ContentsArticles

Morphism 1Homomorphism 3Automorphism 7Endomorphism 9Isomorphism 11Homeomorphism 15Holomorphic function 18Diffeomorphism 21Monomorphism 26Epimorphism 28

ReferencesArticle Sources and Contributors 32Image Sources, Licenses and Contributors 33

Article LicensesLicense 34

Morphism 1

MorphismIn mathematics, a morphism is an abstraction derived from structure-preserving mappings between twomathematical structures. 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; intopology, continuous functions, and so on.The study of morphisms and of the structures (called objects) over which they are defined, is central to categorytheory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concretecategories, where the objects are simply sets with some additional structure, and morphisms are structure-preservingfunctions.

DefinitionA category C consists of two classes, one of objects and the other of morphisms.There are two operations which are defined on every morphism, the domain (or source) and the codomain (ortarget).If a morphism f has domain X and codomain Y, we write f : X → Y. Thus a morphism is represented by an arrowfrom its domain to its codomain. The collection of all morphisms from X to Y is denoted homC(X,Y) or simplyhom(X, Y) and called the hom-set between X and Y. Some authors write MorC(X,Y) or Mor(X, Y). Note that the termhom-set is a bit of a misnomer as the collection of morphisms is not required to be a set.For every three objects X, Y, and Z, there exists a binary operation hom(X, Y) × hom(Y, Z) → hom(X, Z) calledcomposition. The composite of f : X → Y and g : Y → Z is written g ∘ f or gf. The composition of morphisms is oftenrepresented by a commutative diagram. For example,

Morphisms satisfy two axioms:• Identity: for every object X, there exists a morphism idX : X → X called the identity morphism on X, such that for

every morphism f : A → B we have idB ∘ f = f = f ∘ idA.• Associativity: h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever the operations are defined.When C is a concrete category, the identity morphism is just the identity function, and composition is just theordinary composition of functions. Associativity then follows, because the composition of functions is associative.Note that the domain and codomain are in fact part of the information determining a morphism. For example, in thecategory of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may havethe same range), while having different codomains. The two functions are distinct from the viewpoint of categorytheory. Thus many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problembecause if this disjointness does not hold, it can be assured by appending the domain and codomain to themorphisms, (say, as the second and third components of an ordered triple).

Morphism 2

Some specific morphisms• Monomorphism: f : X → Y is called a monomorphism if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 :

Z → X. It is also called a mono or a monic.[1]

• The morphism f has a left inverse if there is a morphism g:Y → X such that g ∘ f = idX. The left inverse g isalso called a retraction of f.[1] Morphisms with left inverses are always monomorphisms, but the converse isnot always true in every category; a monomorphism may fail to have a left-inverse.

• A split monomorphism h : X → Y is a monomorphism having a left inverse g : Y → X, so that g ∘ h = idX.Thus h ∘ g : Y → Y is idempotent, so that (h ∘ g)2 = h ∘ g.

• In concrete categories, a function that has a left inverse is injective. Thus in concrete categories,monomorphisms are often, but not always, injective. The condition of being an injection is stronger than thatof being a monomorphism, but weaker than that of being a split monomorphism.

• Epimorphism: Dually, f : X → Y is called an epimorphism if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1,g2 : Y → Z. It is also called an epi or an epic.[1]

• The morphism f has a right-inverse if there is a morphism g : Y → X such that f ∘ g = idY. The right inverse gis also called a section of f.[1] Morphisms having a right inverse are always epimorphisms, but the converse isnot always true in every category, as an epimorphism may fail to have a right inverse.

• A split epimorphism is an epimorphism having a right inverse. Note that if a monomorphism f splits withleft-inverse g, then g is a split epimorphism with right-inverse f.

• In concrete categories, a function that has a right inverse is surjective. Thus in concrete categories,epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that ofbeing an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, everysurjection has a section, a result equivalent to the axiom of choice.

• A bimorphism is a morphism that is both an epimorphism and a monomorphism.• Isomorphism: f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f ∘ g = idY and g

∘ f = idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is anisomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g isalso an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic orequivalent. Note that while every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism.For example, in the category of commutative rings the inclusion Z → Q is a bimorphism, which is not anisomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both amonomorphism and a split epimorphism, must be an isomorphism. A category, such as Set, in which everybimorphism is an isomorphism is known as a balanced category.

• Endomorphism: f : X → X is an endomorphism of X. A split endomorphism is an idempotent endomorphism fif f admits a decomposition f = h ∘ g with g ∘ h = id. In particular, the Karoubi envelope of a category splits everyidempotent morphism.

• An automorphism is a morphism that is both an endomorphism and an isomorphism.

Examples• In the concrete categories studied in universal algebra (groups, rings, modules, etc.), morphisms are usually

homomorphisms. Likewise, the notions of automorphism, endomorphism, epimorphism, homeomorphism,isomorphism, and monomorphism all find use in universal algebra.

• In the category of topological spaces, morphisms are continuous functions and isomorphisms are calledhomeomorphisms.

• In the category of smooth manifolds, morphisms are smooth functions and isomorphisms are calleddiffeomorphisms.

Morphism 3

• In the category of small categories, functors can be thought of as morphisms.• In a functor category, the morphisms are natural transformations.For more examples, see the entry category theory.

Notes[1][1] Jacobson (2009), p. 15.

References• Jacobson, Nathan (2009), Basic algebra, 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7.

External links• Category (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=965), PlanetMath.org.• TypesOfMorphisms (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=8114), PlanetMath.org.

HomomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such asgroups, rings, or vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos)meaning "same" and μορφή (morphe) meaning "shape". Isomorphisms, automorphisms, and endomorphisms are alltypes of homomorphism.

Definition and illustration

DefinitionThe definition of homomorphism depends on the type of algebraic structure under consideration. Particulardefinitions of homomorphism include the following:• A group homomorphism is a homomorphism between two groups.• A ring homomorphism is a homomorphism between two rings.• A linear map is a homomorphism between two vector spaces.• An algebra homomorphism is a homomorphism between two algebras.• A functor is a homomorphism between two categories.The common theme is that a homomorphism is a function between two algebraic objects that respects the algebraicstructure.For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfyingcertain axioms. If and are groups, a homomorphism from to is a function ƒ: 

 →  such that for any elements g1, g2 ∈ G.When an algebraic structure includes more than one operation, homomorphisms are required to preserve eachoperation. For example, a ring possesses both addition and multiplication, and a homomorphism from the ring

to the ring is a function such that

for any elements r and s of the domain ring.The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a homomorphism ƒ: A → B is a function between two

Homomorphism 4

algebraic structures of the same type such that

for each n-ary operation μ and for all elements a1,...,an ∈ A.

Basic examplesThe real numbers are a ring, having both addition and multiplication. The set of all 2 × 2 matrices is also a ring,under matrix addition and matrix multiplication. If we define a function between these rings as follows:

where r is a real number. Then ƒ is a homomorphism of rings, since ƒ preserves both addition:

and multiplication:

For another example, the nonzero complex numbers form a group under the operation of multiplication, as do thenonzero real numbers. (Zero must be excluded from both groups since it does not have a multiplicative inverse,which is required for elements of a group.) Define a function ƒ from the nonzero complex numbers to the nonzeroreal numbers by

That is, ƒ(z) is the absolute value (or modulus) of the complex number z. Then ƒ is a homomorphism of groups, sinceit preserves multiplication:

Note that ƒ cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since itdoes not preserve addition:

Informal discussionBecause abstract algebra studies sets endowed with operations that generate interesting structure or properties on theset, functions which preserve the operations are especially important. These functions are known as homomorphisms.For example, consider the natural numbers with addition as the operation. A function which preserves additionshould have this property: f(a + b) = f(a) + f(b). For example, f(x) = 3x is one such homomorphism, since f(a + b) =3(a + b) = 3a + 3b = f(a) + f(b). Note that this homomorphism maps the natural numbers back into themselves.Homomorphisms do not have to map between sets which have the same operations. For example,operation-preserving functions exist between the set of real numbers with addition and the set of positive realnumbers with multiplication. A function which preserves operation should have this property: f(a + b) = f(a) * f(b),since addition is the operation in the first set and multiplication is the operation in the second. Given the laws ofexponents, f(x) = ex satisfies this condition : 2 + 3 = 5 translates into e2 * e3 = e5.If we are considering multiple operations on a set, then all operations must be preserved for a function to beconsidered as a homomorphism. Even though the set may be the same, the same function might be a homomorphism,say, in group theory (sets with a single operation) but not in ring theory (sets with two related operations), because itfails to preserve the additional operation that ring theory considers.

Homomorphism 5

Relation to category theorySince homomorphisms are morphisms, the following specific kinds of morphisms defined in any category aredefined for homomorphisms as well. However, the definitions in category theory are somewhat technical. In theimportant special case of module homomorphisms, and for some other classes of homomorphisms, there are muchsimpler descriptions, as follows:• An isomorphism is a bijective homomorphism.• An epimorphism (sometimes called a cover) is a surjective homomorphism.• A monomorphism (sometimes called an embedding or extension) is an injective homomorphism.• An endomorphism is a homomorphism from an object to itself.• An automorphism is an endomorphism which is also an isomorphism, i.e., an isomorphism from an object to

itself.These descriptions may be used in order to derive several interesting properties. For instance, since a function isbijective if and only if it is both injective and surjective, a module homomorphism is an isomorphism if and only if itis both a monomorphism and an epimorphism.For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; thefirst three descriptions do not. For instance, the precise definition for a homomorphism f to be iso is not only that it isbijective, and thus has an inverse f-1, but also that this inverse is a homomorphism, too. This has the importantconsequence that two objects are completely indistinguishable as far as the structure in question is concerned, ifthere is an isomorphism between them. Two such objects are said to be isomorphic.Actually, in the algebraic setting (at least within the context of universal algebra) this extra condition onisomorphisms is automatically satisfied. However, the same is not true for epimorphisms; for instance, the inclusionof Z as a (unitary) subring of Q is not surjective, but an epimorphic ring homomorphism.[1] This inclusion thus alsois an example of a ring homomorphism which is both mono and epi, but not iso.

