In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is "right-cancellable" in the following sense: In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

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 C^op). Image File history File links Epimorphism-01. ... A surjective function. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...

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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

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: In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ... Partial plot of a function f. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

• 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 → {0,1} of the image f(X) and the map g₂: Y → {0,1} that is constant 1.
• Rel, sets with binary relations 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 sets and monotone functions. If f : (X,≤) → (Y,≤) is not surjective, pick y₀ in Y f(X) and let g₁ : Y → {0,1} be the characteristic function of {y | y₀ ≤ y} and g₂ : Y → {0,1} the characteristic function of {y | y₀ < y}. These maps are monotone if {0,1} 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 {0,1} 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 {0,1} 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 mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... Otto Schreier (born March 3, 1901 in Vienna, Austria; died June 2, 1929 in Hamburg, Germany) was an Austrian mathematician who made major contributions in combinatorial group theory. ... This article is about equalisers in mathematics. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a module is a generalization of a vector space. ... If B is a submodule of a module A of a ring R, then the quotient space A/B is also a module of R. Its called the quotient module. ... The category Top has topological spaces as objects and continuous maps as morphisms. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

• 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. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, inclusion is a partial order on sets. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ... The Éléments de géométrie algébrique (Elements of Algebraic Geometry) by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, are an unfinished 1500-page treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related areas of mathematics a subset A of a topological space X is called dense (in X) if the only closed subset of X containing A is X itself. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

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.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... This article just presents the basic definitions. ...

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 an epimorphism. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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. In mathematics, a subcategory S of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the subset Y of objects contains the source and target of all the f in...

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. 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. ... 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. ...

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 In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

being a monomorphism in the functor category Set^C. In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...

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 mathematics, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. ... In abstract algebra, the cokernel of a homomorphism f : X &#8594; Y is the quotient of Y by the image of f. ...

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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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. In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... Illustration of a unit circle. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

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. In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...

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. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... A surjective function. ... Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...

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

• Surjective function
• Monomorphism
• Isomorphism

A surjective function. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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 [1].