In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z,
Epimorphisms are categorical analogues of onto or surjective functions (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 is a ring epimorphism. 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 § Terminology below.
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 that are surjective on the underlying sets:
However, there are also many concrete categories of interest where epimorphisms fail to be surjective. A few examples are:
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:
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: X → Y is easily seen to be an epimorphism. 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 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 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 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(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 § Examples (though not in all concrete categories).
Among other useful concepts are regular epimorphism, extremal epimorphism, immediate epimorphism, strong epimorphism, and split epimorphism.
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 S1 (thought of as a subspace of the complex plane) that 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. Similarly, in the category of rings, the map Z → Q 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 j f1 = f2. This is an equivalence relation, and the equivalence classes are defined to be the quotient objects of X.
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.
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