Morphism

In mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. 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 spaces (or objects) on which they are defined forms a branch of mathematics called 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 the so-called 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, for any two objects X and Y, a set of morphisms from X to Y. Morphisms are often depicted as arrows between those objects, e.g. a morphism f from X to Y is denoted f : X → Y. The set of all morphisms from X to Y is denoted Mor_C(X,Y) or sometimes Hom_C(X,Y).

For every three objects X, Y, and Z there exists a binary operation Mor(X,Y) × Mor(Y,Z) → Mor(X,Z) called composition. The composition of f : X → Y and g : Y → Z is written as 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 by definition).

There are two operations defined on every morphism, the domain (or source) which assigns the morphism f : X → Y the object X:

dom(f) = X,

and the codomain (or target) which assigns the morphism f : X → Y the object Y:

cod(f) = Y.

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.

• A morphism f : X → Y is called an 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 bimorphism but, in general, not every bimorphism is an isomorphism.

• 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.