In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Mathematics, a structure on a set, or more generally a type, consists of additional Mathematical objects that in some manner attach to the

The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are functions preserving this structure. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving Nevertheless, morphisms are not necessarily functions, and objects over which morphisms are defined are not necessarily sets. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Instead, a morphism is often thought of as an arrow linking an object called the domain to another object called the codomain. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Hence morphisms do not so much map sets into sets, as embody a relationship between some posited domain and codomain.

The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in topology, continuous functions; in universal algebra, homomorphisms; in group theory, group homomorphisms. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

Contents 1 Definition 2 Some specific morphisms 3 Examples 4 See also 5 External links

Definition

A category C consists of two classes, one of objects and the other of morphisms. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

There are two operations defined on every morphism, the domain (or source) and the codomain (or target). In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the codomain, or target, of a function f: X → Y is the set

If a morphism f has domain X and codomain Y, we write f : X → Y. Thus a morphism is an arrow from its domain to its codomain. The set of all morphisms from X to Y is denoted hom_C(X,Y) or simply hom(X, Y) and called the hom-set between X and Y. (Some authors write Mor_C(X,Y) or Mor(X, Y)).

For every three objects X, Y, and Z, there exists a binary operation hom(X, Y) × hom(Y, Z) → hom(X, Z) called composition. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, a composite function represents the application of one function to the results of another The composite of f : X → Y and g : Y → Z is written g o f, gf, or even fg. The composition of morphisms is often represented by a commutative diagram. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also For example,

Morphisms satisfy two axioms:

• Identity: for every object X, there exists a morphism id_X : X → X called the identity morphism on X, such that for every morphism f : A → B we have id_B o f = f = f o id_A. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject
• Associativity: h o (g o f) = (h o g) o f whenever the operations are defined. In Mathematics, associativity is a property that a Binary operation can have

When C is a concrete category, the identity morphism is just the identity function, and composition is just the ordinary composition of functions. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that In Mathematics, a composite function represents the application of one function to the results of another 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 the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. In Mathematics, the range of a function is the set of all "output" values produced by that function The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms, (say, as the second and third components of an ordered triple).

Some specific morphisms

• Monomorphism: f : X → Y is called a monomorphism if f o g₁ = f o g₂ implies g₁ = g₂ for all morphisms g₁, g₂ : Z → X. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.

  It is also called a mono or a monic. The morphism f has a left inverse if there is a morphism g:Y → X such that g o f = id_X. The left inverse g is also called a retraction of f. Morphisms with left inverses are always monomorphisms, but the converse is not 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 o h = id_X. Thus h o g : Y → Y is idempotent, so that (h o g)² = h o g. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

  In concrete categories, a function which has left inverse is injective. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.

• Epimorphism: Dually, f : X → Y is called an epimorphism if g₁ o f = g₂ o f implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which It is also called an epi or an epic. The morphism f has a right-inverse if there is a morphism g : Y → X such that f o g = id_Y. The right inverse g is also called a section of f. Morphisms having a right inverse are always epimorphisms, but the converse is not 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.

  In concrete categories, a function which has a right inverse is surjective. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section, a result equivalent to the axiom of choice. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

  Note that if a split monomorphism f has a left-inverse g, then g is a split epimorphism and has right-inverse f.

• 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 o g = id_Y and g o f = id_X. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

  If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

  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 an isomorphism. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.

• Endomorphism: f : X → X is an endomorphism of X. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself

  A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h o g with g o h = id. In particular, the Karoubi envelope of a category splits every idempotent morphism. In Mathematics the Karoubi envelope (or Cauchy completion, but that term has other meanings of a category C is a classification of the

• An automorphism is a morphism that is both an endomorphism and an isomorphism. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

Examples

• In the concrete categories studied in universal algebra (groups, rings, modules, etc. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars ), morphisms are called homomorphisms. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector Likewise, the notions of automorphism, endomorphism, epimorphism, homeomorphism, isomorphism, and monomorphism all find use in universal algebra. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

• In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms. In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Topological equivalence redirects here see also Topological equivalence (dynamical systems.

• In the category of smooth manifolds, morphisms are smooth functions and isomorphisms are called diffeomorphisms. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

• In the category of small categories, functors can be thought of as morphisms. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

• In a functor category, the morphisms are natural transformations. In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal

For more examples, see the entry category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

See also

• anamorphism
• automorphism
• catamorphism
• category theory
• concrete category
• diffeomorphism
• endomorphism
• epimorphism
• holomorphic function
• homeomorphism
• homomorphism
• hylomorphism
• isomorphism
• monomorphism
• normal morphism
• paramorphism
• zero morphism

External links

• Category on PlanetMath
• TypesOfMorphisms on PlanetMath

Anamorphism is a concept from Functional programming grounded in Category theory. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself The concept of a catamorphism is grounded in Category theory, and has been applied to Functional programming. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Computer science, and in particular Functional programming, a hylomorphism is a Recursive function corresponding to the composition of an In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory and its applications to Mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of Morphism A paramorphism (from Greek παρα, meaning "close together" is an extension of the concept of Catamorphism to deal with a form which “eats In Category theory, a zero morphism is a special kind of "trivial" Morphism. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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