Relationships between different kinds of module homomorphisms. H = set of Homomorphisms, M = set of Monomorphisms, P = set of Epimorphisms, S = set of Isomorphisms, N = set of Endomorphism, A = set of Automorphisms. Notice that: M ∩ P = S, S ∩ N = A,

Homomorphism 6

(M ∩ N) \ A and (P ∩ N) \ A contain only homomorphisms from some infinite modules to themselves.

Kernel of a homomorphismAny homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b if and only if f(a) = f(b). The relation~ is called the kernel of f. It is a congruence relation on X. The quotient set X/~ can then be given an object-structurein a natural way, i.e. [x] * [y] = [x * y]. In that case the image of X in Y under the homomorphism f is necessarilyisomorphic to X/~; this fact is one of the isomorphism theorems. Note in some cases (e.g. groups or rings), a singleequivalence class K suffices to specify the structure of the quotient; so we can write it X/K. (X/K is usually read as "Xmod K".) Also in these cases, it is K, rather than ~, that is called the kernel of f (cf. normal subgroup).

Homomorphisms of relational structuresIn model theory, the notion of an algebraic structure is generalized to structures involving both operations andrelations. Let L be a signature consisting of function and relation symbols, and A, B be two L-structures. Then ahomomorphism from A to B is a mapping h from the domain of A to the domain of B such that• h(FA(a1,…,an)) = FB(h(a1),…,h(an)) for each n-ary function symbol F in L,• RA(a1,…,an) implies RB(h(a1),…,h(an)) for each n-ary relation symbol R in L.In the special case with just one binary relation, we obtain the notion of a graph homomorphism.

Homomorphisms and e-free homomorphisms in formal language theoryHomomorphisms are also used in the study of formal languages[2] (although within this context, often they arebriefly referred to as morphisms[3]). Given alphabets and , a function h : → such that

for all u and v in is called a homomorphism (or simply morphism) on .[4] Let e denotethe empty word. If h is a homomorphism on and for all in , then h is called an e-free

homomorphism.This type of homomorphism can be thought of as (and is equivalent to) a monoid homomorphism where the setof all words over a finite alphabet is a monoid (in fact it is the free monoid on ) with operation concatenationand the empty word as the identity.

References[1] Exercise 4 in section I.5, in Saunders Mac Lane, Categories for the Working Mathematician, ISBN 0-387-90036-5[2] Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, ISBN 0-7204-2506-9.[3] T. Harju, J. Karhumӓki, Morphisms in Handbook of Formal Languages, Volume I, edited by G. Rozenberg, A. Salomaa, Springer, 1997,

ISBN 3-540-61486-9.[4] In homomorphisms on formal languages, the * operation is the Kleene star operation. The and are both concatenation, commonly

denoted by juxtaposition.

A monograph available free online:• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. (http:/ / www. thoralf.

uwaterloo. ca/ htdocs/ ualg. html) Springer-Verlag. ISBN 3-540-90578-2.

Automorphism 7

AutomorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, asymmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of allautomorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetrygroup of the object.

DefinitionThe exact definition of an automorphism depends on the type of "mathematical object" in question and what,precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaningis an abstract branch of mathematics called category theory. Category theory deals with abstract objects andmorphisms between those objects.In category theory, an automorphism is an endomorphism (i.e. a morphism from an object to itself) which is also anisomorphism (in the categorical sense of the word).This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren'tnecessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and themorphisms will be functions preserving that structure.In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring,or vector space. An isomorphism is simply a bijective homomorphism. (The definition of a homomorphism dependson the type of algebraic structure; see, for example: group homomorphism, ring homomorphism, and linearoperator).The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other(non-identity) automorphisms are called nontrivial automorphisms.

Automorphism groupIf the automorphisms of an object X form a set (instead of a proper class), then they form a group under compositionof morphisms. This group is called the automorphism group of X. That this is indeed a group is simple to see:• Closure: composition of two endomorphisms is another endomorphism.• Associativity: composition of morphisms is always associative.• Identity: the identity is the identity morphism from an object to itself which exists by definition.• Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is

also an endomorphism of the same object it is an automorphism.The automorphism group of an object X in a category C is denoted AutC(X), or simply Aut(X) if the category is clearfrom context.

Examples• In set theory, an automorphism of a set X is an arbitrary permutation of the elements of X. The automorphism

group of X is also called the symmetric group on X.• In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial

automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking,negation is an automorphism of any abelian group, but not of a ring or field.

• A group automorphism is a group isomorphism from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group G there is a natural group homomorphism G → Aut(G) whose image is the group Inn(G) of inner automorphisms and whose kernel is the center of G. Thus, if

Automorphism 8

G has trivial center it can be embedded into its own automorphism group.[1]

• In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is aninvertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is thesame as the general linear group, GL(V).

• A field automorphism is a bijective ring homomorphism from a field to itself. In the cases of the rational numbers(Q) and the real numbers (R) there are no nontrivial field automorphisms. Some subfields of R have nontrivialfield automorphisms, which however do not extend to all of R (because they cannot preserve the property of anumber having a square root in R). In the case of the complex numbers, C, there is a unique nontrivialautomorphism that sends R into R: complex conjugation, but there are infinitely (uncountably) many "wild"automorphisms (assuming the axiom of choice).[2] Field automorphisms are important to the theory of fieldextensions, in particular Galois extensions. In the case of a Galois extension L/K the subgroup of allautomorphisms of L fixing K pointwise is called the Galois group of the extension.

• In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. Inparticular, if two nodes are joined by an edge, so are their images under the permutation.

• For relations, see relation-preserving automorphism.• In order theory, see order automorphism.

• In geometry, an automorphism may be called a motion of the space. Specialized terminology is also used:• In metric geometry an automorphism is a self-isometry. The automorphism group is also called the isometry

group.• In the category of Riemann surfaces, an automorphism is a bijective biholomorphic map (also called a

conformal map), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbiustransformations.

• An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphismgroup is sometimes denoted Diff(M).

• In topology, morphisms between topological spaces are called continuous maps, and an automorphism of atopological space is a homeomorphism of the space to itself, or self-homeomorphism (see homeomorphismgroup). In this example it is not sufficient for a morphism to be bijective to be an isomorphism.

HistoryOne of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points)was given by the Irish mathematician William Rowan Hamilton in 1856, in his Icosian Calculus, where hediscovered an order two automorphism,[3] writing:

so that is a new fifth root of unity, connected with the former fifth root by relations of perfectreciprocity.

Inner and outer automorphismsIn some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into twotypes, called "inner" and "outer" automorphisms.In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For eachelement a of a group G, conjugation by a is the operation φa : G → G given by φa(g) = aga−1 (or a−1ga; usagevaries). One can easily check that conjugation by a is a group automorphism. The inner automorphisms form anormal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.The other automorphisms are called outer automorphisms. The quotient group Aut(G) / Inn(G) is usually denoted byOut(G); the non-trivial elements are the cosets that contain the outer automorphisms.

Automorphism 9

The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras thedefinition is slightly different.

References[1] PJ Pahl, R Damrath (2001). "§7.5.5 Automorphisms" (http:/ / books. google. com/ ?id=kvoaoWOfqd8C& pg=PA376). Mathematical

foundations of computational engineering (Felix Pahl translation ed.). Springer. p. 376. ISBN 3-540-67995-2. .[2] Yale, Paul B. (May 1966). "Automorphisms of the Complex Numbers" (http:/ / mathdl. maa. org/ images/ upload_library/ 22/ Ford/

PaulBYale. pdf). Mathematics Magazine 39 (3): 135–141. doi:10.2307/2689301. JSTOR 2689301. .[3] Sir William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity" (http:/ / www. maths. tcd. ie/ pub/

HistMath/ People/ Hamilton/ Icosian/ NewSys. pdf). Philosophical Magazine 12: 446. .

External links• Weisstein, Eric W., " Automorphism (http:/ / mathworld. wolfram. com/ Automorphism. html)" from MathWorld.

EndomorphismIn mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. Forexample, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is agroup homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category. In the category ofsets, endomorphisms are simply functions from a set S into itself.In any category, the composition of any two endomorphisms of X is again an endomorphism of X. It follows that theset of all endomorphisms of X forms a monoid, denoted End(X) (or EndC(X) to emphasize the category C).An invertible endomorphism of X is called an automorphism. The set of all automorphisms is a subset of End(X)with a group structure, called the automorphism group of X and denoted Aut(X). In the following diagram, thearrows denote implication:

automorphism isomorphism

endomorphism (homo)morphism

Any two endomorphisms of an abelian group A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under thisaddition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set ofendomorphisms of Zn is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space ormodule also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of anonabelian group generate an algebraic structure known as a nearring. Every ring with one is the endomorphism ringof its regular module, and so is a subring of an endomorphism ring of an abelian group,[1] however there are ringswhich are not the endomorphism ring of any abelian group.

Endomorphism 10

Operator theoryIn any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may beinterpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits ofelements, etc.Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can haveproperties like continuity, boundedness, and so on. More details should be found in the article about operator theory.

Endofunctions in mathematicsIn mathematics, an endofunction is a function whose codomain is equal to its domain. A homomorphicendofunction is an endomorphism.Let S be an arbitrary set. Among endofunctions on S one finds permutations of S and constant functions associatingto each a given . Every permutation of S has the codomain equal to its domain and is bijective andinvertible. A constant function on S, if S has more than 1 element, has a codomain that is a proper subset of itsdomain, is not bijective (and non invertible). The function associating to each natural integer n the floor of n/2 has itscodomain equal to its domain and is not invertible.Finite endofunctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to1, and can be easily described. For sets of size n, there are n^n endofunctions on the set.Particular bijective endofunctions are the involutions, i.e. the functions coinciding with their inverses.

