</sub> and the zero function {{math|''g''<sub>2</sub>: ''Y'' → [0,1].}}
However, there are also many concrete categories of interest where epimorphisms fail to be surjective. A few examples are:
*In the [[Monoid (category theory)|category of monoids]], '''Mon''', the [[inclusion map]] '''N''' → '''Z''' is a non-surjective epimorphism.  To see this, suppose that ''g''<sub>1</sub> and ''g''<sub>2</sub> are two distinct maps from '''Z''' to some monoid ''M''.  Then for some ''n'' in '''Z''', ''g''<sub>1</sub>(''n'') ≠ ''g''<sub>2</sub>(''n''), so ''g''<sub>1</sub>(−''n'') ≠ ''g''<sub>2</sub>(−''n'').  Either ''n'' or −''n'' is in '''N''', so the restrictions of ''g''<sub>1</sub> and ''g''<sub>2</sub> to '''N''' are unequal.
*In the category of algebras over commutative ring '''R''', take '''R'''['''N'''] → '''R'''['''Z'''], where '''R'''['''G'''] is the [[monoid ring]] of the monoid '''G''' and the morphism is induced by the inclusion '''N''' → '''Z''' as in the previous example. This follows from the observation that '''1''' generates the algebra '''R'''['''Z'''] (note that the unit in '''R'''['''Z'''] is given by '''0''' of '''Z'''), and the inverse of the element represented by '''n''' in '''Z''' is just the element represented by −'''n'''. Thus any homomorphism from '''R'''['''Z'''] is uniquely determined by its value on the element represented by '''1''' of '''Z'''.
*In the [[category of rings]], '''Ring''', the inclusion map '''Z''' → '''Q''' is a non-surjective epimorphism; to see this, note that any [[ring homomorphism]] on '''Q''' is determined entirely by its action on '''Z''', similar to the previous example. A similar argument shows that the natural ring homomorphism from any [[commutative ring]] ''R'' to any one of its [[localization of a ring|localizations]] is an epimorphism.
*In the [[category of commutative rings]], a [[Finitely generated object|finitely generated]] homomorphism of rings ''f'' : ''R'' → ''S'' is an epimorphism if and only if for all [[prime ideal]]s ''P'' of ''R'', the ideal ''Q'' generated by ''f''(''P'') is either ''S'' or is prime, and if ''Q'' is not ''S'', the induced map [[Field of fractions|Frac]](''R''/''P'') → Frac(''S''/''Q'') is an [[isomorphism]] ([[Éléments de géométrie algébrique|EGA]] IV 17.2.6).
*In the category of Hausdorff spaces, '''Haus''',  the epimorphisms are precisely the continuous functions with [[dense set|dense]] images.  For example, the inclusion map '''Q''' → '''R''', is a non-surjective epimorphism.

The above differs from the case of monomorphisms where it is more frequently true that monomorphisms are precisely those whose underlying functions are [[injective]].

As for examples of epimorphisms in non-concrete categories:
* If a [[monoid]] or [[ring (mathematics)|ring]] is considered as a category with a single object (composition of morphisms given by multiplication), then the epimorphisms are precisely the right-cancellable elements.
* If a [[directed graph]] is considered as a category (objects are the vertices, morphisms are the paths, composition of morphisms is the concatenation of paths), then ''every'' morphism is an epimorphism.