# Map (mathematics)

In mathematics, the term mapping, sometimes shortened as map, is a general function between two mathematical objects or structures.[1][2] It can be thought of as the mathematical abstraction of the process of making a geographical map.[3]

Maps may either be functions or morphisms, though the terms share some overlap.[3][4] In the sense of a function, a map is often associated with some sort of structure, particularly a set constituting the codomain.[5][6] 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.[3] 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,[7][2][8] 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,[9] 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.[3] 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.

## Maps as morphisms

In category theory, "map" is often used as a synonym for "morphism" or "arrow", and thus is more general than "function".[10] For example, a morphism ${\displaystyle f:\,X\to Y}$ in a concrete category (i.e. a morphism which can be viewed as functions) carries with it the information of its domain (the source ${\displaystyle X}$ of the morphism) and its codomain (the target ${\displaystyle Y}$). In the widely used definition of a function ${\displaystyle f:X\to Y}$, ${\displaystyle f}$ is a subset of ${\displaystyle X\times Y}$ consisting of all the pairs ${\displaystyle (x,f(x))}$ for ${\displaystyle x\in X}$. In this sense, the function doesn't capture the information of which set ${\displaystyle Y}$ is used as the codomain; only the range ${\displaystyle f(X)}$ is determined by the function.

## Other uses

### In logic

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

### In graph theory

In graph theory, a map is a drawing of a graph on a surface without overlapping edges (an embedding). If the surface is a plane then a map is a planar graph, similar to a political map.[11]

### In computer science

In the communities surrounding programming languages that treat functions as first-class citizens, a map is often referred to as the binary higher-order function that takes a function f and a list [v0, v1, ..., vn] as arguments and returns [f(v0), f(v1), ..., f(vn)] (where n ≥ 0).

## References

1. "The Definitive Glossary of Higher Mathematical Jargon — Mapping". Math Vault. 2019-08-01. Retrieved 2019-12-06.
2. Weisstein, Eric W. "Map". mathworld.wolfram.com. Retrieved 2019-12-06.
3. "Mapping | mathematics". Encyclopedia Britannica. Retrieved 2019-12-06.
4. The words map, mapping, transformation, correspondence, and operator are often used synonymously. Halmos 1970, p. 30. In many authors, the term 'map' is with a more general meaning than 'function', which may be restricted to having domains of sets of numbers only.
5. Apostol, T. M. (1981). Mathematical Analysis. Addison-Wesley. p. 35. ISBN 0-201-00288-4.
6. Stacho, Juraj (October 31, 2007). "Function, one-to-one, onto" (PDF). cs.toronto.edu. Retrieved 2019-12-06.
7. "Functions or Mapping | Learning Mapping | Function as a Special Kind of Relation". Math Only Math. Retrieved 2019-12-06.
8. "Mapping, Mathematical | Encyclopedia.com". www.encyclopedia.com. Retrieved 2019-12-06.
9. Lang, Serge (1971). Linear Algebra (2nd ed.). Addison-Wesley. p. 83. ISBN 0-201-04211-8.
10. Simmons, H. (2011). An Introduction to Category Theory. Cambridge University Press. p. 2. ISBN 978-1-139-50332-7.
11. Gross, Jonathan; Yellen, Jay (1998). Graph Theory and its applications. CRC Press. p. 294. ISBN 0-8493-3982-0.