Map (mathematics)

A map or mapping in mathematics is a type of function. The words come from making a geographical map, where the surface of the Earth is shown on paper.

The word map can be used for special kinds of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces. In category theory, a map can mean a morphism. The word transformation can also mean the same thing, but it often means a function that sends a set to itself. There are also some less common uses in logic and graph theory.

Maps as functions

Main article: Function (mathematics)

In mathematics, a map is just another way to say "function." Sometimes, people call special kinds of functions "maps," depending on what part of math they 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 special and important properties.

As morphisms

Main article: Morphism

In category theory, the word "map" often means "morphism" or "arrow." This term describes a special kind of function. A morphism shows how things change and tells us what starts the change and what ends it. This helps us understand relationships between different mathematical ideas more clearly.