Notes[1][1] Jacobson (2009), p. 162, Theorem 3.2.

References• Jacobson, Nathan (2009), Basic algebra, 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

External links• Endomorphism (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=7462), PlanetMath.org.

Isomorphism 11

IsomorphismIn abstract algebra, an isomorphism[1] is a bijective homomorphism.[2] Two mathematical structures are said to beisomorphic if there is an isomorphism between them.In category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y→ X, with the property that both f −1f = idX and f f −1 = idY.[3]

PurposeIsomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objectsare isomorphic, then any property that is preserved by an isomorphism and that is true of one of the objects, is alsotrue of the other. If an isomorphism can be found from a relatively unknown part of mathematics into some wellstudied division of mathematics, where many theorems are already proved, and many methods are already availableto find answers, then the function can be used to map whole problems out of unfamiliar territory over to "solidground" where the problem is easier to understand and work with.

Practical examplesThe following are examples of isomorphisms from ordinary algebra.• Consider the logarithm function: For any fixed base b, the logarithm function logb maps from the positive real

numbers R+ onto the real numbers R; formally:

This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithmfunction. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations.Specifically, consider the group (R+,×) of positive real numbers under ordinary multiplication. The logarithmfunction obeys the following identity:

But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphismfrom the group (R+,×) to the group (R,+). Logarithms can therefore be used to simplify multiplication of positivereal numbers. By working with logarithms, multiplication of positive real numbers is replaced by addition of logs.This way it is possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with alogarithmic scale.

• Consider the group (Z6, +), the integers from 0 to 5 with addition modulo 6. Also consider the group (Z2 × Z3, +),the ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition inthe x-coordinate is modulo 2 and addition in the y-coordinate is modulo 3. These structures are isomorphic underaddition, if you identify them using the following scheme:

(0,0) → 0(1,1) → 1(0,2) → 2(1,0) → 3(0,1) → 4(1,2) → 5

or in general (a,b) → (3a + 4b) mod 6. For example note that (1,1) + (1,0) = (0,1), which translates in the other system as 1 + 3 = 4. Even though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are exactly the same. More generally, the direct product of two cyclic

Isomorphism 12

groups Zm and Zn is isomorphic to Zmn if and only if m and n are coprime.

Abstract examples

A relation-preserving isomorphismIf one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relationS then an isomorphism from X to Y is a bijective function ƒ : X → Y such that[4]:

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, totalorder, strict weak order, total preorder (weak order), an equivalence relation, or a relation with any other specialproperties, if and only if R is.For example, R is an ordering ≤ and S an ordering , then an isomorphism from X to Y is a bijective function ƒ : X→ Y such that

Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism.If X = Y, then this is a relation-preserving automorphism.

An operation-preserving isomorphismSuppose that on these sets X and Y, there are two binary operations and that happen to constitute the groups (X,

) and (Y, ). Note that the operators operate on elements from the domain and range, respectively, of the"one-to-one" and "onto" function ƒ. There is an isomorphism from X to Y if the bijective function ƒ : X → Y happensto produce results, that sets up a correspondence between the operator and the operator .

for all u, v in X.

ApplicationsIn abstract algebra, two basic isomorphisms are defined:• Group isomorphism, an isomorphism between groups• Ring isomorphism, an isomorphism between rings. (Note that isomorphisms between fields are actually ring

isomorphisms)Just as the automorphisms of an algebraic structure form a group, the isomorphisms between two algebras sharing acommon structure form a heap. Letting a particular isomorphism identify the two structures turns this heap into agroup.In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easieralgebraic equations.In category theory, Iet the category C consist of two classes, one of objects and the other of morphisms. Then ageneral definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism ƒ: a → b that has an inverse, i.e. there exists a morphism g : b → a with ƒg = 1b and gƒ = 1a. For example, a bijectivelinear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is alsocontinuous is an isomorphism between topological spaces, called a homeomorphism.In graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to thevertices of H that preserves the "edge structure" in the sense that there is an edge from vertex u to vertex v in G ifand only if there is an edge from ƒ(u) to ƒ(v) in H. See graph isomorphism.

Isomorphism 13

In mathematical analysis, an isomorphism between two Hilbert spaces is a bijection preserving addition, scalarmultiplication, and inner product.In early theories of logical atomism, the formal relationship between facts and true propositions was theorized byBertrand Russell and Ludwig Wittgenstein to be isomorphic. An example of this line of thinking can be found inRussell's Introduction to Mathematical Philosophy.In cybernetics, the Good Regulator or Conant-Ashby theorem is stated "Every Good Regulator of a system must be amodel of that system". Whether regulated or self-regulating an isomorphism is required between regulator part andthe processing part of the system.

Relation with equalityIn certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the onehand and isomorphism on the other.[5] Equality is when two objects are exactly the same, and everything that's trueabout one object is true about the other, while an isomorphism implies everything that's true about one object'sstructure is true about the other's. For example, the sets

and are equal; they are merely different presentations—the first an intensional one (in set builder notation), and thesecond extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets {A,B,C} and{1,2,3} are not equal – the first has elements that are letters, while the second has elements that are numbers. Theseare isomorphic as sets, since finite sets are determined up to isomorphism by their cardinality (number of elements)and these both have three elements, but there are many choices of isomorphism – one isomorphism is

while another is and no one isomorphism is intrinsically better than any other.[6][7] On this view and in this sense, these two sets arenot equal because one cannot consider them identical: one can choose an isomorphism between them, but that is aweaker claim than identity—and valid only in the context of the chosen isomorphism.Sometimes the isomorphisms can seem obvious and compelling, but are still not equalities. As a simple example, thegenealogical relationships among Joe, John, and Bobby Kennedy are, in a real sense, the same as those among theAmerican football quarterbacks in the Manning family: Archie, Peyton, and Eli. The father-son pairings and theelder-brother-younger-brother pairings correspond perfectly. That similarity between the two family structuresillustrates the origin of the word isomorphism (Greek iso-, "same," and -morph, "form" or "shape"). But because theKennedys are not the same people as the Mannings, the two genealogical structures are merely isomorphic and notequal.Another example is more formal and more directly illustrates the motivation for distinguishing equality fromisomorphism: the distinction between a finite-dimensional vector space V and its dual space V* = { φ : V → K} oflinear maps from V to its field of scalars K. These spaces have the same dimension, and thus are isomorphic asabstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified bycardinality), but there is no "natural" choice of isomorphism . If one chooses a basis for V, then this yields anisomorphism: For all u. v ∈ V,

.This corresponds to transforming a column vector (element of V) to a row vector (element of V*) by transpose, but adifferent choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis". Moresubtly, there is a map from a vector space V to its double dual V** = { x : V* → K} that does not depend on thechoice of basis: For all v ∈ V and φ ∈ V*,

.

Isomorphism 14

This leads to a third notion, that of a natural isomorphism: while V and V** are different sets, there is a "natural"choice of isomorphism between them. This intuitive notion of "an isomorphism that does not depend on an arbitrarychoice" is formalized in the notion of a natural transformation; briefly, that one may consistently identify, or moregenerally map from, a vector space to its double dual, , for any vector space in a consistent way.Formalizing this intuition is a motivation for the development of category theory.If one wishes to draw a distinction between an arbitrary isomorphism (one that depends on a choice) and a naturalisomorphism (one that can be done consistently), one may write ≈ for an unnatural isomorphism and ≅ for a naturalisomorphism, as in V ≈ V* and V ≅ V**. This convention is not universally followed, and authors who wish todistinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction.Generally, saying that two objects are equal is reserved for when there is a notion of a larger (ambient) space thatthese objects live in. Most often, one speaks of equality of two subsets of a given set (as in the integer set exampleabove), but not of two objects abstractly presented. For example, the 2-dimensional unit sphere in 3-dimensionalspace

and the Riemann sphere which can be presented as the one-point compactification of the complex plane C ∪ {∞} or as the complexprojective line (a quotient space)

are three different descriptions for a mathematical object, all of which are isomorphic, but not equal because they arenot all subsets of a single space: the first is a subset of R3, the second is C ≅ R2[8] plus an additional point, and thethird is a subquotient of C2

In the context of category theory, objects are usually at most isomorphic – indeed, a motivation for the developmentof category theory was showing that different constructions in homology theory yielded equivalent (isomorphic)groups. Given maps between two objects X and Y, however, one asks if they are equal or not (they are both elementsof the set Hom(X, Y), hence equality is the proper relationship), particularly in commutative diagrams.

