In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. 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 analysis and topology, continuous functions, and so on.

In category theory, morphism is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism.^[1]

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. A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as ${\displaystyle f\colon X\to Y}$ or ${\textstyle X\,{\overset {f}{\longrightarrow }}\,Y,}$ the latter form being better suited for commutative diagrams.

For many common categories, objects are sets (often with some additional structure) and morphisms are functions from an object to another object. Therefore, the source and the target of a morphism are often called domain and codomain respectively.

Morphisms are equipped with a partial binary operation, called composition. The composition of two morphisms f and g is defined precisely when the target of f is the source of g, and is denoted g ∘ f (or sometimes simply gf). The source of g ∘ f is the source of f, and the target of g ∘ f is the target of g. 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 f : A → B we have id_B ∘ f = f = f ∘ id_A.

Associativity

h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever all the compositions are defined, i.e. when the target of f is the source of g, and the target of g is the source of h.

For a concrete category (a category in which the objects are sets, possibly with additional structure, and the morphisms are structure-preserving functions), the identity morphism is just the identity function, and composition is just ordinary composition of functions.

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). The term hom-set is something 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. Because hom-sets may not be sets, some people prefer to use the term "hom-class".

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

Examples

• For algebraic structures commonly considered in algebra, such as groups, rings, modules, etc., the morphisms are usually the homomorphisms, and the notions of isomorphism, automorphism, endomorphism, epimorphism, and monomorphism are the same as the above defined ones. However, in the case of rings, "epimorphism" is often considered as a synonym of "surjection", although there are ring epimorphisms that are not surjective (e.g., when embedding the integers in the rational numbers).
• In the category of topological spaces, the morphisms are the continuous functions and isomorphisms are called homeomorphisms. There are bijections (that is, isomorphisms of sets) that are not homeomorphisms.
• In the category of smooth manifolds, the morphisms are the smooth functions and isomorphisms are called diffeomorphisms.
• In the category of small categories, the morphisms are functors.
• In a functor category, the morphisms are natural transformations.

For more examples, see Category theory.

See also

• Normal morphism
• Zero morphism

Notes

References

• Jacobson, Nathan (2009), Basic algebra, vol. 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7.
• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6. Now available as free on-line edition (4.2MB PDF).

External links

• "Morphism", Encyclopedia of Mathematics, EMS Press, 2001 [1994]