mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a morphism is a concept of

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

that generalizes structure-preserving maps such as

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

between

algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...

s, functions from a set to another set, and

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

s between

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar to

function composition In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...

. Morphisms and objects are constituents of a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

. Morphisms, also called ''maps'' or ''arrows'', relate two objects called the ''source'' and the ''target'' of the morphism. There is a

partial operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation o ...

, called ''composition'', on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition (

associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

of composition when it is defined, and existence of an

identity morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...

for every object). Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

and

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. They belong to the foundational tools of Grothendieck's

scheme theory In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different s ...

, a generalization of

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

that applies also to

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

Definition

''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'' ''from'' ''X'' ''to'' ''Y'' is a morphism with source ''X'' and target ''Y''; it is commonly written as or the latter form being better suited for

commutative diagram 350px, The commutative diagram used in the proof of the five lemma In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...

s. For many common categories, an object is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

(often with some additional structure) and a morphism is a function 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 (''partial'' because the composition is not necessarily defined over every pair of morphisms of a category). The composition of two morphisms ''f'' and ''g'' is defined precisely when the target of ''f'' is the source of ''g'', and is denoted (or sometimes simply ''gf''). The source of is the source of ''f'', and the target of 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 or ...

s: ; : For every object ''X'', there exists a morphism called the identity morphism on ''X'', such that for every morphism we have . ;

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

: 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

. For example, : The collection of all morphisms from ''X'' to ''Y'' is denoted or simply and called the hom-set between ''X'' and ''Y''. Some authors write , or . The term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set; a category where is a set for all objects ''X'' and ''Y'' is called

locally small In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...

. Because hom-sets may not be sets, some people prefer to use the term "hom-class". 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 by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...

, where morphisms are functions, two functions may be identical as sets of ordered pairs, while having different codomains. The two functions are distinct from the viewpoint of category theory. Many authors require that the hom-classes 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 is called a monomorphism if implies for all morphisms ''g''₁, . 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 such that . Thus is

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

; that is, . The left inverse ''g'' is also called a retraction of ''f''. Morphisms with left inverses are always monomorphisms ( implies , where is the left inverse of ), but the converse is not true in general; a monomorphism may fail to have a left inverse. In concrete categories, a morphism 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 of its codomain; that is, implies (equivalently by contraposition, impl ...

, and a morphism that is injective is a monomorphism. In concrete categories, monomorphisms are often, but not always, injective; thus 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 is called an epimorphism if implies for all morphisms ''g''₁, . 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 such that . The right inverse ''g'' is also called a section of ''f''. Morphisms having a right inverse are always epimorphisms ( implies where is the right inverse of ), 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 In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

. 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, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

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

Isomorphisms

A morphism is called an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...

if there exists a morphism 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...

or equivalent. While every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

s the inclusion 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 (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 with . In particular, the Karoubi envelope of a category splits every idempotent morphism. An

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all 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

s commonly considered in

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, 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 such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

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

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 (for example, The set of all ...

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

s and isomorphisms are called

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

s. There are

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

s (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 may ...

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 (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

s and isomorphisms are called

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

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

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

s. * In a

functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...

, 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 natur ...

s. For more examples, see

Category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

Definition

Some special morphisms

Monomorphisms and epimorphisms

Isomorphisms

Endomorphisms and automorphisms

Examples

See also

Notes

References

External links