# Map (mathematics)

In mathematics, the term **mapping**, sometimes shortened to **map**, is a relationship between mathematical objects or structures.

Maps may either be *functions* or *morphisms*, though the terms share some overlap.^{[1]} In the sense of a function, a map is often associated with some sort of structure, particularly a set constituting the codomain.^{[2]} Alternatively, a map may be described by 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.

## Contents

## Maps as functionsEdit

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,^{[3]} 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 morphismsEdit

In category theory, "map" is often used as a synonym for morphism or arrow, thus for something more general than a function.^{[4]} For example, morphisms , in a concrete category, in other words morphisms that can be viewed as functions, carry with them the information of both its domain (the source of the morphism), but also its co-domain (the target ). In the widely used definition of function , this is a subset of consisting of all the pairs for . In this sense, the function doesn't capture the information of which set is used as the co-domain. Only the range is determined by the function.

## Other usesEdit

### In logicEdit

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 theoryEdit

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.^{[5]}

### In computer scienceEdit

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.

## See alsoEdit

## ReferencesEdit

**^**The words**map**,**mapping**,**transformation**,**correspondence**, and**operator**are often used synonymously. Halmos 1970, p. 30. In many authors, the term 'map' is with a more general meaning than 'function', which may be restricted to having domains of sets of numbers only.**^**T. M. Apostol (1981).*Mathematical Analysis*. Addison-Wesley. p. 35.**^**Lang, Serge (1971),*Linear Algebra*(2nd ed.), Addison-Wesley, p. 83**^**Simmons, H. (2011),*An Introduction to Category Theory*, Cambridge University Press, p. 2, ISBN 9781139503327**^**Gross, Jonathan; Yellen, Jay (1998),*Graph Theory and its applications*, CRC Press, p. 294, ISBN 0-8493-3982-0