In mathematics, particularly in category theory, a morphism is a structure-preserving map from one

mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...

to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

, morphisms are functions; in

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

, linear transformations; in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, group homomorphisms; in

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to

function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...

. A morphism in category theory is an abstraction of a

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. 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 ''structure-preserving functions''. In category theory, morphisms are sometimes also called arrows.

Definition

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

''C'' consists of two classes, one of and the other of . There are two objects that are associated to every morphism, the and the . A morphism ''f'' with source ''X'' and target ''Y'' is written ''f'' : ''X'' → ''Y'', and is represented diagrammatically by an from ''X'' to ''Y''. For many common categories, objects are sets (often with some additional structure) and morphisms are functions from an object to another object. Therefore, the source and the target of a morphism are often called and respectively. Morphisms are equipped with a partial binary operation, called . The composition of two morphisms ''f'' and ''g'' is defined precisely when the target of ''f'' is the source of ''g'', and is denoted ''g'' ∘ ''f'' (or sometimes simply ''gf''). The source of ''g'' ∘ ''f'' is the source of ''f'', and the target of ''g'' ∘ ''f'' is the target of ''g''. The composition satisfies two

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...

s: ;: For every object ''X'', there exists a morphism id_''X'' : ''X'' → ''X'' called the identity morphism on ''X'', such that for every morphism we have id_''B'' ∘ ''f'' = ''f'' = ''f'' ∘ id_''A''. ;

Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

: ''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f'' whenever all the compositions are defined, i.e. when the target of ''f'' is the source of ''g'', and the target of ''g'' is the source of ''h''. For a concrete category (a category in which the objects are sets, possibly with additional structure, and the morphisms are structure-preserving functions), the identity morphism is just the

identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

, and composition is just ordinary composition of functions. The composition of morphisms is often represented by a commutative diagram. For example, : The collection 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''), Mor(''X'', ''Y'') or C(''X'', ''Y''). Note that the term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set; a category where Hom(''X'', ''Y'') is a set for all objects ''X'' and ''Y'' is called locally small. Because hom-sets may not be sets, some people prefer to use the term "hom-class". Note that the domain and codomain are in fact part of the information determining a morphism. For example, in the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...

, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. 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 special morphisms

Monomorphisms and epimorphisms

A morphism ''f'': ''X'' → ''Y'' is called a monomorphism if ''f'' ∘ ''g''₁ = ''f'' ∘ ''g''₂ implies ''g''₁ = ''g''₂ for all morphisms ''g''₁, ''g''₂: ''Z'' → ''X''. A monomorphism can be called a ''mono'' for short, and we can use ''monic'' as an adjective.Jacobson (2009), p. 15. A morphism ''f'' has a left inverse or is a split monomorphism if there is a morphism ''g'': ''Y'' → ''X'' such that ''g'' ∘ ''f'' id_''X''. Thus ''f'' ∘ ''g'': ''Y'' → ''Y'' is idempotent; that is, (''f'' ∘ ''g'')² ''f'' ∘ (''g'' ∘ ''f'') ∘ ''g'' ''f'' ∘ ''g''. The left inverse ''g'' is also called a

retraction Retraction or retract(ed) may refer to: Academia * Retraction in academic publishing, withdrawals of previously published academic journal articles Mathematics * Retraction (category theory) * Retract (group theory) * Retraction (topology) Huma ...

of ''f''. Morphisms with left inverses are always monomorphisms, but the converse is not true in general; a monomorphism may fail to have a left inverse. In concrete categories, a function that has a left inverse is

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

. 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 to monomorphisms, a morphism ''f'': ''X'' → ''Y'' is called an epimorphism if ''g''₁ ∘ ''f'' = ''g''₂ ∘ ''f'' implies ''g''₁ = ''g''₂ for all morphisms ''g''₁, ''g''₂: ''Y'' → ''Z''. An epimorphism can be called an ''epi'' for short, and we can use ''epic'' as an adjective. A morphism ''f'' has a right inverse or is a split epimorphism if there is a morphism ''g'': ''Y'' → ''X'' such that ''f'' ∘ ''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 true in general, as an epimorphism may fail to have a right inverse. If a monomorphism ''f'' splits with left 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 of being an epimorphism, but weaker than that of being a split epimorphism. In the

, the statement that every surjection has a section is equivalent to the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

. A morphism that is both an epimorphism and a monomorphism is called a bimorphism.

Isomorphisms

A morphism ''f'': ''X'' → ''Y'' is called an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

if there exists a morphism ''g'': ''Y'' → ''X'' such that ''f'' ∘ ''g'' = id_''Y'' and ''g'' ∘ ''f'' = id_''X''. 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. 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 that 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, such as a Set, in which every bimorphism is an isomorphism is known as a balanced category.

Endomorphisms and automorphisms

A morphism ''f'': ''X'' → ''X'' (that is, a morphism with identical source and target) is an endomorphism of ''X''. A split endomorphism is an idempotent endomorphism ''f'' if ''f'' admits a decomposition ''f'' = ''h'' ∘ ''g'' with ''g'' ∘ ''h'' = id. In particular, the Karoubi envelope of a category splits every idempotent morphism. An automorphism is a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form a group, called the automorphism group of the object.

Examples

* For

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

s commonly considered in

algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, such as groups, rings, modules, etc., the morphisms are usually the

s, and the notions of isomorphism, automorphism, endomorphism, epimorphism, and monomorphism are the same as the above defined ones. However, in the case of rings, "epimorphism" is often considered as a synonym of "

surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

", although there are ring epimorphisms that are not surjective (e.g., when embedding the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s in the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s). * In the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...

, the morphisms are the continuous functions and isomorphisms are called

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

s. There are bijections (that is, isomorphisms of sets) that are not homeomorphisms. * In the category of

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...

s, the morphisms are the

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

s and isomorphisms are called

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...

s. * In the category of small categories, the morphisms are

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

s. * In a functor category, the morphisms are

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a na ...

s. For more examples, see Category theory.

Notes

References

* . * Now available as free on-line edition (4.2MB PDF).

External links

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