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In mathematics, a map or mapping is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
in its general sense. These terms may have originated as from the process of making a
geographical map Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s. For example, a linear map is a homomorphism of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, while the term
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but '' transformation'' often refers to a function from a set to itself. There are also a few less common uses in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
and
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
.

# Maps as functions

In many branches of mathematics, the term ''map'' is used to mean a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a " continuous function" in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a "
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
" in linear algebra, etc. Some authors, such as Serge Lang, use "function" only to refer to maps in which the
codomain 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 in the notation . The term range is sometimes ambiguously used to refer to either th ...
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 mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
,
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
in analysis and
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. In the theory of dynamical systems, a map denotes an
evolution function In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
used to create discrete dynamical systems. A ''partial map'' is a ''
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
''. 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", which is a structure-respecting function and thus may imply more structure than "function" does. For example, a morphism $f:\, X \to Y$ in a
concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of t ...
(i.e. a morphism that can be viewed as a function) carries with it the information of its domain (the source $X$ of the morphism) and its codomain (the target $Y$). In the widely used definition of a function $f:X\to Y$, $f$ is a subset of $X\times Y$ consisting of all the pairs $\left(x,f\left(x\right)\right)$ for $x\in X$. In this sense, the function does not capture the set $Y$ that is used as the codomain; only the range $f\left(X\right)$ is determined by the function.

* * Arrow notation – e.g., $x\mapsto x+1$, also known as ''map'' * * * List of chaotic maps * Maplet arrow (↦) – commonly pronounced "maps to" * * *