Notes[1] From the Greek: ἴσος isos "equal", and μορφή morphe "shape"[2] Buchmann, Johannes (2004). Introduction to cryptography (http:/ / books. google. com/ books?id=JEpVP9FC4o4C& pg=PA54). Springer.

p. 54. ISBN 9780387207568. .[3] Awodey, Steve (2006). "Isomorphisms" (http:/ / books. google. com/ books?id=IK_sIDI2TCwC& pg=PA11). Category theory. Oxford

University Press. p. 11. ISBN 9780198568612. .[4] Vinberg, Ėrnest Borisovich (2003). A Course in Algebra (http:/ / books. google. com/ books?id=kd24d3mwaecC& pg=PA3). American

Mathematical Society. p. 3. ISBN 9780821834138. .[5][5] (Mazur 2007)[6] The careful reader may note that A, B, C have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the

integers, and thus one particular isomorphism is "natural", namely

.More formally, as sets these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as ordered setsthey are naturally isomorphic (there is a unique isomorphism, given above), since finite total orders are uniquely determined up to uniqueisomorphism by cardinality.</math>

This intuition can be formalized by saying that any two finite totally ordered sets of the same cardinality have anatural isomorphism, the one that sends the least element of the first to the least element of the second, the leastelement of what remains in the first to the least element of what remains in the second, and so forth, but in general,pairs of sets of a given finite cardinality are not naturally isomorphic because there is more than one choice of map –except if the cardinality is 0 or 1, where there is a unique choice.[7] In fact, there are precisely different isomorphisms between two sets with three elements. This is equal to the number of

automorphisms of a given three-element set (which in turn is equal to the order of the symmetric group on three letters), and more generally one has that the set of isomorphisms between two objects, denoted is a torsor for the automorphism group of A,

Isomorphism 15

and also a torsor for the automorphism group of B. In fact, automorphisms of an object are a key reason to be concerned with the distinctionbetween isomorphism and equality, as demonstrated in the effect of change of basis on the identification of a vector space with its dual or withits double dual, as elaborated in the sequel.

[8][8] Being precise, the identification of the complex numbers with the real plane,

depends on a choice of one can just as easily choose , which yields a different identification – formally,complex conjugation is an automorphism – but in practice one often assumes that one has made such anidentification.

References

Further reading• Mazur, Barry (12 June 2007), When is one thing equal to some other thing? (http:/ / www. math. harvard. edu/

~mazur/ preprints/ when_is_one. pdf)

External links• Isomorphism (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=1936), PlanetMath.org.• Weisstein, Eric W., " Isomorphism (http:/ / mathworld. wolfram. com/ Isomorphism. html)" from MathWorld.

Homeomorphism

A continuous deformation between a coffee mugand a donut illustrating that they are

homeomorphic. But there need not be acontinuous deformation for two spaces to be

homeomorphic — only a continuous mappingwith a continuous inverse.

In the mathematical field of topology, a homeomorphism ortopological isomorphism or bicontinuous function is a continuousfunction between topological spaces that has a continuous inversefunction. Homeomorphisms are the isomorphisms in the category oftopological spaces—that is, they are the mappings that preserve all thetopological properties of a given space. Two spaces with ahomeomorphism between them are called homeomorphic, and from atopological viewpoint they are the same.

Roughly speaking, a topological space is a geometric object, and thehomeomorphism is a continuous stretching and bending of the objectinto a new shape. Thus, a square and a circle are homeomorphic toeach other, but a sphere and a donut are not. An often-repeatedmathematical joke is that topologists can't tell their coffee cup fromtheir donut,[1] since a sufficiently pliable donut could be reshaped tothe form of a coffee cup by creating a dimple and progressivelyenlarging it, while shrinking the hole into a handle.

Topology is the study of those properties of objects that do not changewhen homeomorphisms are applied. As Henri Poincaré famously said, mathematics is not the study of objects, butinstead, the relations (isomorphisms for instance) between them.[2]

Homeomorphism 16

DefinitionA function f: X → Y between two topological spaces (X, TX) and (Y, TY) is called a homeomorphism if it has thefollowing properties:• f is a bijection (one-to-one and onto),• f is continuous,• the inverse function f −1 is continuous (f is an open mapping).A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Yare homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. Thehomeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalenceclasses are called homeomorphism classes.

Examples

A trefoil knot is homeomorphic to a circle.Continuous mappings are not always realizable asdeformations. Here the knot has been thickened

to make the image understandable.

• The unit 2-disc D2 and the unit square in R2 are homeomorphic.• The open interval (a, b) is homeomorphic to the real numbers R for

any a < b.• The product space S1 × S1 and the two-dimensional torus are

homeomorphic.• Every uniform isomorphism and isometric isomorphism is a

homeomorphism.• The 2-sphere with a single point removed is homeomorphic to the

set of all points in R2 (a 2-dimensional plane).• Let A be a commutative ring with unity and let S be a multiplicative

subset of A. Then Spec(AS) is homeomorphic to {p ∈ Spec(A) : p ∩S = ∅}.

• Rm and Rn are not homeomorphic for m ≠ n.• The Euclidean real line is not homeomorphic to the unit circle as a subspace of R2 as the unit circle is compact as

a subspace of Euclidean R2 but the real line is not compact.

NotesThe third requirement, that f −1 be continuous, is essential. Consider for instance the function f: [0, 2π) → S1 definedby f(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism (S1 is compact but [0,2π) is not).Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of twohomeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X forms a group, calledthe homeomorphism group of X, often denoted Homeo(X); this group can be given a topology, such as thecompact-open topology, making it a topological group.For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one canreduce this group to the mapping class group.Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphismsbetween them, Homeo(X, Y), is a torsor for the homeomorphism groups Homeo(X) and Homeo(Y), and given aspecific homeomorphism between X and Y, all three sets are identified.

Homeomorphism 17

Properties• Two homeomorphic spaces share the same topological properties. For example, if one of them is compact, then

the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then theother is as well; their homotopy & homology groups will coincide. Note however that this does not extend toproperties defined via a metric; there are metric spaces that are homeomorphic even though one of them iscomplete and the other is not.

• A homeomorphism is simultaneously an open mapping and a closed mapping; that is, it maps open sets to opensets and closed sets to closed sets.

• Every self-homeomorphism in can be extended to a self-homeomorphism of the whole disk (Alexander'strick).

Informal discussionThe intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice toapply correctly—it may not be obvious from the description above that deforming a line segment to a point isimpermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.This characterization of a homeomorphism often leads to confusion with the concept of homotopy, which is actuallydefined as a continuous deformation, but from one function to another, rather than one space to another. In the caseof a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points onspace X correspond to which points on Y—one just follows them as X deforms. In the case of homotopy, thecontinuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of themaps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence.There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cuttingand regluing are required) an isotopy between the identity map on X and the homeomorphism from X to Y.

References[1] Hubbard, John H.; West, Beverly H. (1995). Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems

(http:/ / books. google. com/ books?id=SHBj2oaSALoC& pg=PA204& dq="coffee+ cup"+ topologist+ joke#v=onepage& q="coffee cup"topologist joke& f=false). Texts in Applied Mathematics. 18. Springer. p. 204. ISBN 978-0-387-94377-0. .

[2] Poincaré, Henri. "Chapter II: Mathematical Magnitude and Experiment" (https:/ / en. wikisource. org/ wiki/ Science_and_Hypothesis/PART_I#b). Science and Hypothesis. .

External links• Homeomorphism (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=912), PlanetMath.org.

Holomorphic function 18

Holomorphic function

A rectangular grid (top) and its image under aconformal map f (bottom).

In mathematics, holomorphic functions are the central objects ofstudy in complex analysis. A holomorphic function is acomplex-valued function of one or more complex variables that iscomplex differentiable in a neighborhood of every point in its domain.The existence of a complex derivative is a very strong condition, for itimplies that any holomorphic function is actually infinitelydifferentiable and equal to its own Taylor series.

The term analytic function is often used interchangeably with“holomorphic function”, although the word “analytic” is also used in abroader sense to describe any function (real, complex, or of moregeneral type) that is equal to its Taylor series in a neighborhood ofeach point in its domain. The fact that the class of complex analyticfunctions coincides with the class of holomorphic functions is a majortheorem in complex analysis.

Holomorphic functions are also sometimes referred to as regularfunctions[1] or as conformal maps. A holomorphic function whosedomain is the whole complex plane is called an entire function. Thephrase "holomorphic at a point z0" means not just differentiable at z0,but differentiable everywhere within some neighborhood of z0 in thecomplex plane.

Definition

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z0 in its domain isdefined by the limit

This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. Inparticular, the limit is taken as the complex number z approaches z0, and must have the same value for any sequenceof complex values for z that approach z0 on the complex plane. If the limit exists, we say that ƒ iscomplex-differentiable at the point z0. This concept of complex differentiability shares several properties with realdifferentiability: it is linear and obeys the product rule, quotient rule, and chain rule.If ƒ is complex differentiable at every point z0 in an open set  U, we say that ƒ is holomorphic on U. We say that ƒ isholomorphic at the point z

0 if it is holomorphic on some neighborhood of z0. We say that ƒ is holomorphic on some

non-open set A if it is holomorphic in an open set containing A.The relationship between real differentiability and complex differentiability is the following. If a complex functionƒ(x + i y) = u(x, y) + i v(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, andsatisfy the Cauchy–Riemann equations:

or, equivalently, the Wirtinger derivative of ƒ with respect to the complex conjugate of z is zero: which is

to say that, roughly, ƒ is functionally independendent from the complex conjugate of z.

Holomorphic function 19

If continuity is not a given, the converse is not necessarily true. A simple converse is that if u and v have continuousfirst partial derivatives and satisfy the Cauchy–Riemann equations, then ƒ is holomorphic. A more satisfyingconverse, which is much harder to prove, is the Looman–Menchoff theorem: if ƒ is continuous, u and v have firstpartial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then ƒ isholomorphic.

TerminologyThe word "holomorphic" was introduced by two of Cauchy's students, Briot (1817–1882) and Bouquet (1819–1895),and derives from the Greek ὅλος (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".[2]

Today, the term "holomorphic function" is sometimes preferred to "analytic function", as the latter is a more generalconcept. This is also because an important result in complex analysis is that every holomorphic function is complexanalytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use.

PropertiesBecause complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products andcompositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions isholomorphic wherever the denominator is not zero.

