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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map. 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 == A category ''C'' consists of two classes, one of ''objects'' and the other of ''morphisms''. There are two objects that are associated to every morphism, the ''source'' and the ''target''. For many common categories, objects are sets (usually with more structure) and morphisms are functions from an object to another object. Therefore the source and the target of a morphism are often called respectively ''domain'' and ''codomain''. A morphism ''f'' with source ''X'' and target ''Y'' is written ''f'' : ''X'' → ''Y''. Thus a morphism is represented by an ''arrow'' from its source to its target. Morphisms are equipped with a partial binary operation, called ''composition''. The composition of two morphism ''f'' and ''g'' is defined if and only if the target of ''g'' is the source of ''f'', and is denoted ''f''∘''g''. The source of ''f''∘''g'' is the source of ''g'', and the target of ''f''∘''g'' is the target of ''f''. The composition satisfies 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 we have id''B'' ∘ ''f'' = ''f'' = ''f'' ∘ id''A''. :Associativity: ''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f'' whenever the operations are defined, that is when the target of ''f'' is the source of ''g'', and the target of ''g'' is the source of ''h''. For a concrete category (that is the objects are sets with additional structure, and of the morphisms as structure-preserving functions), the identity morphism is just the identity function, and composition is just the ordinary composition of functions. ''Associativity'' then follows, because the composition of functions is associative. 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 a bit 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. Note that 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 (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 == ;Monomorphism: ''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''. It is also called a ''mono'' or a ''monic''.〔Jacobson (2009), p. 15.〕 ;Epimorphism: Dually, ''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''. It is also called an ''epi'' or an ''epic''.〔 ;Bimorphism: A morphism that is both an epimorphism and a monomorphism. ;Isomorphism: ''f'': ''X'' → ''Y'' is called an isomorphism 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. Note that 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 Set, in which every bimorphism is an isomorphism is known as a balanced category. ;Endomorphism: ''f'': ''X'' → ''X'' 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. ;Automorphism: A morphism that is both an endomorphism and an isomorphism. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Morphism」の詳細全文を読む スポンサード リンク
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