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 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 have been given specific names. These include homomorphisms in 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 $f:\,X\to Y$ 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 $X$ of the morphism) and its codomain (the target $Y$ ). In the widely used definition of a function $f:X\to Y$ , $f$ is a subset of $X\times Y$ consisting of all the pairs $(x,f(x))$ for $x\in X$ . In this sense, the function does not capture the set $Y$ that is used as the codomain; only the range $f(X)$ is determined by the function.

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.
^ 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" (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.
^ 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.