In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) 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 every area of mathematics where category theory is applied.

The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word.

Let C and D be categories. A functor F from C to D is a mapping that:

That is, functors must preserve identity morphisms and composition of morphisms.

There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor F from C to D as a mapping that

Variance of functor (composite)

Note that contravariant functors reverse the direction of composition.

Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the opposite category ${\displaystyle C^{\mathrm {op} }}$. Some authors prefer to write all expressions covariantly. That is, instead of saying ${\displaystyle F\colon C\to D}$ is a contravariant functor, they simply write ${\displaystyle F\colon C^{\mathrm {op} }\to D}$ (or sometimes ${\displaystyle F\colon C\to D^{\mathrm {op} }}$) and call it a functor.