The derivative can be written as a contour integral using Cauchy's differentiation formula:

for any simple loop positively winding once around , and

for infinitesimal positive loops around .If one identifies C with R2, then the holomorphic functions coincide with those functions of two real variables withcontinuous first derivatives which solve the Cauchy-Riemann equations, a set of two partial differential equations.Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution ofLaplace's equation on R2. In other words, if we express a holomorphic function f(z) as u(x, y) + i v(x, y) both u and vare harmonic functions, where v is the harmonic conjugate of u and vice-versa.In regions where the first derivative is not zero, holomorphic functions are conformal in the sense that they preserveangles and the shape (but not size) of small figures.Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its valueson the disk's boundary.Every holomorphic function is analytic. That is, a holomorphic function f has derivatives of every order at each pointa in its domain, and it coincides with its own Taylor series at a in a neighborhood of a. In fact, f coincides with itsTaylor series at a in any disk centered at that point and lying within the domain of the function.From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and acomplex vector space. In fact, it is a locally convex topological vector space, with the seminorms being the supremaon compact subsets.From a geometric perspective, a function f is holomorphic at z0 if and only if its exterior derivative df in aneighborhood U of z0 is equal to f′(z) dz for some continuous function f′. It follows from

Holomorphic function 20

that df′ is also proportional to dz, implying that the derivative f′ is itself holomorphic and thus that f is infinitelydifferentiable. Similarly, the fact that d(f dz) = f′ dz ∧ dz = 0 implies that any function f that is holomorphic on thesimply connected region U is also integrable on U. (For a path γ from z0 to z lying entirely in U, define

;in light of the Jordan curve theorem and the generalized Stokes' theorem, Fγ(z) is independent of the particularchoice of path γ, and thus F(z) is a well-defined function on U having F(z0) = F0 and dF = f dz.)

ExamplesAll polynomial functions in z with complex coefficients are holomorphic on C, and so are sine, cosine and theexponential function. (The trigonometric functions are in fact closely related to and can be defined via theexponential function using Euler's formula). The principal branch of the complex logarithm function is holomorphicon the set C \ {z ∈ R : z ≤ 0}. The square root function can be defined as

and is therefore holomorphic wherever the logarithm log(z) is. The function 1/z is holomorphic on {z : z ≠ 0}.As a consequence of the Cauchy–Riemann equations, a real-valued holomorphic function must be constant.Therefore, the absolute value of z, the argument of z, the real part of z and the imaginary part of z are notholomorphic. Another typical example of a continuous function which is not holomorphic is complex conjugation.

Several variablesA complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it islocally expandable (within a polydisk, a Cartesian product of disks, centered at that point) as a convergent powerseries in the variables. This condition is stronger than the Cauchy–Riemann equations; in fact it can be stated asfollows:A function of several complex variables is holomorphic if and only if it satisfies the Cauchy–Riemann equations andis locally square-integrable.

Extension to functional analysisThe concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. Forinstance, the Fréchet or Gâteaux derivative can be used to define a notion of a holomorphic function on a Banachspace over the field of complex numbers.

References[1] Springer Online Reference Books (http:/ / eom. springer. de/ a/ a012240. htm), Wolfram MathWorld (http:/ / mathworld. wolfram. com/

RegularFunction. html)[2] Markushevich, A.I.; Silverman, Richard A. (ed.) (2005) [1977]. Theory of functions of a Complex Variable (http:/ / books. google. com/

books?id=H8xfPRhTOcEC& dq) (2nd ed. ed.). New York: American Mathematical Society. p. 112. ISBN 0-8218-3780-X. .

Diffeomorphism 21

DiffeomorphismIn mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertiblefunction that maps one differentiable manifold to another, such that both the function and its inverse are smooth.

The image of a rectangular grid on a square undera diffeomorphism from the square onto itself.

Definition

Given two manifolds M and N, a bijective map f from M to N is calleda diffeomorphism if both

and its inverse

are differentiable (if these functions are r times continuouslydifferentiable, f is called a -diffeomorphism).

Two manifolds M and N are diffeomorphic (symbol usually being ) if there is a smooth bijective map f from M to N with a smoothinverse. They are diffeomorphic if there is an r timescontinuously differentiable bijective map between them whose inverseis also r times continuously differentiable.

Diffeomorphisms of subsets of manifoldsGiven a subset X of a manifold M and a subset Y of a manifold N, a function f : X→ Y is said to be smooth if for all pin X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree

(note that g is an extension of f). We say that f is a diffeomorphism if it is bijective, smooth, andif its inverse is smooth.

Local descriptionModel example: if U and V are two connected open subsets of Rn such that V is simply connected, a differentiablemap f: U → V is a diffeomorphism if it is proper and if• the differential Dfx: R

n → Rn is bijective at each point x in U.Remarks• It is essential for U to be simply connected for the function f to be globally invertible (under the sole condition

that its derivative is a bijective map at each point).

• For example, consider the map (which is the "realification" ofthe complex square function) where U = V = R2 \ {(0,0)}. Then the map f is surjective and its satisfies

(thus Dfx is bijective at each point) yet f is not invertible, because it fails to beinjective, e.g., f(1,0) = (1,0) = f(-1,0).

• Since the differential at a point (for a differentiable function) is a linear map it has awell defined inverse if, and only if, Dfx is a bijection. The matrix representation of Dfx is the n × n matrix of firstorder partial derivatives whose entry in the i-th row and j-th colomn is . We often use this so-calledJacobian matrix for explicit computations.

• Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine that f were going fromdimension n to dimension k. If n < k then Dfx could never be surjective, and if n > k then Dfx could never beinjective. So in both cases Dfx fails to be a bijection.

Diffeomorphism 22

• If Dfx is a bijection at x then we say that f is a local diffeomorphism (since by continuity Dfy will also be bijectivefor all y sufficiently close to x).

• Given a smooth map from dimension n to dimension k, if Df (resp. Dfx) is surjective then we say that f is asubmersion (resp. local submersion), and if Df (resp. Dfx) is injective we say that f is an immersion (resp. localimmersion).

• A differentiable bijection is not necessarily a diffeomorphism, e.g. f(x) = x3 is not a diffeomorphism from R toitself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of ahomeomorphism that is not a diffeomorphism.

• f being a diffeomorphism is a stronger condition than f being a homeomorphism (when f is a map betweendifferentiable manifolds). For a diffeomorphism we need f and its inverse to be differentiable. For ahomeomorphism we only require that f and its inverse be continuous. Thus every diffeomorphism is ahomeomorphism, but the converse is false: not every homeomorphism is a diffeomorphism.

Now, f: M → N is called a diffeomorphism if in coordinates charts it satisfies the definition above. More precisely,pick any cover of M by compatible coordinate charts, and do the same for N. Let φ and ψ be charts on M and Nrespectively, with U being the image of φ and V the image of ψ. Then the conditions says that the map ψ f φ−1: U →V is a diffeomorphism as in the definition above (whenever it makes sense). One has to check that for every coupleof charts φ, ψ of two given atlases, but once checked, it will be true for any other compatible chart. Again we seethat dimensions have to agree.

ExamplesSince any manifold can be locally parametrised, we can consider some explicit maps from two-space into two-space.

• Let . We can calculate the Jacobian matrix:

The Jacobian matrix has zero determinant if, and only if xy = 0. We see that f is a diffeomorphism away from thex-axis and the y-axis.

• Let where the and arearbitrary real numbers, and the omitted terms are of degree at least two in x and y. We can calculate the Jacobianmatrix at 0:

We see that g is a local diffeomorphism at 0 if, and only if, , i.e. the linear terms in thecomponents of g are linearly independent as polynomials.• Now let . We can calculate the Jacobian matrix:

The Jacobian matrix has zero determinant everywhere! In fact we see that the image of h is the unit circle.

Diffeomorphism 23

Diffeomorphism groupLet M be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of M is thegroup of all Cr diffeomorphisms of M to itself, and is denoted by Diffr(M) or Diff(M) when r is understood. This is a'large' group, in the sense that it is not locally compact (provided M is not zero-dimensional).

TopologyThe diffeomorphism group has two natural topologies, called the weak and strong topology (Hirsch 1997). When themanifold is compact, these two topologies agree. The weak topology is always metrizable. When the manifold is notcompact, the strong topology captures the behavior of functions "at infinity", and is not metrizable. It is, however,still Baire.Fixing a Riemannian metric on M, the weak topology is the topology induced by the family of metrics

as K varies over compact subsets of M. Indeed, since M is σ-compact, there is a sequence of compact subsets Knwhose union is M. Then, define

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of Cr vector fields(Leslie 1967). Over a compact subset of M, this follows by fixing a Riemannian metric on M and using theexponential map for that metric. If r is finite and the manifold is compact, the space of vector fields is a Banachspace. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphismgroup into a Banach manifold. If r = ∞ or if the manifold is σ-compact, the space of vector fields is a Fréchet space.Moreover, the transition maps are smooth, making the diffeomorphism group into a Fréchet manifold.

Lie algebraIn particular, the Lie algebra of the diffeomorphism group of M consists of all vector fields on M, equipped with theLie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinate x at eachpoint in space:

so the infinitesimal generators are the vector fields

Examples• When M = G is a Lie group, there is a natural inclusion of G in its own diffeomorphism group via left-translation.

Let Diff(G) denote the diffeomorphism group of G, then there is a splitting Diff(G) ≃ G × Diff(G,e) whereDiff(G,e) is the subgroup of Diff(G) that fixes the identity element of the group.

• The diffeomorphism group of Euclidean space Rn consists of two components, consisting of the orientationpreserving and orientation reversing diffeomorphisms. In fact, the general linear group is a deformation retract ofsubgroup Diff(Rn,0) of diffeomorphisms fixing the origin under the map ƒ(x) ↦ ƒ(tx)/t, t ∈ (0,1]. Hence, inparticular, the general linear group is also a deformation retract of the full diffeomorphism group as well.

