Map (mathematics)

For other uses, see Map (disambiguation).

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.

In graph theory, a map is a drawing of a graph on a surface without overlapping edges (a planar graph), similar to a political map.

Maps as functions

In many branches of mathematics, the term is used to mean a function 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.

In contrast, in category theory, "map" is often used as a synonym for morphism or arrow, thus for something more general than a function.

Some authors, such as Serge Lang, use "map" as a general term for an association of an element in the range with each element in the domain, and "function" only to refer to maps in which the range is a field.

Sets of maps of special kinds are the subjects of many important theories: see for instance Lie group, mapping class group, permutation group.

In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.

A partial map is a partial function, and a total map is a total function. Related terms like 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.

See also

• Bijection, injection and surjection
• Category theory
• Correspondence (mathematics)
• Homeomorphism
• Homomorphism
• List of chaotic maps
• Mapping class group
• Morphism
• Projection (mathematics)
• Topology