Map (mathematics)

[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.

[3][4] In category theory, a map may refer to a morphism.

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.

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.

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.

Related terminology such as domain, codomain, injective, and continuous can be applied equally to maps and functions, with the same meaning.

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.

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