• For a finite set of points, the diffeomorphism group is simply the symmetric group. Similarly, if M is any manifold there is a group extension 0 → Diff0(M) → Diff(M) → Σ(π0M). Here Diff0(M)is the subgroup of Diff(M) that preserves all the components of M, and Σ(π0M) is the permutation group of the set π0M (the

Diffeomorphism 24

components of M). Moreover, the image of the map Diff(M) → Σ(π0M) is the bijections of π0M that preservediffeomorphism classes.

TransitivityFor a connected manifold M the diffeomorphism group acts transitively on M. More generally, the diffeomorphismgroup acts transitively on the configuration space CkM. If the dimension of M is at least two the diffeomorphismgroup acts transitively on the configuration space FkM: the action on M is multiply transitive (Banyaga 1997, p. 29).

Extensions of diffeomorphismsIn 1926, Tibor Radó asked whether the harmonic extension of any homeomorphism (or diffeomorphism) of the unitcircle to the unit disc yields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards byHellmuth Kneser and a completely different proof was discovered in 1945 by Gustave Choquet, apparently unawarethat the theorem was already known.The (orientation-preserving) diffeomorphism group of the circle is pathwise connected. This can be seen by notingthat any such diffeomorphism can be lifted to a diffeomorphism f of the reals satisfying f(x+1) = f(x) +1; this space isconvex and hence path connected. A smooth eventually constant path to the identity gives a second more elementaryway of extending a diffeomorphism from the circle to the open unit disc (this is a special case of the Alexandertrick). Moreover, the diffeomorphism group of the circle has the homotopy-type of the orthogonal group O(2).The corresponding extension problem for diffeomorphisms of higher dimensional spheres Sn−1 was much studied inthe 1950s and 1960s, with notable contributions from René Thom, John Milnor and Stephen Smale. An obstructionto such extensions is given by the finite Abelian group Γn, the "group of twisted spheres", defined as the quotient ofthe Abelian component group of the diffeomorphism group by the subgroup of classes extending to diffeomorphismsof the ball Bn.

ConnectednessFor manifolds the diffeomorphism group is usually not connected. Its component group is called the mapping classgroup. In dimension 2, i.e. for surfaces, the mapping class group is a finitely presented group, generated by Dehntwists (Dehn, Lickorish, Hatcher). Max Dehn and Jakob Nielsen showed that it can be identified with the outerautomorphism group of the fundamental group of the surface.William Thurston refined this analysis by classifying elements of the mapping class group into three types: thoseequivalent to a periodic diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curveinvariant; and those equivalent to pseudo-Anosov diffeomorphisms. In the case of the torus S1 x S1 = R2/Z2, themapping class group is just the modular group SL(2,Z) and the classification reduces to the classical one in terms ofelliptic, parabolic and hyperbolic matrices. Thurston accomplished his classification by observing that the mappingclass group acted naturally on a compactification of Teichmüller space; since this enlarged space was homeomorphicto a closed ball, the Brouwer fixed-point theorem became applicable.If M is an oriented smooth closed manifold, it was conjectured by Smale that the identity component of the group oforientation-preserving diffeomorphisms is simple. This had first been proved for a product of circles by MichelHerman; it was proved in full generality by Thurston.

Diffeomorphism 25

Homotopy types• The diffeomorphism group of S2 has the homotopy-type of the subgroup O(3). This was proven by Steve

Smale.[1]

• The diffeomorphism group of the torus has the homotopy-type of its linear automorphisms: (S1)2 × GL(2, Z).• The diffeomorphism groups of orientable surfaces of genus g > 1 have the homotopy-type of their mapping class

groups—i.e.: the components are contractible.•• The homotopy-type of the diffeomorphism groups of 3-manifolds are fairly well-understood via the work of

Ivanov, Hatcher, Gabai and Rubinstein although there are a few outstanding open cases, primarily 3-manifoldswith finite fundamental groups.

• The homotopy-type of diffeomorphism groups of n-manifolds for n > 3 are poorly undersood. For example, it isan open problem whether or not Diff(S4) has more than two components. But via the work of Milnor, Kahn andAntonelli it's known that Diff(Sn) does not have the homotopy-type of a finite CW-complex provided n > 6.

Homeomorphism and diffeomorphismIt is easy to find a homeomorphism that is not a diffeomorphism, but it is more difficult to find a pair ofhomeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2, 3, any pair of homeomorphic smoothmanifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairshave been found. The first such example was constructed by John Milnor in dimension 7. He constructed a smooth7-dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7-sphere but notdiffeomorphic to it. There are in fact 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere(each of them is a total space of the fiber bundle over the 4-sphere with the 3-sphere as the fiber).Much more extreme phenomena occur for 4-manifolds: in the early 1980s, a combination of results due to SimonDonaldson and Michael Freedman led to the discovery of exotic R4s: there are uncountably many pairwisenon-diffeomorphic open subsets of R4 each of which is homeomorphic to R4, and also there are uncountably manypairwise non-diffeomorphic differentiable manifolds homeomorphic to R4 that do not embed smoothly in R4.

Notes[1] Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959) 621–626.

ReferencesChaudhuri, Shyamoli, Hakuru Kawai and S.-H Henry Tye. "Path-integral formulation of closed strings," Phys. Rev.D, 36: 1148, 1987.• Banyaga, Augustin (1997), The structure of classical diffeomorphism groups, Mathematics and its Applications,

400, Kluwer Academic, ISBN 0-7923-4475-8• Duren, Peter L. (2004), Harmonic Mappings in the Plane, Cambridge Mathematical Tracts, 156, Cambridge

University Press, ISBN 0-521-64121-7• Hirsch, Morris (1997), Differential Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90148-0• Kriegl, Andreas; Michor, Peter (1997), The convenient setting of global analysis, Mathematical Surveys and

Monographs, 53, American Mathematical Society, ISBN 0-8218-0780-3• Leslie, J. A. (1967), "On a differential structure for the group of diffeomorphisms", Topology. an International

Journal of Mathematics 6 (2): 263–271, doi:10.1016/0040-9383(67)90038-9, ISSN 0040-9383, MR0210147• Milnor, John W. (2007), Collected Works Vol. III, Differential Topology, American Mathematical Society,

ISBN 0-8218-4230-7

Diffeomorphism 26

• Omori, Hideki (1997), Infinite-dimensional Lie groups, Translations of Mathematical Monographs, 158,American Mathematical Society, ISBN 0-8218-4575-6

• Kneser, Hellmuth (1926), "Lösung der Aufgabe 41." (in German), Jahresbericht der DeutschenMathematiker-Vereinigung 35 (2): 123f.

MonomorphismIn the context of abstract algebra or universal algebra, amonomorphism is an injective homomorphism. Amonomorphism from X to Y is often denoted with the notation

.

In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is aleft-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X,

Monomorphisms are a categorical generalization of injective functions; in some categories the notions coincide, butmonomorphisms are more general, as in the examples below.The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is anepimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism.

TerminologyThe companion terms monomorphism and epimorphism were originally introduced by Nicolas Bourbaki; Bourbakiuses monomorphism as shorthand for an injective function. Early category theorists believed that the correctgeneralization of injectivity to the context of categories was the cancellation property given above. While this is notexactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms.Saunders Mac Lane attempted to make a distinction between what he called monomorphisms, which were maps in aconcrete category whose underlying maps of sets were injective, and monic maps, which are monomorphisms in thecategorical sense of the word. This distinction never came into general use.Another name for monomorphism is extension, although this has other uses too.

Relation to invertibilityLeft invertible maps are necessarily monic: if l is a left inverse for f (meaning ), then f is monic, as

A left invertible map is called a split mono.

A map f : X → Y is monic if and only if the induced map f∗ : Hom(Z, X) → Hom(Z, Y), defined by for all morphisms h : Z → X , is injective for all Z.

Monomorphism 27

ExamplesEvery morphism in a concrete category whose underlying function is injective is a monomorphism. In the categoryof sets, the converse also holds so the monomorphisms are exactly the injective morphisms. The converse also holdsin most naturally occurring categories of algebras because of the existence of a free object on one generator. Inparticular, it is true in the categories of groups and rings, and in any abelian category.It is not true in general, however, that all monomorphisms must be injective in other categories. For example, in thecategory Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms thatare not injective: consider, for example, the quotient map q : Q → Q/Z. This is not an injective map; nevertheless, itis a monomorphism in this category. This follows from (and in fact, is equivalent to) the implication q ∘ h = 0 ⇒ h =0, which we now prove. (NB: The converse of this last implication, namely, h = 0 ⇒ q ∘ h = 0, is trivially true, but itis not needed here). If h : G → Q, where G is some divisible group, and q ∘ h = 0, then h(x) ∈ Z, ∀ x ∈ G. Now fixsome x ∈ G. Without loss of generality, we may assume that h(x) ≥ 0 (otherwise, choose −x instead). Then, since Gis a divisible group, for some y ∈ G, x = (h(x) + 1) y, so h(x) = (h(x) + 1) h(y). From this, and 0 ≤ h(x) < h(x) + 1, itfollows that

Since h(y) ∈ Z, it follows that h(y) = 0, and thus h(x) = 0 = h(−x), ∀ x ∈ G. This says that h = 0, as desired. Now, ifq ∘ f = q ∘ g for some morphisms f, g : G → Q, where G is some divisible group then q ∘ (f − g) = 0, where (f − g) : x↦ f(x) − g(x). (Since (f − g)(0) = 0, and (f - g)(x + y) = (f - g)(x) + (f - g)(y), it follows that (f - g) ∈ Hom(G, Q)).From the result just proved, q ∘ (f − g) = 0 ⇒ f − g = 0 ⇔ ∀ x ∈ G, f(x) = g(x) ⇔ f = g. Hence q is a monomorphism,as claimed.

Properties• In a topos, every monic is an equalizer, and any map that is both monic and epic is an isomorphism.•• Every isomorphism is monic.

