Morphisms

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. 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.

Morphisms and objects are constituents of a category. Morphisms, also called ''maps'' or ''arrows'', relate two objects called the ''source'' and the ''target'' of the morphism. There is a partial operation, 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 of composition when it is defined, and existence of an identity morphism for every object).

Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for homological algebra and algebraic topology. They belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that applies also to algebraic number theory.

Definition

A category ''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 diagrams.

For many common categories, an object is a set (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 axioms:
; : For every object ''X'', there exists a morphism called the identity morphism on ''X'', such that for every morphism we have .
; Associativity : 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, 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 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. 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, 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; 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, 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. 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, the statement that every surjection has a section is equivalent to the axiom of choice.

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

Isomorphisms

A morphism is called an isomorphism 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 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 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 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 structures commonly considered in algebra, such as groups, rings, modules, etc., the morphisms are usually the homomorphisms, 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", although there are ring epimorphisms that are not surjective (e.g., when embedding the integers in the rational numbers).
* In the category of topological spaces, the morphisms are the continuous functions and isomorphisms are called homeomorphisms. There are bijections (that is, isomorphisms of sets) that are not homeomorphisms.
* In the category of smooth manifolds, the morphisms are the smooth functions and isomorphisms are called diffeomorphisms.
* In the category of small categories, the morphisms are functors.
* In a functor category, the morphisms are natural transformations.

For more examples, see Category theory.

See also

* Normal morphism
* Zero morphism

Notes

References

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

External links

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