Covariance and contravariance of functors

In mathematics, a functor is a type of mapping between categories arising in category theory. Functors can be thought of as homomorphisms between categories. In the category of small categories, functors can be thought of more generally as morphisms.

Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areas within mathematics that category theory can make an abstraction of.

The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap, who used the term 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

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 ${\displaystyle F:C\rightarrow D}$ is a contravariant functor, they simply write ${\displaystyle F:C^{\mathrm {op} }\rightarrow D}$ (or sometimes ${\displaystyle F:C\rightarrow D^{\mathrm {op} }}$) and call it a functor.

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