Last updated
Epimorphism scenarios.svg

In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,


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 : YX such that fj = idY, then f: XY 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 : CD, 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 : XY 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 : GH, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism GK that is defined like f, followed by the injective homomorphism KH 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: AB 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 : XY1 and f2 : XY2 are said to be equivalent if there exists an isomorphism j : Y1Y2 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.

See also


Related Research Articles

Group homomorphism

In mathematics, given two groups, and, a group homomorphism from to is a function h : GH such that for all u and v in G it holds that

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

Ring homomorphism

In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : RS such that f is

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

Category (mathematics) Mathematical object that generalizes the standard notions of sets and functions

In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation .

Exact sequence Sequence of homomorphisms such that each kernel equals the preceding image

An exact sequence is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

Homological algebra Branch of mathematics

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,

Category of groups

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written

This is a glossary of properties and concepts in category theory in mathematics.

In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question Does every monad arise from an adjunction? The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli.

In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair:

Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

Nodal decomposition

In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphism.