# Covariant and Contravariant Functors

In Category Theory, we have covariant and contravariant functors. Mathematically I know the difference between the two - I can picture that.

What I would like to know why these concepts are named covariant and contravariant. How does the meaning in English link to the mathematical concepts?

NB. I am a computer programmer not a maths student.

• I think contra-x and co-x are often used to denote concepts where contra-x reverses the order of something and co-x does not. In this case contra variant functors reverse the order of the composition of maps while co-x does not. See for example en.wikipedia.org/wiki/Covariance_and_contravariance – math635 Dec 30 '15 at 22:32
• what about the variance part of the words? what is varying? – M.K. Dec 30 '15 at 22:45
• @Mika'il With a covariant functor F, you have a map (A -> B) -> F A -> F B whereas with a contravariant functor G, the map goes (A -> B) -> G B -> G A. – gallais Dec 31 '15 at 13:08

What is varying is which object you're looking at. Thinking of a category as a generalization of a monoid, the image of a functor is like a set Wu an action. If you think that applying an action of an algebraic object changes something (e.g. rotation when the category is a group) then there is variation going on, and covariant functor said are those whose images vary in the same direction (are, say COordinated with) as the domain, and vice versa.

It's worth noting that in every other instance in category the prefix "co" indicates the result of reversing arrows, which could be confusing-perhaps a comodule should be called a contramodule!