Morphism

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 topology, continuous functions, and so on. 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 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. 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. 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 with source X and target Y is written f : X → Y, and is represented diagrammatically by an arrow from X to Y. 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: 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 HomC(X,Y) or simply Hom(X, Y) and called the hom-set between X and Y. Some authors write MorC(X,Y), Mor(X, Y) or C(X, Y). Note that 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.