09}}In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. 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.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.

A category C consists of two classes, one of objects and the other of morphisms.There are two operations which are defined on every morphism, the domain (or source) and the codomain (or target).If a morphism f has domain X and codomain Y, we write f

X ? Y. Thus a morphism is represented by an arrow from its domain to its codomain. 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) or Mor(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.

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 o f = f = f o id_A.

Associativity: h o (g o f) = (h o g) o f whenever the operations are defined.

Monomorphism: f

X ? Y is called a monomorphism if f o g₁ = f o g₂ implies g₁ = g₂ for all morphisms g₁, g₂

Z ? X. It is also called a mono or a monic. 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 split monomorphism h

X ? Y is a monomorphism having a left inverse g

Y ? X, so that g o h = id_X. Thus h o g

Y ? Y is idempotent, so that (h o g)² = h o g.

In concrete categories, a function that has a 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.

Epimorphism: Dually, f

X ? Y is called an epimorphism if g₁ o f = g₂ o f implies g₁ = g₂ for all morphisms g₁, g₂

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 f o 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 always true in every category, as an epimorphism may fail to have a right inverse.

A split epimorphism is an epimorphism having a right inverse. Note that 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, every surjection has a section, a result equivalent to the axiom of choice.

A bimorphism is 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 o g = id_Y and g o 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, 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, such as Set, in which every bimorphism is an isomorphism is known a
Examples


In the concrete categories studied in universal algebra (groups, rings, modules, etc.), morphisms are called homomorphisms. Likewise, the notions of automorphism, endomorphism, epimorphism, homeomorphism, isomorphism, and monomorphism all find use in universal algebra.

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.

In the category of small categories, functors can be thought of as morphisms.

In a functor category, the morphisms are natural transformations.

For more examples, see the entry category theory.




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See also


normal morphism

zero morphism




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Notes





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