Editing Map (mathematics)

{{Short description|Function, homomorphism, or morphism}}
{{Other uses|map (disambiguation)}}
[[File:Function_color_example_3.svg|thumb|A 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''' or '''mapping''' is a [[function (mathematics)|function]] in its general sense.<ref>The words ''map'', ''mapping'', ''correspondence'', and ''operator'' are often used synonymously. {{harvnb|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.</ref>  These terms may have originated as from the process of making a [[map|geographical map]]: ''mapping'' the Earth surface to a sheet of paper.<ref name=":1">{{Cite web|url=https://www.britannica.com/science/mapping|title=Mapping {{!}} mathematics|website=Encyclopedia Britannica|language=en|access-date=2019-12-06}}</ref>

The term ''map'' may be used to distinguish some special types of functions, such as [[homomorphism]]s. For example, a [[linear map]] is a homomorphism of [[vector space]]s, while the term [[linear function]] may have this meaning or it may mean a [[linear polynomial]].<ref>{{cite book |first=T. M. |last=Apostol |author-link=Tom M. Apostol |title=Mathematical Analysis |year=1981 |publisher=Addison-Wesley |isbn=0-201-00288-4 |page=35 }}</ref><ref>{{Cite web|url=http://www.cs.toronto.edu/~stacho/macm101-2.pdf|title=Function, one-to-one, onto|last=Stacho|first=Juraj|date=October 31, 2007|website=cs.toronto.edu|access-date=2019-12-06}}</ref> In [[category theory]], a map may refer to a [[morphism]].<ref name=":1" /> The term ''transformation'' can be used interchangeably,<ref name=":1" /> but ''[[transformation (function)|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==
{{Main article|Function (mathematics)}}

In many branches of mathematics, the term ''map'' is used to mean a [[Function (mathematics)|function]],<ref>{{Cite web|url=https://www.math-only-math.com/functions-or-mapping.html|title=Functions or Mapping {{!}} Learning Mapping {{!}} Function as a Special Kind of Relation|website=Math Only Math|access-date=2019-12-06}}</ref><ref name=":0">{{Cite web|url=http://mathworld.wolfram.com/Map.html|title=Map|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-06}}</ref><ref>{{Cite web|url=https://www.encyclopedia.com/education/news-wires-white-papers-and-books/mapping-mathematical|title=Mapping, Mathematical {{!}} Encyclopedia.com|website=www.encyclopedia.com|access-date=2019-12-06}}</ref> sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "[[continuous function]]" in [[topology]], a "[[Linear map|linear transformation]]" in [[linear algebra]], etc.

Some authors, such as [[Serge Lang]],<ref>{{cite book |first=Serge |last=Lang |title=Linear Algebra |edition=2nd |year=1971 |page=83 |publisher=Addison-Wesley |isbn=0-201-04211-8 }}</ref> use "function" only to refer to maps in which the [[codomain]] is a set of numbers (i.e. a subset of [[real numbers|'''R''']] or [[complex numbers|'''C''']]), and reserve the term ''mapping'' for more general functions.

Maps of certain kinds have been given specific names. These include [[homomorphism]]s in [[algebra]], [[isometries]] in [[geometry]], [[Operator (mathematics)|operators]] in [[Mathematical analysis|analysis]] and [[Group representation|representations]] in [[group theory]].<ref name=":1" />  

In the theory of [[dynamical system]]s, a map denotes an [[Discrete-time dynamical system|evolution function]] used to create [[Dynamical system#Maps|discrete dynamical systems]]. 

A ''partial map'' is a ''[[partial function]]''. Related terminology such as ''[[Domain of a function|domain]]'', ''[[codomain]]'', ''[[Injective function|injective]]'', and ''[[Continuous function|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==
{{Main article|Morphism}}

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.<ref>{{cite book |title=An Introduction to Category Theory |first=H. |last=Simmons |publisher=Cambridge University Press |year=2011 |isbn=978-1-139-50332-7 |page=2 |url=https://books.google.com/books?id=VOCQUC_uiWgC&pg=PA2 }}</ref> For example, a morphism <math>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>X</math> of the morphism) and its codomain (the target <math>Y</math>). In the widely used definition of a function <math>f:X\to Y</math>, <math>f</math> is a subset of <math>X\times Y</math> consisting of all the pairs <math>(x,f(x))</math> for <math>x\in X</math>. In this sense, the function does not capture the set <math>Y</math> that is used as the codomain; only the range <math>f(X)</math> is determined by the function.

==See also==
* {{annotated link|Apply|Apply function}}
* [[Function (mathematics)#Arrow notation|Arrow notation]] – e.g., <math>x\mapsto x+1</math>, also known as ''map''
* {{annotated link|Bijection, injection and surjection}}
* {{annotated link|Homeomorphism}}
* [[List of chaotic maps]]
* [[Maplet arrow|Maplet arrow (↦)]] – commonly pronounced "maps to"
* {{annotated link|Mapping class group}}
* {{annotated link|Permutation group}}
* {{annotated link|Regular map (algebraic geometry)}}

==References==
{{Reflist}}

===Works cited===
* {{cite book |last=Halmos |first=Paul R. |author-link=Paul Halmos |year=1970 |title=Naive Set Theory |publisher=Springer-Verlag |isbn=978-0-387-90092-6 |url=https://books.google.com/books?id=x6cZBQ9qtgoC}}

==External links==
{{Mathematical logic}}
{{authority control}}
[[Category:Functions and mappings| ]]
[[Category:Basic concepts in set theory]]