# Map (mathematics)

One type of map is a function, as in the association of any of the four colored shapes in X to its color in Y

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[citation needed], 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.

## Maps as functions

In many branches of mathematics, the term map is used to mean a function,[6][2][7] sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc.

Some authors, such as Serge Lang,[8] use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of R or C), and reserve the term mapping for more general functions.

Maps of certain kinds are the subjects of many important theories. These include homomorphisms in abstract algebra, isometries in geometry, operators in analysis and representations in group theory.[3]

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems.

A partial map is a partial function. Related terms such as domain, codomain, injective, and continuous can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.

## As morphisms

In category theory, "map" is often used as a synonym for "morphism" or "arrow", and thus is more general than "function".[9] For example, a morphism ${\displaystyle f:\,X\to Y}$ in a concrete category (i.e. a morphism which can be viewed as functions) carries with it the information of its domain (the source ${\displaystyle X}$ of the morphism) and its codomain (the target ${\displaystyle Y}$). In the widely used definition of a function ${\displaystyle f:X\to Y}$, ${\displaystyle f}$ is a subset of ${\displaystyle X\times Y}$ consisting of all the pairs ${\displaystyle (x,f(x))}$ for ${\displaystyle x\in X}$. In this sense, the function does not capture the information of which set ${\displaystyle Y}$ is used as the codomain; only the range ${\displaystyle f(X)}$ is determined by the function.