Morphism

In mathematics, a morphism is an abstraction of a structure-preserving process 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.

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). (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 g O 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 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.

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 set of ordered pairs, 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.

Types of morphisms

• Let f : X → Y be a morphism. If there exists a morphism g : Y → X such that f O g = id_Y and g O f = id_X then f is called an isomorphism and g is said to be an inverse morphism of f. Inverse morphisms, if they exist, are unique. It is easy to see that g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent. Isomorphisms are the most important kinds of morphisms in category theory.

• A morphism 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. Epimorphisms in concrete categories are typically surjective morphisms, although this is not always the case.

• A morphism 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. Monomorphisms in concrete categories are typically injective morphisms.

• If f is both an epimorphism and a monomorphism then f is called a bimorphism. Note that every isomorphism is a bimorphism but, in general, not every bimorphism is 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.

• If f : X → Y and g : Y → X satisfy f O g = id_Y one can show that f is epic and g is monic and that g O f : X → X is idempotent. In this case f and g are said to be split. f is called a retraction of g and g is called a section of f. Any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism.

See also:

• zero morphisms
• normal morphisms

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.de:Morphismus es:Morfismo fr:Morphisme it:Morfismo sv:Morfism