Inmathematics, amap ormapping is afunction in its general sense.[1] These terms may have originated as from the process of making ageographical map:mapping the Earth surface to a sheet of paper.[2]
The termmap may be used to distinguish some special types of functions, such ashomomorphisms. For example, alinear map is a homomorphism ofvector spaces, while the termlinear function may have this meaning or it may mean alinear polynomial.[3][4] Incategory theory, a map may refer to amorphism.[2] The termtransformation can be used interchangeably,[2] buttransformation often refers to a function from a set to itself. There are also a few less common uses inlogic andgraph theory.
In many branches of mathematics, the termmap is used to mean afunction,[5][6][7] sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" intopology, a "linear transformation" inlinear algebra, etc.
Some authors, such asSerge Lang,[8] use "function" only to refer to maps in which thecodomain is a set of numbers (i.e. a subset ofR orC), and reserve the termmapping for more general functions.
Maps of certain kinds have been given specific names. These includehomomorphisms inalgebra,isometries ingeometry,operators inanalysis andrepresentations ingroup theory.[2]
In the theory ofdynamical systems, a map denotes anevolution function used to creatediscrete dynamical systems.
Apartial map is apartial function. Related terminology such asdomain,codomain,injective, andcontinuous 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.
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 aconcrete 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.
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