- Epimorphism
In
category theory an epimorphism (also called an epic morphism or an epi) is amorphism "f" : "X" → "Y" which is right-cancellative in the following sense: :"g"1 o "f" = "g"2 o "f" implies "g"1 = "g"2 for all morphisms "g"1, "g"2 : "Y" → "Z".Epimorphisms are analogues of
surjective function s, but they are not exactly the same. The dual of an epimorphism is amonomorphism (i.e. an epimorphism in a category "C" is a monomorphism in the dual category "C"op).Many authors in
abstract algebra anduniversal algebra define an epimorphism simply as an "onto" orsurjective 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 the section on Terminology below.Examples
Every morphism in a
concrete category whose underlying function issurjective 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 which are surjective on the underlying sets:
*Set, sets and functions. To prove that every epimorphism "f": "X" → "Y" in Set is surjective, we compose it with both the characteristic function "g"1: "Y" → {0,1} of the image "f"("X") and the map "g"2: "Y" → {0,1} that is constant 1.
*Rel, sets withbinary 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}×{0,1}.
*Pos,partially ordered set s andmonotone function s. If "f" : ("X",≤) → ("Y",≤) is not surjective, pick "y"0 in "Y" "f"("X") and let "g"1 : "Y" → {0,1} be the characteristic function of {"y" | "y"0 ≤ "y"} and "g"2 : "Y" → {0,1} the characteristic function of {"y" | "y"0 < "y"}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
*Grp, groups andgroup homomorphism s. The result that every epimorphism in Grp is surjective is due toOtto Schreier (he actually proved more, showing that every subgroup is an equalizer using thefree 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.
*Ab,abelian group s and group homomorphisms.
*"K"-Vect,vector space s over a field "K" and "K"-linear transformations.
*Mod-"R", right modules over a ring "R" andmodule 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 map "g" 1: "Y" → "Y"/"f"("X") and the zero map "g"2: "Y" → "Y"/"f"("X").
*Top,topological spaces andcontinuous function s. To prove that every epimorphism in Top is surjective, we proceed exactly as in Set, giving {0,1} the indiscrete topology which ensures that all considered maps are continuous.
*HComp, compactHausdorff space s and continuous functions. Here we proceed as in Set, but give {0,1} thediscrete topology so that it becomes a compact Hausdorff space. The map "g"1 is continuous because the image of "f" is a closed subset of "Y".However there are also many concrete categories of interest where epimorphisms fail to be surjective. A few examples are:
*In thecategory of monoids , Mon, theinclusion map N → Z is a non-surjective epimorphism. To see this, suppose that "g"1 and "g"2 are two distinct maps from Z to some monoid "M". Then for some "n" in Z, "g"1("n") ≠ "g"2("n"), so "g"1("-n") ≠ "g"2("-n"). Either "n" or "-n" is in N, so the restrictions of "g"1 and "g"2 to N are unequal.
*In thecategory of rings , Ring, the inclusion map Z → Q is a non-surjective epimorphism; to see this, note that anyring 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 anycommutative ring "R" to any one of its localizations is an epimorphism.
*In thecategory of commutative rings , afinitely generated homomorphism of rings "f" : "R" → "S" is an epimorphism if and only if for allprime 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 Frac("R"/"P") → Frac("S"/"Q") is anisomorphism (EGA IV 17.2.6).
*In the category ofHausdorff space s, Haus, the epimorphisms are precisely the continuous functions with 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 to examples of epimorphisms in non-concrete categories:
* If amonoid or 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 adirected graph is considered as a category (objects are the vertices, morphisms are the paths, composition of morphisms is the concatenation of paths), then the epimorphisms are precisely the paths that end in a vertex "y" from which no two different paths can reach the same vertex "z".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"Y", then "f" is easily seen to be an epimorphism. A map with such a right-sided inverse is called a split epi.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" which is an epimorphism when considered as a morphism in "C" is also an epimorphism in "D"; the converse, however, need not hold; the smaller category can (and often will) have more epimorphisms.As for most concepts in category theory, epimorphisms are preserved under equivalences of categories: given an equivalence "F" : "C" → "D", then a morphism "f" is an epimorphism in the category "C" if and only if "F"("f") is an epimorphism in "D". A 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:are
injective for every choice of "Z". This in turn is equivalent to the inducednatural transformation :being a monomorphism in thefunctor category Set"C".Every
coequalizer is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in particular that everycokernel 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 a monomorphism followed by an epimorphism. For instance, given a group homomorphism "f" : "G" → "H", we can define the group "K" = im("f") = "f"("G") and then write "f" as the composition of the surjective homomorphism "G" → "K" which is defined like "f", followed by the injective homomorphism "K" → "H" which 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 the Examples section (though not in all concrete categories).
Related concepts
Among other useful concepts are "regular epimorphism", "extremal epimorphism", "strong epimorphism", and "split epimorphism". A regular epimorphism coequalizes some parallel pair of morphisms. An extremal epimorphism is an epimorphism that has no monomorphism as a second factor, unless that monomorphism is an
isomorphism . A strong epimorphism satisfies a certain lifting property with respect to commutative squares involving a monomorphism.A split epimorphism is a morphism which has a right-sided inverse.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 thehalf-open interval [0,1) to theunit circle S1 (thought of as a subspace of thecomplex plane ) which sends "x" to exp(2π"ix") (seeEuler's formula ) is continuous and bijective but not ahomeomorphism 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→R in the category Haus; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism.Epimorphisms are used to define abstract
quotient object s in general categories: two epimorphisms "f"1 : "X" → "Y"1 and "f"2 : "X" → "Y"2 are said to be "equivalent" if there exists an isomorphism "j" : "Y"1 → "Y"2 with "j" "f"1 = "f"2. This is anequivalence relation , and the equivalence classes are defined to be the quotient objects of "X".Terminology
The companion terms "epimorphism" and "
monomorphism " were first introduced byBourbaki . Bourbaki uses "epimorphism" as shorthand for asurjective 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 References
*Adámek, Jií, Herrlich, Horst, & Strecker, George E. (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf "Abstract and Concrete Categories"] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
*Bergman, George M. (1998), "An Invitation to General Algebra and Universal Constructions", Harry Helson Publisher, Berkeley. ISBN 0-9655211-4-1.
*Linderholm, Carl (1970). A Group Epimorphism is Surjective. "American Mathematical Monthly " 77, pp. 176–177. Proof summarized by Arturo Magidin in [http://groups.google.com/group/sci.math/msg/6d4023d93a2b4300] .
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