In "mathematics, the term **mapping**, usually shortened to **map**, refers to either a "function, often with some sort of special structure, or a "morphism in category theory, which generalizes the idea of a function. There are also a few, less common uses in logic and graph theory.

## Maps as functions[edit]

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,^{[1]} use "function" only to refer to maps in which the "codomain is a set of numbers (i.e. a subset of the "fields **R** or **C**) and the term *mapping* for more general functions.

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

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.

## Maps as morphisms[edit]

In "category theory, "map" is often used as a synonym for "morphism or arrow, thus for something more general than a function.^{[2]}

## Other uses[edit]

### In logic[edit]

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[edit]

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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.^{[3]}

### In computer science[edit]

In the communities surrounding "programming languages that treat functions as "first-class citizens, a "map often refers to the "binary "higher-order function that takes a function *f* and a "list [*v*_{0}, *v*_{1}, ..., *v*_{n}] as "arguments and returns [*f*(*v*_{0}), *f*(*v*_{1}), ..., *f*(*v*_{n})], where *n* ≥ 0.

## See also[edit]

## References[edit]

**^** Lang, Serge (1971), *Linear Algebra* (2nd ed.), Addison-Wesley, p. 83
**^** Simmons, H. (2011), *An Introduction to Category Theory*, Cambridge University Press, p. 2, "ISBN "9781139503327
**^** Gross, Jonathan; Yellen, Jay (1998), *Graph Theory and its applications*, CRC Press, p. 294, "ISBN "0-8493-3982-0

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