contravariantly

Contravariantly is an adjective used in mathematics and computer science to describe a relationship between types or categories that is opposite to the direction of covariance. In the context of type theory, contravariance refers to a relationship where a type A is a subtype of B if B is a subtype of A. This is in contrast to covariance, where the subtype relationship is in the same direction.

In functional programming, contravariance is often encountered in the context of function types. For example, if a function f has a parameter of type A, and another function g has a parameter of type B, where B is a subtype of A, then g can be used wherever f is expected, because g can handle any input that f can. This is because the parameter type of g is contravariant in the function type.

In category theory, contravariance is a fundamental concept. It is used to define contravariant functors, which are functors that reverse the direction of the morphisms. Contravariant functors are used to model various mathematical structures, such as the dual of a category.

In summary, contravariance is a concept that describes a relationship between types or categories that is opposite to the direction of covariance. It is a fundamental concept in type theory, functional programming, and category theory.