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In mathematics, a 'morphism' is an abstraction of a structure-preserving mapping between two mathematical structures.
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.
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.
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.

Contents

Definition

Examples

Some remarkable morphisms

See also

External links

Definition

A category ''C'' is given by two pieces of data: a class of ''objects'' and a class of ''morphisms''.
There are two operations defined on every morphism, the 'domain' (or 'source') and the 'codomain' (or 'target').
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 $gcirc\; f$ or ''gf'' (Some authors write it as ''fg''.) Composition of morphisms is often denoted by means of a commutative diagram. For example,

Morphisms must satisfy two axioms:

★ 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 $\{$
m id}_Bcirc f=f=fcirc{
m id}_A.

★ ASSOCIATIVITY: $hcirc(gcirc\; f)=(hcirc\; g)circ\; f$ 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.)
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.

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.

Some remarkable morphisms

• 
  A morphism ''f'' : ''X'' → ''Y'' is called a 'monomorphism' if $fcirc\; g\_1=fcirc\; g\_2$ implies ''g''₁ = ''g''₂ for all morphisms ''g''₁, ''g''₂ : ''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 $gcirc\; f=\{$
  m id}_X. 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 $g\_1circ\; f=g\_2circ\; f$ implies ''g''₁ = ''g''₂ for all morphisms ''g''₁, ''g''₂ : ''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 $fcirc\; g=\{$
  m id}_Y. 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''.
  



• 
  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 $fcirc\; g=\{$
  m id}_Y and $gcirc\; f=\{$
  m 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.
  

  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.
  



• 
  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 $gcirc\; h=\{$
  m id}_X, thus $hcirc\; g:,\; Y\; o\; Y$ is idempotent, which means that $(hcirc\; g)^2=hcirc\; g$. More generally, any idempotent endomorphism ''f'' is said to be 'split' if it admits a decomposition $f=hcirc\; g$ with $gcirc\; h=mathrm\{id\}$. In particular, the Karoubi envelope of a category splits every idempotent.
  

See also

★ zero morphism

★ normal morphism

★ holomorphic function

★ homomorphism

★ anamorphism

★ catamorphism

★ hylomorphism

★ paramorphism

External links

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