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{{Short description|Surjective homomorphism}}
{{about|the mathematical function|the biological phenomenon|Epimorphosis}}
[[Image:Epimorphism scenarios.svg|right|thumb|220px]]

In [[category theory]], an '''epimorphism''' is a [[morphism]] ''f'' : ''X'' → ''Y'' that is [[Cancellation property|right-cancellative]] in the sense that, for all objects ''Z'' and all morphisms {{nobreak|''g''<sub>1</sub>, ''g''<sub>2</sub>: ''Y'' → ''Z''}},
: <math>g_1 \circ f = g_2 \circ f \implies g_1 = g_2.</math>

Epimorphisms are categorical analogues of [[surjective function|onto or surjective function]]s (and in the [[category of sets]] the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion <math> \mathbb{Z}\to\mathbb{Q} </math> is a ring epimorphism. The [[dual (category theory)|dual]] of an epimorphism is a [[monomorphism]] (i.e. an epimorphism in a [[category (mathematics)|category]] ''C'' is a monomorphism in the [[Dual (category theory)|dual category]] ''C''<sup>op</sup>).

Many authors in [[abstract algebra]] and [[universal algebra]] define an '''epimorphism''' simply as an ''onto'' or [[surjective]] [[homomorphism]]. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see {{section link||Terminology}} below.

==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]] ''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 ''f'' : (''X'',&nbsp;≤) → (''Y'',&nbsp;≤) is not surjective, pick ''y''<sub>0</sub> in ''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]] ''g'' <sub>1</sub>: ''Y'' → ''Y''/''f''(''X'') and the [[zero map]] ''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 ''y''&nbsp;&isin;&nbsp;''Y''&nbsp;−&nbsp;''fX''. Since ''fX'' is closed, by [[Urysohn's Lemma]] there is a continuous function ''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 ''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.

==Properties==
Every [[isomorphism]] is an epimorphism; indeed only a right-sided inverse is needed: if there exists a morphism ''j'' : ''Y'' → ''X'' such that ''fj'' = id<sub>''Y''</sub>, then ''f'': ''X'' → ''Y'' is easily seen to be an epimorphism.  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 the morphism alone, 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 categories and also in all the concrete categories mentioned above in {{section link||Examples}} (though not in all concrete categories).

== 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 '''Q'''&nbsp;→ '''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 '''Z'''&nbsp;→ '''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 ''j''&nbsp;''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''.

== Terminology ==
The companion terms ''epimorphism'' and ''[[monomorphism]]'' were first introduced by [[Nicolas Bourbaki|Bourbaki]]. Bourbaki uses ''epimorphism'' as shorthand for a [[surjective function]]. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections.  Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.  [[Saunders Mac Lane]] attempted to create a distinction between ''epimorphisms'', which were maps in a concrete category whose underlying set maps were surjective, and ''epic morphisms'', which are epimorphisms in the modern sense.  However, this distinction never caught on.

It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept.  Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior. It is very difficult, for example, to classify all the epimorphisms of rings.  In general, epimorphisms are their own unique concept, related to surjections but fundamentally different.

== See also ==
*[[List of category theory topics]]
*[[Monomorphism]]

==Notes==
{{reflist}}

== References ==
{{ref begin}}
*{{cite book|last1=Adámek|first1=Jiří|last2=Herrlich|first2=Horst|last3=Strecker|first3=George E.|title=Abstract and Concrete Categories |url=http://katmat.math.uni-bremen.de/acc/acc.pdf |year=1990|publisher=John Wiley & Sons.|isbn=0-471-60922-6}}
*{{cite book|last=Bergman|first=George|title=An Invitation to General Algebra and Universal Constructions |url=http://math.berkeley.edu/~gbergman/245/index.html |year=2015|publisher=Springer|isbn=978-3-319-11478-1}}
*{{cite book|last=Borceux|first=Francis|title=Handbook of Categorical Algebra. Volume 1: Basic Category Theory|year=1994|publisher=Cambridge University Press|isbn=978-0521061193}}
*{{cite book |isbn=9780486809038|url=https://math.jhu.edu/~eriehl/context/|title=Category Theory in Context |last1=Riehl |first1=Emily |year=2016|publisher=Dover Publications, Inc Mineola, New York}}
*{{cite book|last1=Tsalenko|first1=M.S.| last2=Shulgeifer|first2=E.G.|title=Foundations of category theory|year=1974|publisher=Nauka|isbn=5-02-014427-4}}
*{{springer|title=Epimorphism|id=p/e035890}}
*{{cite book|last1=Lawvere|first1=F. William|last2=Rosebrugh|first2=Robert|title=Sets for Mathematics|url=https://archive.org/details/setsformathemati0000lawv|year=2015|publisher=Cambridge university press|isbn=978-0-521-80444-8|url-access=registration}}
*{{cite journal|last=Linderholm|first=Carl|title=A Group Epimorphism is Surjective |journal=American Mathematical Monthly |url=https://groups.google.com/group/sci.math/msg/6d4023d93a2b4300|year=1970|volume=77|issue=2 |pages=176–177|doi=10.1080/00029890.1970.11992448}}
{{ref end}}

==External links==

*{{nlab|id=epimorphism}}
*{{nlab|id=strong+epimorphism|title=Strong epimorphism}}


[[Category:Morphisms]]
[[Category:Algebraic properties of elements]]