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 *g*_{1}, *g*_{2}: *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 *C*^{op}).

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:

**Set**: sets and functions. To prove that every epimorphism*f*:*X*→*Y*in**Set**is surjective, we compose it with both the characteristic function*g*_{1}:*Y*→ {0,1} of the image*f*(*X*) and the map*g*_{2}:*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*y*_{0}in*Y*\*f*(*X*) and let*g*_{1}:*Y*→ {0,1} be the characteristic function of {*y*|*y*_{0}≤*y*} and*g*_{2}:*Y*→ {0,1} the characteristic function of {*y*|*y*_{0}<*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*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*g*_{2}:*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. If*f*:*X*→*Y*is not surjective, let*y*∈*Y*−*fX*. Since*fX*is closed, by Urysohn's Lemma there is a continuous function*g*_{1}:*Y*→ [0,1] such that*g*_{1}is 0 on*fX*and 1 on*y*. We compose*f*with both*g*_{1}and the zero function*g*_{2}:*Y*→ [0,1].

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*g*_{1}and*g*_{2}are two distinct maps from**Z**to some monoid*M*. Then for some*n*in**Z**,*g*_{1}(*n*) ≠*g*_{2}(*n*), so*g*_{1}(*-n*) ≠*g*_{2}(−*n*). Either*n*or −*n*is in**N**, so the restrictions of*g*_{1}and*g*_{2}to**N**are unequal. - In the category of algebras over commutative ring
**R**, take**R**[**N**] →**R**[**Z**], where**R**[**G**] is the group ring of the group**G**and the morphism is induced by the inclusion**N**→**Z**as in the previous example. This follows from the observation that**1**generates the algebra**R**[**Z**] (note that the unit in**R**[**Z**] is given by**0**of**Z**), and the inverse of the element represented by**n**in**Z**is just the element represented by −**n**. Thus any homomorphism from**R**[**Z**] is uniquely determined by its value on the element represented by**1**of**Z**. - 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 for 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
*every*morphism is an epimorphism.

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*: *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 **Set**^{C}.

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*.

- An epimorphism is said to be
**regular**if it is a coequalizer of some pair of parallel morphisms. - An epimorphism is said to be
**extremal**^{ [1] }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**^{ [1] }^{ [2] }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 (in this case is called a right-sided inverse for ).

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 S^{1} (thought of as a subspace of the complex plane) that sends *x* to exp(2πi*x*) (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 *f*_{1} : *X* → *Y*_{1} and *f*_{2} : *X* → *Y*_{2} are said to be *equivalent* if there exists an isomorphism *j* : *Y*_{1} → *Y*_{2} with *j* *f*_{1} = *f*_{2}. 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.

In mathematics, given two groups, and, a **group homomorphism** from to is a function *h* : *G* → *H* 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).

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* : *R* → *S* 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". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.

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, intuitively corresponding to a weak form of equivalence between two related categories. 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 .

In mathematics, specifically in category theory, a **pre-abelian category** is an additive category that has all kernels and cokernels.

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** 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, a branch of abstract 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*,

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 branch of mathematics, a **section** is a right inverse of some morphism. Dually, a **retraction** is a left inverse of some morphism. In other words, if *f* : *X* → *Y* and *g* : *Y* → *X* are morphisms whose composition *f*o*g* : *Y* → *Y* is the identity morphism on *Y*, then *g* is a section of *f*, and *f* is a retraction of *g*.

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.

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.

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