Map (mathematics)

In mathematics, a map is often used as a synonym for a function,[1] but may also refer to some generalizations. Originally, this was an abbreviation of mapping, which often refers to the action of applying a function to the elements of its domain. This terminology is not completely fixed, as these terms are generally not formally defined, and can be considered to be jargon.[2] These terms may have originated as a generalization of the process of making a geographical map, which consists of mapping the Earth surface to a sheet of paper.[3]

Maps may either be functions or morphisms, though the terms share some overlap.[3] The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning as well as another one.[4][5] In category theory, a map may refer to a morphism, which is a generalization of the idea of a function. In some occasions, the term transformation can also be used interchangeably.[3] There are also a few less common uses in logic and graph theory.