Morphism

In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. 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, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

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.

For many common categories, objects are sets (usually with more 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.

A morphism f with source X and target Y is written f : X → Y. Thus a morphism is represented by an arrow from its source to its target.

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