Related conceptsThere are also useful concepts of regular monomorphism, strong monomorphism, and extremalmonomorphism. A regular monomorphism equalizes some parallel pair of morphisms. An extremal monomorphismis a monomorphism that cannot be nontrivially factored through an epimorphism: Precisely, if m=g ∘ e with e anepimorphism, then e is an isomorphism. A strong monomorphism satisfies a certain lifting property with respect tocommutative squares involving an epimorphism.

References• Francis Borceux (1994), Handbook of Categorical Algebra 1, Cambridge University Press. ISBN 0-521-44178-1.• George Bergman (1998), An Invitation to General Algebra and Universal Constructions [1], Henry Helson

Publisher, Berkeley. ISBN 0-9655211-4-1.• Jaap van Oosten, Basic Category Theory [2]

References[1] http:/ / math. berkeley. edu/ ~gbergman/ 245/ index. html[2] http:/ / www. math. uu. nl/ people/ jvoosten/ syllabi/ catsmoeder. pdf

Epimorphism 28

EpimorphismIn category theory, an epimorphism (also called an epicmorphism or, colloquially, an epi) is a morphism f : X → Ywhich is right-cancellative in the sense that, for all morphismsg1, g2 : Y → Z,

Epimorphisms are analogues of surjective functions, but they are not exactly the same. The dual of an epimorphismis 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 surjectivehomomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, butthe converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of categorytheory given above. For more on this, see the section on Terminology below.

ExamplesEvery morphism in a concrete category whose underlying function is surjective is an epimorphism. In many concretecategories of interest the converse is also true. For example, in the following categories, the epimorphisms areexactly those morphisms which 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 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 OttoSchreier (he actually proved more, showing that every subgroup is an equalizer using the free product with oneamalgamated 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 quotientmap 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, weproceed exactly as in Set, giving {0,1} the indiscrete topology which ensures that all considered maps arecontinuous.

• HComp, compact Hausdorff spaces and continuous functions. If f: X → Y is not surjective, let y in Y-fX. Since fXis closed, by Urysohn's Lemma there is a continuous function g1:Y → [0,1] such that g1 is 0 on fX and 1 on y. Wecompose f with both g1 and the zero function g2: Y → [0,1].

However there are also many concrete categories of interest where epimorphisms fail to be surjective. A fewexamples 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(n) ≠ g2(n), so g1(-n) ≠

Epimorphism 29

g2(-n). 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 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 theobservation that 1 generates the algebra R[Z] (note that the unit in R[Z] is given by 0 of Z), and the inverse of theelement represented by n in Z is just the element represented by -n. Thus any homomorphism from R[Z] isuniquely 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 thatany ring homomorphism on Q is determined entirely by its action on Z, similar to the previous example. Asimilar argument shows that the natural ring homomorphism from any commutative ring R to any one of itslocalizations is an epimorphism.

• In the category of commutative rings, a finitely generated homomorphism of rings f : R → S is an epimorphism ifand 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, theinduced 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 denseimages. 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 areprecisely those whose underlying functions are injective.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 every morphism is an epimorphism.

PropertiesEvery 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 is easily seen to be an epimorphism. A map with such a right-sided inverse is called asplit 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 anepimorphism, then f must be an epimorphism.As some of the above examples show, the property of being an epimorphism is not determined by the morphismalone, but also by the category of context. If D is a subcategory of C, then every morphism in D which is anepimorphism when considered as a morphism in C is also an epimorphism in D; the converse, however, need nothold; 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 anequivalence F : C → D, then a morphism f is an epimorphism in the category C if and only if F(f) is an epimorphismin 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 theinduced 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.

Epimorphism 30

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 acoequalizer, is not true in all categories.In many categories it is possible to write every morphism as the composition of a monomorphism followed by anepimorphism. For instance, given a group homomorphism f : G → H, we can define the group K = im(f) = f(G) andthen write f as the composition of the surjective homomorphism G → K which is defined like f, followed by theinjective homomorphism K → H which sends each element to itself. Such a factorization of an arbitrary morphisminto an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all theconcrete categories mentioned above in the Examples section (though not in all concrete categories).

Related conceptsAmong other useful concepts are regular epimorphism, extremal epimorphism, strong epimorphism, and splitepimorphism. A regular epimorphism coequalizes some parallel pair of morphisms. An extremal epimorphism is anepimorphism that has no monomorphism as a second factor, unless that monomorphism is an isomorphism. A strongepimorphism satisfies a certain lifting property with respect to commutative squares involving a monomorphism. Asplit epimorphism is a morphism which has a right-sided inverse.A morphism that is both a monomorphism and an epimorphism is called a bimorphism. Every isomorphism is abimorphism but the converse is not true in general. For example, the map from the half-open interval [0,1) to the unitcircle S1 (thought of as a subspace of the complex plane) which sends x to exp(2πix) (see Euler's formula) iscontinuous and bijective but not a homeomorphism since the inverse map is not continuous at 1, so it is an instanceof a bimorphism that is not an isomorphism in the category Top. Another example is the embedding Q → R in thecategory 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 maps Z → Q and Q → R are bimorphisms but not isomorphisms.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 equivalencerelation, and the equivalence classes are defined to be the quotient objects of X.

TerminologyThe companion terms epimorphism and monomorphism were first introduced by Bourbaki. Bourbaki usesepimorphism as shorthand for a surjective function. Early category theorists believed that epimorphisms were thecorrect analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exactanalogue of injections. Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely tosurjections than ordinary epimorphisms. Saunders Mac Lane attempted to create a distinction betweenepimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epicmorphisms, 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 betterconcept. 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 ownunique concept, related to surjections but fundamentally different.

Epimorphism 31

References• Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories [1] (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 [2].• Lawvere & Rosebrugh: Sets for Mathematics, Cambridge university press, 2003. ISBN 0-521-80444-2.

References[1] http:/ / katmat. math. uni-bremen. de/ acc/ acc. pdf[2] http:/ / groups. google. com/ group/ sci. math/ msg/ 6d4023d93a2b4300

Article Sources and Contributors 32

Article Sources and ContributorsMorphism  Source: http://en.wikipedia.org/w/index.php?oldid=502672907  Contributors: 213.253.39.xxx, Alberto da Calvairate, Albmont, Aragorn2, Archelon, AxelBoldt, Bart v M, Blotwell,Bonotake, Brad7777, Bryan Derksen, Cenarium, Ciphergoth, Conversion script, Dysprosia, Edcolins, Elwikipedista, Fropuff, Giftlite, Glenn, Gorobay, Graham87, Greenrd, Helder.wiki, Hotfeba,IstvanWolf, JamesBWatson, Jon Awbrey, Jose Ramos, Jsnx, Kku, Kmarinas86, Larry_Sanger, Lethe, Linas, LokiClock, Looxix, Michael Hardy, Michael K. Edwards, Mikez, Msh210, Ntmatter,Odoncaoa, PV=nRT, Palnot, Phys, Porges, Pthag, Revolver, Salix alba, Sam Hocevar, Semorrison, Simon J Kissane, Sjef, Slawekb, Smimram, TakuyaMurata, The Anome, Tkeu, Weppens,Xantharius, 27 anonymous edits

Homomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=493862815  Contributors: 213.253.39.xxx, Adriaan Joubert, Allanhalme, Andres, AugPi, AxelBoldt, Bloodshedder, BryanDerksen, CRGreathouse, Calle, Classicalecon, Clément Pillias, Conversion script, Dirac1933, Dysprosia, EagleFan, Ed g2s, El C, Electiontechnology, EmilJ, Ex13, FractalFusion, Fropuff,Gandalf61, Giftlite, Glenn, Goochelaar, Graham87, Hadal, Hukkinen, Ishboyfay, JAn Dudík, Jim.belk, JoergenB, Johntyree, Jyoshimi, Kanags, Kevinbsmith, KnightRider, Korako, Kostisl,Larry_Sanger, Ligulem, LokiClock, Martynas Patasius, Mehdi.manshadi, Mesoderm, Mgreenbe, Michael Hardy, Michael Slone, Mlm42, Msh210, Netsnipe, Nishantjr, Oblivious, OlegAlexandrov, PV=nRT, Pcap, Pramcom, Qwfp, R'n'B, Rat144, Rgamble, Riteshsood, Robert K S, Sahar Tomer, Salix alba, Sebastian Goll, Staffwaterboy, The Anome, TheSeven, Toby Bartels,Tong, Vantelimus, Wowulu, XJamRastafire, Youandme, Zero0000, Zundark, 78 anonymous edits

Automorphism  Source: http://en.wikipedia.org/w/index.php?oldid=502140961  Contributors: Algebraist, Altenmann, Amire80, Archelon, AxelBoldt, Bdesham, Brews ohare, Caiodnh,Cbogart2, Chas zzz brown, Conversion script, Crisófilax, David Eppstein, Dogaroon, Dreftymac, Dysprosia, Elwikipedista, Fropuff, Gauge, Giftlite, Gogo Dodo, Goochelaar, Gregbard, HannesEder, Happy-melon, Henning Makholm, Hephaestos, Huppybanny, JRB-Europe, JackSchmidt, Jan Hidders, Keenan Pepper, Ksanyi, LC, MSGJ, Marc van Leeuwen, MatrixFrog, Mcmillin24,Mets501, Michael Hardy, Mild Bill Hiccup, MishaMisha, Mskalak13, Nbarth, Nishantjr, Noisy, Obradovic Goran, Orenburg1, PV=nRT, Patrick, Peruvianllama, Phys, Pixeltoo, Pred, Qwertyus,Rdsmith4, Reyk, Rgdboer, Rich Farmbrough, Salgueiro, Sam Hocevar, Shadowjams, SirJective, Tarquin, TechnoGuyRob, Thehotelambush, TimothyRias, Tommy Jantarek, Topology Expert,Tosha, VKokielov, William M. Connolley, Xezbeth, Youssefsan, Zaslav, 23 anonymous edits

Endomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=504334772  Contributors: Altenmann, Andres, Arskwad, AugPi, AxelBoldt, CitizenB, Conversion script, David Eppstein,Dbenbenn, Doctorfree, Dysprosia, Elwikipedista, Ewlyahoocom, Fropuff, Giftlite, Graham87, Helder.wiki, Ilmari Karonen, JackSchmidt, Javy tahu, Jim.belk, Joshua Davis, JoshuaGrosse,Kmarinas86, MFH, Magmalex, Melchoir, Oleg Alexandrov, PV=nRT, Point-set topologist, Quiddity, Salix alba, Smimram, Tarquin, Thomas T Howard, Tosha, Vonkje, Zundark, Zyqqh, 21anonymous edits

Isomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=502956121  Contributors: 16@r, Albmont, Andre Engels, Anonymous Dissident, Army1987, Avaya1, AxelBoldt, Bahar101,Banus, Bender2k14, Bigdumbdinosaur, Bkell, Borhan0, Brad7777, Brews ohare, Bryanmcdonald, CRGreathouse, Charles Matthews, Conversion script, Coolhandscot, Cpiral, Cronholm144, DanGranahan, Debresser, Dysprosia, Edemaine, Elwikipedista, EmilJ, Ezequiels.90, Fantomdrives, Fly by Night, Ghakko, Giftlite, Glenn, Grubber, Gryllida, Hannes Eder, Iamthedeus,Ioannis.Demetriou, Isomorphic, Ivan Štambuk, Jim.belk, Jlittlet, KSmrq, Kmarinas86, Koberozendaal, LOL, Lars Washington, LokiClock, Lubos, M hariprasad, Marc Venot, MarkSweep,MathKnight, Mathwiz777, Mattmacf, Mcole13, Melongrower, Mesoderm, Mets501, Michael Angelkovich, Michael Hardy, Michael Slone, Mike4ty4, Msh210, Mxn, Nabla, Nbarth, Netsnipe,Nick Green, Oleg Alexandrov, Omnipedian, PV=nRT, Patrick, Paul August, PaulTanenbaum, Peruvianllama, Philip Cross, PhotoBox, Phys, Pokus9999, QuiteUnusual, Quondum, Revolver,Richard Molnár-Szipai, Rljacobson, Rlupsa, Rschwieb, Ryguasu, Sam Staton, Slawekb, Smite-Meister, Spacepotato, Spinality, Spinningspark, Ssavelan, Stevertigo, Stuhacking,Subversive.sound, Superbatfish, Sławomir Biały, TK-925, Taemyr, Thaddeus Slamp, The Rhymesmith, Thehotelambush, Tosha, Trumpet marietta 45750, Uncle Dick, Wrp103, YnnusOiramo,Youandme, Youssefsan, Yuide, Yurik, Zell08v, Zero0000, Zundark, Канеюку, 89 ,کاشف عقیل anonymous edits

Homeomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=498554773  Contributors: 9258fahsflkh917fas, Adeliine, Ahoerstemeier, Andrew Delong, Arjun r acharya, Asztal,AxelBoldt, BenFrantzDale, Brians, Charles Matthews, ClamDip, Conversion script, Crasshopper, DekuDekuplex, Dominus, Dr.K., Dysprosia, Elwikipedista, EmilJ, Explorall, Frencheigh,Fropuff, Giftlite, Glenn, Grondemar, Hadal, Hakeem.gadi, Henry Delforn (old), Joule36e5, Juan Marquez, Kieff, Linas, LokiClock, Lorem Ip, Mark Foskey, MathKnight, MathMartin,Mchipofya, Michael Angelkovich, Michael Hardy, Miguel, MisfitToys, Ms2ger, Msh210, Myasuda, Nbarth, Ojigiri, Oleg Alexandrov, Orionus, Orthografer, Pandaritrl, Parudox, Paul August,Pengo, PhilKnight, Prijutme4ty, Renamed user 1, RyanEberhart, Salix alba, Seqsea, Set theorist, SingingDragon, Steelpillow, The cattr, TheObtuseAngleOfDoom, ThomasWinwood, TobiasBergemann, Toby Bartels, Topology Expert, Tosha, Vonkje, Weialawaga, Wikomidia, Wjmallard, XJamRastafire, Xmlizer, Youandme, Zero sharp, Zhaoway, Zundark, 69 ,ماني anonymous edits

Holomorphic function  Source: http://en.wikipedia.org/w/index.php?oldid=505855894  Contributors: 212.242.115.xxx, Aaronbrick, Abiola Lapite, Acepectif, Adoniscik, Almit39, Altenmann,AxelBoldt, Bdmy, Ben pcc, Brad7777, CRGreathouse, Charles Matthews, Chinju, Christian.Mercat, Conversion script, Crasshopper, CrniBombarder!!!, Crust, Damian Yerrick, DavidCBryant,DomenicDenicola, Dougher, DragonflySixtyseven, Dratman, Duckbill, Dysprosia, Dzordzm, Eequor, Email4mobile, EmilJ, Fakhredinblog, FilipeS, Frencheigh, FuriousScribble, Giftlite,Graham87, HannsEwald, Hesam7, Howard McCay, Irigi, Isocliff, JamesBWatson, Jao, Jim.belk, Jmath666, Jobh, Jusjih, KnightRider, Laurent MAYER, Lethe, Lhf, Linas, Lunch, Maxim Razin,Michael Hardy, Michael K. Edwards, Mike40033, Mokhtari34, Naddy, NickBush24, ObsessiveMathsFreak, Oleg Alexandrov, PV=nRT, Patrick, PierreAbbat, Prim Ethics, Reaverdrop,Riwnodennyk, Robert Illes, Rvollmert, Saleemsan, Salix alba, Silly rabbit, Slawekb, Sligocki, Some jerk on the Internet, Stephen Bain, TakuyaMurata, Tarquin, Tcnuk, Tetracube, The DiagonalPrince, Thorfinn, Tobias Bergemann, Tobias Hoevekamp, Unco, Vanished User 0001, Weierstraß, XJamRastafire, 60 anonymous edits

Diffeomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=504516901  Contributors: 0, AugPi, Charles Matthews, Chester Markel, Connelly, CryptoDerk, Dharma6662000, Dreadstar,Dysprosia, Ensign beedrill, Fly by Night, Fropuff, Gaius Cornelius, Geometry guy, Giftlite, Headbomb, Helder.wiki, Herve.lombaert, JeLuF, JerroldPease-Atlanta, Kuszi, LachlanA, Lethe,Lmatt, MathMartin, Mathsci, Maury Markowitz, Med, Mhwu, Michael Hardy, Msh210, Muses' house, Nakon, Nosirrom, Oleg Alexandrov, PV=nRT, Pascalromon, Paul August, Physicistjedi,Point-set topologist, Poor Yorick, R.e.b., Redrose64, Rybu, Silly rabbit, Slawekb, Sławomir Biały, TakuyaMurata, Topology Expert, Tosha, Uni.Liu, Whitepaw, Woseph, Åkebråke, 42anonymous edits

Monomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=503687785  Contributors: Adler F, Albmont, Altenmann, Archelon, Blotwell, CRGreathouse, Ckhenderson, Classicalecon,Crasshopper, E-boy, Eric Kvaalen, Fropuff, Gauge, Giftlite, Hairy Dude, Hans Adler, Le schtroumpf, Lethe, Magidin, MarSch, Nbarth, Nick Number, Obradovic Goran, PV=nRT, Perturbationist,Rich Farmbrough, Sam Hocevar, Tesseran, Tobias Bergemann, Toby Bartels, Tosha, Xantharius, YouRang?, 14 anonymous edits

Epimorphism  Source: http://en.wikipedia.org/w/index.php?oldid=487833533  Contributors: Albmont, Altenmann, Anonymous Dissident, AxelBoldt, Beroal, Ckhenderson, Daniel5Ko, Esap,Fropuff, Gauge, Giftlite, Icey, Itai, Kmarinas86, Kompik, Magidin, Mszamotulski, Nbarth, PV=nRT, Paul August, Pit-trout, Salix alba, Sam Hocevar, Silvonen, Slawekb, Thehotelambush, TobyBartels, Tosha, VeryVerily, Zoicon5, 28 anonymous edits

Image Sources, Licenses and Contributors 33

Image Sources, Licenses and ContributorsImage:Commutative diagram for morphism.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Commutative_diagram_for_morphism.svg  License: Public Domain  Contributors:User:CepheusImage:Morphisms.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Morphisms.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: User:EmilJ,based on a png image by en:User:AugPiImage:Mug and Torus morph.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Mug_and_Torus_morph.gif  License: Public Domain  Contributors: Abnormaal, Durova, Howcheng,Kieff, Kri, Manco Capac, Maximaximax, Rovnet, SharkD, Takabeg, 16 anonymous editsImage:Trefoil knot arb.png  Source: http://en.wikipedia.org/w/index.php?title=File:Trefoil_knot_arb.png  License: GNU Free Documentation License  Contributors: Editor at Large, Ylebru, 2anonymous editsImage:Conformal map.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Conformal_map.svg  License: Public Domain  Contributors: Oleg AlexandrovImage:Diffeomorphism of a square.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Diffeomorphism_of_a_square.svg  License: Public Domain  Contributors: Oleg AlexandrovImage:Monomorphism-01.png  Source: http://en.wikipedia.org/w/index.php?title=File:Monomorphism-01.png  License: Public Domain  Contributors: Derlay, Ico83, Samulili, 1 anonymouseditsImage:Epimorphism-01.png  Source: http://en.wikipedia.org/w/index.php?title=File:Epimorphism-01.png  License: Public Domain  Contributors: Darapti, Derlay, Maksim

License 34

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/