# Map (mathematics)

In mathematics, the term **mapping**, usually shortened to **map**, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function. 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, 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,^{[1]} use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of the fields **R** or **C**) and the term *mapping* for more general functions.

Sets of maps of special kinds are the subjects of many important theories: see for instance Lie group, mapping class group, permutation group.

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

A *partial map* is a *partial function*, and a *total map* is a *total function*. Related terms like *domain*, *codomain*, *injective*, *continuous*, etc. 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.

## Maps as morphisms

In category theory, "map" is often used as a synonym for morphism or arrow, thus for something more general than a function.^{[2]}

## Other uses

### In logic

In formal logic, the term **map** is sometimes used for a *functional predicate*, whereas a function is a model of such a predicate in set theory.

### In graph theory

In graph theory, a **map** is a drawing of a graph on a surface without overlapping edges (an embedding). If the surface is a plane then a map is a planar graph, similar to a political map.^{[3]}

### In computer science

In the communities surrounding programming languages that treat functions as first-class citizens, a map often refers to the binary higher-order function that takes a function *f* and a list [*v*_{0}, *v*_{1}, ..., *v*_{n}] as arguments and returns [*f*(*v*_{0}), *f*(*v*_{1}), ..., *f*(*v*_{n})], where *n* ≥ 0.