In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ...

The most common example occurs when the process is a function or map which preserves the structure in some sense. In set theory, for example, morphisms are just functions; in group theory they are group homomorphisms; while in topology they are continuous functions. In the context of universal algebra morphisms are generically known as homomorphisms. Partial plot of a function f. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Group theory is that branch of mathematics concerned with the study of groups. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... This word should not be confused with homeomorphism. ...

The abstract study of morphisms and the structures (or objects) between which they are defined forms part of category theory. In category theory, morphisms need not be functions at all and are usually thought as arrows between two different objects (which need not be sets). Rather than mapping elements of one set to another they simply represent some sort of relationship between the domain and codomain. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

Despite the abstract nature of morphisms, most people's intuition about them (and indeed much of the terminology) comes from the case of 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 a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...

Definition

A category C is given by two pieces of data: a class of objects and a class of morphisms. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

There are two operations defined on every morphism, the domain (or source) and the codomain (or target). In mathematics, the domain of a function is the set of all input values to the function. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...

Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. 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. The composite of f : X → Y and g : Y → Z is written or gf (Some authors write it as fg.) Composition of morphisms is often denoted by means of a commutative diagram. For example, In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Morphisms must satisfy two axioms: Image File history File links Commutative_diagram_for_morphism. ... This article does not adequately cite its references or sources. ...

• 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 .
• ASSOCIATIVITY: whenever the operations are defined.

When C is a concrete category, composition is just ordinary composition of functions, the identity morphism is just the identity function, and associativity is automatic. (Functional composition is associative.) In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

Note that the domain and codomain are really part of the information determining the morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (having the same range), but have different codomains. These functions are considered distinct for the purposes of category theory. For this reason, many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problem, because if they are not disjoint, the domain and codomain can be appended to the morphisms, (say, as the second and third components of an ordered triple), making them disjoint. In mathematics, the range of a function is the set of all output values produced by that function. ...

Some remarkable morphisms

• A morphism f : X → Y is called a monomorphism if implies g₁ = g₂ for all morphisms g₁, g₂ : Z → X. 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 . 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 monomorphism which does have a left-inverse is called a split monomorphism. In concrete categories, a function which has left-inverse is injective. 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.
• Dually, a morphism f : X → Y is called an epimorphism if implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z. 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 . The right-inverse g is also called a section of f. Morphisms with right-inverse are always epimorphisms, but the converse is not always true in every category; an epimorphism may fail to have a right-inverse. An epimorphism which does have a right-inverse is called a split epimorphism. In concrete categories, a function which has 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 of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section. This result is equivalent to the axiom of choice.

  Note that if a split monomorphism f has a left-inverse g, then g is a split epimorphism and has right-inverse f. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

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

  Note that every isomorphism is a bimorphism but, in general, not every bimorphism is an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism which is not an isomorphism. 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 in which every bimorphism is an isomorphism is a balanced category. For example, Set is a balanced category. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

• Any morphism f : X → X is called an endomorphism of X.
• An endomorphism that is also an isomorphism is called an automorphism.
• A split monomorphism h : X → Y has left-inverse g : Y → X, so that , thus is idempotent, which means that . More generally, any idempotent endomorphism f is said to be split if it admits a decomposition with . In particular, the Karoubi envelope of a category splits every idempotent.

See also: In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... The Karoubi envelope is a classification of the idempotents of a category. ...

• zero morphisms
• normal morphisms
• holomorphic function

In category theory, a zero morphism is a special kind of trivial morphism. ... In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Examples

• In the concrete categories studied in universal algebra (such as those of groups, rings, modules, etc.), morphisms are called homomorphisms. The terms isomorphism, epimorphism, monomorphism, endomorphism, and automorphism are all used in that specialized context as well.

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

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

• Functors can be thought of as morphisms in the category of small categories.

• In a functor category the morphisms are natural transformations.

For more examples see the article on category theory. Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... This picture illustrates how the hours in a clock form a group. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra, a homomorphism is a structure-preserving map. ... The category Top has topological spaces as objects and continuous maps as morphisms. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...