Map (mathematics)

For other uses, see Map (disambiguation).

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

Main article: Function (mathematics)

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.

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 ƒ and a list [v0,v1,...,vn] as arguments and returns [ƒ(v0),ƒ(v1),...,ƒ(vn)], s.t. n ≥ 0.

Maps as morphisms

Main article: Morphism

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

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

An example of a map 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.^[3]

References

↑ 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

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory	Set Empty set Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Range Map Function Relation Ordered pair

Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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