Map (mathematics)

In mathematics, a map or mapping is a function in its general sense. These terms may have originated from the process of making a geographical map: mapping the Earth surface to a sheet of paper.

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 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 and graph theory.

Maps as functions

Main article: Function (mathematics)

In mathematics, a map is another way to talk about a function. Sometimes, special kinds of functions are called maps, depending on what area of math you are studying. For example, in topology, a map might be a continuous function, and in linear algebra, it might be a linear transformation.

Some types of maps have special names, like homomorphisms in algebra or isometries in geometry. These names help mathematicians talk about functions that have particular important properties.

As morphisms

Main article: Morphism

In category theory, the word "map" is often used to mean "morphism" or "arrow." This term describes a special kind of function that keeps certain rules in mind. For example, a morphism shows not just how things change, but also tells us what starts the change and what ends it. This helps us understand relationships between different mathematical ideas more clearly.