# User Contributed Dictionary

### Noun

- A map f such that whenever g composed with f equals h composed with f, then g=h. In most everyday categories, a map is an epimorphism iff it's surjective

### See also

# Extensive Definition

In category
theory an epimorphism (also called an epic morphism or an epi)
is a morphism f : X
→ Y which is right-cancellative
in the following sense:

- g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z.

Epimorphisms are analogues of surjective
functions, but they are not exactly the same. The dual
of an epimorphism is a monomorphism (i.e. an
epimorphism in a category C is a monomorphism in the dual category
Cop).

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 the section
on Terminology
below.

## Examples

Every morphism in a concrete category whose underlying 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 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 g1: Y → of the image f(X) and the map g2: Y → that is constant 1.
- Rel, sets with binary relations and relation preserving functions. Here we can use the same proof as for Set, equipping with the full relation ×.
- Pos, partially ordered sets and monotone functions. If f : (X,≤) → (Y,≤) is not surjective, pick y0 in Y \ f(X) and let g1 : Y → be the characteristic function of and g2 : Y → the characteristic function of . These maps are monotone if is given the standard ordering 0 < 1.
- Grp, groups and group homomorphisms. The result that every epimorphism in Grp is surjective is due to Otto Schreier (he actually proved more, showing that every subgroup is an 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.
- Ab, abelian groups and group homomorphisms.
- K-Vect, vector spaces over a field K and K-linear transformations.
- Mod-R, right modules over a ring R and module homomorphisms. 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 g2: Y → Y/f(X).
- Top, topological spaces and continuous functions. To prove that every epimorphism in Top is surjective, we proceed exactly as in Set, giving the indiscrete topology which ensures that all considered maps are continuous.
- HComp, compact Hausdorff spaces and continuous functions. Here we proceed as in Set, but give the discrete topology so that it becomes a compact Hausdorff space. The map g1 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 the category of monoids, Mon, the inclusion map N → Z is a non-surjective epimorphism. To see this, suppose that g1 and g2 are two distinct maps from Z to some monoid M. Then for some n in Z, g1(n) ≠ g2(n), so g1(-n) ≠ g2(-n). Either n or -n is in N, so the restrictions of g1 and g2 to N are unequal.
- 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 localizations is an epimorphism.
- In the category of commutative rings, a finitely generated homomorphism of rings f : R → S is an epimorphism if and only if for all prime ideals 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 an isomorphism (EGA IV 17.2.6).
- In the category of Hausdorff spaces, 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 a monoid 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 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 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 = idY, 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

- \begin\operatorname(Y,Z) &\rightarrow& \operatorname(X,Z)\\

- \begin\operatorname(Y,-) &\rightarrow& \operatorname(X,-)\end

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 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 the half-open
interval [0,1) to the unit circle
S1 (thought of as a subspace
of the complex
plane) which sends x to exp(2πix) (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→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
objects in general categories: two epimorphisms f1 : X
→ Y1 and f2 : X → Y2 are said to be equivalent if
there exists an isomorphism j : Y1 → Y2 with j f1
= f2. 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 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

## References

- Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). 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.

epimorphism in German: Epimorphismus

epimorphism in French: Épimorphisme

epimorphism in Polish: Epimorfizm

epimorphism in Finnish:
Epimorfismi