# Map (mathematics)

In mathematics, a **map** or **mapping** is a function in its general sense.^{[1]} These terms may have originated as from the process of making a geographical map: *mapping* the Earth surface to a sheet of paper.^{[2]}

The term *map* may be used to distinguish some special types of functions, such as

^{[3]}

^{[4]}In category theory, a map may refer to a morphism.

^{[2]}The term

*transformation*can be used interchangeably,

^{[2]}but

*transformation*often refers to a function from a set to itself. 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,^{[5]}^{[6]}^{[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

**C**

*mapping*for more general functions.

Maps of certain kinds have been given specific names. These include

In the theory of

A *partial map* is a *partial function*. Related terminology 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", which is a structure-respecting function and thus may imply more structure than "function" does.^{[9]} For example, a morphism in a concrete category (i.e. a morphism that can be viewed as a function) carries with it the information of its domain (the source of the morphism) and its codomain (the target ). In the widely used definition of a function , is a subset of consisting of all the pairs for . In this sense, the function does not capture the set that is used as the codomain; only the range is determined by the function.

## See also

- Apply function – Function that maps a function and its arguments to the function value
- Arrow notation – e.g., , also known as
*map* - Bijection, injection and surjection – Properties of mathematical functions
- Homeomorphism – Mapping which preserves all topological properties of a given space
- List of chaotic maps
- Maplet arrow (↦)– commonly pronounced "maps to"
- Mapping class group – Group of isotopy classes of a topological automorphism group
- Permutation group – Group whose operation is composition of permutations
- Regular map (algebraic geometry)– Morphism of algebraic varieties

## References

**^**The words*map*,*mapping*,*correspondence*, and*operator*are often used synonymously. Halmos 1970, p. 30. Some authors use the term*function*with a more restricted meaning, namely as a map that is restricted to apply to numbers only.- ^
^{a}^{b}^{c}^{d}"Mapping | mathematics".*Encyclopedia Britannica*. Retrieved 2019-12-06. - ISBN 0-201-00288-4.
**^**Stacho, Juraj (October 31, 2007). "Function, one-to-one, onto" (PDF).*cs.toronto.edu*. Retrieved 2019-12-06.**^**"Functions or Mapping | Learning Mapping | Function as a Special Kind of Relation".*Math Only Math*. Retrieved 2019-12-06.**^**Weisstein, Eric W. "Map".*mathworld.wolfram.com*. Retrieved 2019-12-06.**^**"Mapping, Mathematical | Encyclopedia.com".*www.encyclopedia.com*. Retrieved 2019-12-06.- ISBN 0-201-04211-8.
**.**

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**### Works cited

## External links