# Map (mathematics)

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In mathematics, a map is often used as a synonym for a function, [1] but may also refer to some generalizations. Originally, this was an abbreviation of mapping, which often refers to the action of applying a function to the elements of its domain. This terminology is not completely fixed, as these terms are generally not formally defined, and can be considered to be jargon. [2] [3] These terms may have originated as a generalization of the process of making a geographical map, which consists of mapping the Earth surface to a sheet of paper. [4]

## Contents

Maps may either be functions or morphisms , though the terms share some overlap. [4] The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning as well as another one. [5] [6] In category theory, a map may refer to a morphism, which is a generalization of the idea of a function. In some occasions, the term transformation can also be used interchangeably. [4] 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] [3] [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. [4]

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

A partial map is a partial function . Related terms such as domain , codomain, injective , and continuous 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.

## 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 does not 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).

## Related Research Articles

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: XY is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. The term one-to-one correspondence must not be confused with one-to-one function.

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape".

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

In mathematics, a surjective function is a function f that maps an element x to every element y; that is, for every y, there is an x such that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

In mathematics, an injective function is a function f that maps distinct elements to distinct elements; that is, f(x1) = f(x2) implies x1 = x2. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.

In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: XY. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.

In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In category theory, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

The cokernel of a linear mapping of vector spaces f : XY is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f.

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a.

In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another. A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

## References

1. The words map, mapping, transformation , correspondence, and operator are often used synonymously. Halmos 1970 , p. 30. Some authors use the term map with a more general meaning than function, which may be restricted to apply to numbers only.
2. "The Definitive Glossary of Higher Mathematical Jargon — Mapping". Math Vault. 2019-08-01. Retrieved 2019-12-06.
3. Weisstein, Eric W. "Map". mathworld.wolfram.com. Retrieved 2019-12-06.
4. "Mapping | mathematics". Encyclopedia Britannica. Retrieved 2019-12-06.
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.