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Encyclopedia > Morphism

In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

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... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, a continuous function is a function 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. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

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. Category theory is a mathematical theory that 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. ...

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 : XY. The set of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y) and called the hom-set between X and Y. (Some authors write MorC(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 : XY and g : YZ is written $gcirc f$ 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, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... 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. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...

• IDENTITY: for every object X, there exists a morphism idX : XX called the identity morphism on X, such that for every morphism f : AB we have ${rm id}_Bcirc f=f=fcirc{rm id}_A$.
• 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 set of ordered pairs, 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.

## Types of morphisms

• Let f : XY be a morphism. If there exists a morphism g : YX such that and $gcirc f={rm id}_X$ then f is called an isomorphism and g is said to be an inverse morphism of f. Inverse morphisms, if they exist, are unique. It is easy to see that g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent. Isomorphisms are the most important kinds of morphisms in category theory.
• A morphism f : XY is called an epimorphism if $g_1circ f=g_2circ f$ implies g1 = g2 for all morphisms g1, g2 : YZ. It is also called an epi or an epic. Epimorphisms in concrete categories are typically surjective morphisms, although this is not always the case.
• A morphism f : XY is called a monomorphism if implies g1 = g2 for all morphisms g1, g2 : ZX. It is also called a mono or a monic. Monomorphisms in concrete categories are typically injective morphisms.
• If f is both an epimorphism and a monomorphism then f is called a bimorphism. 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 ZQ is a bimorphism which is not an isomorphism. A category in which every bimorphism is an isomorphism is a balanced category. For example, Set is a balanced category.
• An endomorphism that is also an isomorphism is called an automorphism.
• If f : XY and g : YX satisfy one can show that f is epic and g is monic and that $gcirc f$ : XX is idempotent. In this case f and g are said to be split. f is called a retraction of g and g is called a section of f. Any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. (E.g. if the latter holds for f, then $fcirc gcirc f={rm id}_Y circ f = f circ {rm id}_X$, and therefore, by the monic property of f, $gcirc f={rm id}_X$.)

See also: 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. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... 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 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 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. ...

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.

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. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... 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 notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... 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 in which arbitrarily small changes in the input produce arbitrarily 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 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. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Results from FactBites:

 Morphism - definition of Morphism in Encyclopedia (801 words) In the context of universal algebra morphisms are generically known as homomorphisms. The abstract study of morphisms and the spaces (or objects) on which they are defined forms a branch of mathematics called category theory. Despite the abstract nature of morphisms, most people's intution about them (and indeed much of the terminology) comes from the case of the so-called concrete categories where the objects are simply sets with some additional structure and morphisms are functions preserving this structure.
 Morphism - Wikipedia, the free encyclopedia (924 words) In set theory, for example, morphisms are just functions, in group theory they are group homomorphisms, while in topology they are continuous functions. 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. Morphisms are often depicted as arrows from their domain to their codomain, e.g.
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