Epimorphism


In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms,
Some authors use the adjective epi. Epimorphisms are categorical analogues of onto or surjective functions, but they may not exactly coincide in all contexts. The dual of an epimorphism is a monomorphism.
Epimorphism can be a subtly weaker condition than surjectivity. For example, in the category of rings, the inclusion of integers into rational numbers is an epimorphism, since the images of integers under a homomorphism also determine the images of quotients of integers. In the category of Hausdorff spaces, an epimorphism is precisely a continuous function with dense image, since the image of a Cauchy sequence determines the image of its limit point: for example the inclusion of the metric space of rational numbers into the real number line.
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 below.

Examples

In a concrete category, if the underlying function of a morphism is surjective, then the morphism is epi. 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:
  • Set: sets and functions. To prove that every epimorphism in Set is surjective, we compose it with both the characteristic function of the image and the map 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 is not surjective, pick in and let be the characteristic function of and the characteristic function of. These maps are monotone if is given the standard ordering.
  • Grp: groups and group homomorphisms. The result that every epimorphism in Grp is surjective is due to Otto Schreier ; an elementary proof can be found in.
  • FinGrp: finite groups and group homomorphisms. Also due to Schreier; the proof given in establishes this case as well.
  • Ab: abelian groups and group homomorphisms.
  • -Vect: vector spaces over a field and -linear transformations.
  • Mod-: right modules over a ring and module homomorphisms. This generalizes the two previous examples; to prove that every epimorphism in Mod- is surjective, we compose it with both the canonical quotient map and the zero map.
  • 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. If is not surjective, let Since is closed, by Urysohn's Lemma there is a continuous function such that is on and on. We compose with both and the zero function
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 ≠ g2, so g1 ≠ g2. Either n or −n is in N, so the restrictions of g1 and g2 to N are unequal.
  • In the category of algebras over commutative ring, take the polyomials over included in the Laurent polynomials. This is an epimorphism since any homomorphism of algebras respects multiplicative inverse whenever it is defined, so the image of determines the image of any Laurent polynomial.
  • 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 is either S or is prime, and if Q is not S, the induced map Frac → Frac is an isomorphism.
  • 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 for examples of epimorphisms in non-concrete categories:
  • If a monoid or ring is considered as a category with a single object, then the epimorphisms are precisely the right-cancellable elements.
  • If a directed graph is considered as a category, then every morphism is an epimorphism.

    Properties

Every isomorphism is an epimorphism; indeed only a right-sided inverse is needed: suppose there exists a morphism such that fj = idY. For any morphisms where, you have that. A map with such a right-sided inverse is called a 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 have more epimorphisms.
As for most concepts in category theory, epimorphisms are preserved under equivalences of categories: given an equivalence F : C → D, a morphism f is an epimorphism in the category C if and only if 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 induced natural transformation
being a monomorphism in the functor category SetC.
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 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 .

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 f is said to be strict if it is a coequalizer of every pair of morphisms g, h such that.
  • An epimorphism is said to be extremal if in each representation, where is a monomorphism, the morphism is automatically an isomorphism.
  • An epimorphism is said to be immediate if in each representation, where is a monomorphism and is an epimorphism, the morphism is automatically an isomorphism.
  • An epimorphism is said to be strong if for any monomorphism and any morphisms and such that, there exists a morphism such that and.
  • An epimorphism is said to be split if there exists a morphism such that .
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 : D → D.
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 that sends x to exp 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 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 rings, the map is a bimorphism but 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 This is an equivalence relation, and the equivalence classes are defined to be the quotient objects of X.