# Map (mathematics)

__: Function, homomorphism, or morphism__

**Short description**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 homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial.^{[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 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.^{[2]}

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 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 [math]\displaystyle{ f:\, X \to Y }[/math] 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 [math]\displaystyle{ X }[/math] of the morphism) and its codomain (the target [math]\displaystyle{ Y }[/math]). In the widely used definition of a function [math]\displaystyle{ f:X\to Y }[/math], [math]\displaystyle{ f }[/math] is a subset of [math]\displaystyle{ X\times Y }[/math] consisting of all the pairs [math]\displaystyle{ (x,f(x)) }[/math] for [math]\displaystyle{ x\in X }[/math]. In this sense, the function does not capture the set [math]\displaystyle{ Y }[/math] that is used as the codomain; only the range [math]\displaystyle{ f(X) }[/math] 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., [math]\displaystyle{ x\mapsto x+1 }[/math], 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

## 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. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}"Mapping | mathematics" (in en). https://www.britannica.com/science/mapping. - ↑ Apostol, T. M. (1981).
*Mathematical Analysis*. Addison-Wesley. p. 35. ISBN 0-201-00288-4. - ↑ Stacho, Juraj (October 31, 2007). "Function, one-to-one, onto". http://www.cs.toronto.edu/~stacho/macm101-2.pdf.
- ↑ "Functions or Mapping | Learning Mapping | Function as a Special Kind of Relation". https://www.math-only-math.com/functions-or-mapping.html.
- ↑ Weisstein, Eric W.. "Map" (in en). http://mathworld.wolfram.com/Map.html.
- ↑ "Mapping, Mathematical | Encyclopedia.com". https://www.encyclopedia.com/education/news-wires-white-papers-and-books/mapping-mathematical.
- ↑ Lang, Serge (1971).
*Linear Algebra*(2nd ed.). Addison-Wesley. p. 83. ISBN 0-201-04211-8. - ↑ Simmons, H. (2011).
*An Introduction to Category Theory*. Cambridge University Press. p. 2. ISBN 978-1-139-50332-7. https://books.google.com/books?id=VOCQUC_uiWgC&pg=PA2.

### Works cited

- Halmos, Paul R. (1970).
*Naive Set Theory*. Springer-Verlag. ISBN 978-0-387-90092-6. https://books.google.com/books?id=x6cZBQ9qtgoC.

## External links

Original source: https://en.wikipedia.org/wiki/Map (mathematics).
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