In mathematics, the term mapping, sometimes shortened as map, is a general function between two mathematical objects or structures. It can be thought of as the mathematical abstraction of the process of making a geographical map.
Maps may either be functions or morphisms, though the terms share some overlap. In the sense of a function, a map is often associated with some sort of structure, particularly a set constituting the codomain. Alternatively, a map may be described by a morphism in category theory, which generalizes the idea of a function. In some occasions, the term “transformation” can also be used interchangeably. There are also a few less common uses in logic and graph theory.
In many branches of mathematics, the term map is used to mean a function, 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, 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. For more, see Lie group, mapping class group and permutation group.
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See Poincaré map for more.
A partial map is a partial function, and a total map is a total function. Related terms such as domain, codomain, injective, continuous, etc. 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.