# Contra- and Co- variance

December 7, 2015

A small little note to remember about contra- and co- variants for the future, as well as a mnemonic to remember which is which.

The jist of it is that a contravariant can be thought of a wildcard of a class and all of it’s inherited types, while a covariant is a wildcard of a class and all of its subtypes. This leads to a couple of conclusions; the first being that, in a language like java, the contravariant has a terminal type - Object - while the covariant has a non-terminal type. In this way this makes it impossible for the type checker to reason about the type of a covariant – so it would be impossible to assign the type and all we can hope to do is read its value. Alternatively, with a contravariant we can reason about everything above the given type – making it assignable, however we could only use the greatest common type, Object in the case of java, in in assignment; making it virtually useless.

Also, keep in mind that this makes abstract or inherited types more powerful than the contra- or covariants as we have access to more information. Especially in java we will wind up using contra-/co-variant with the intent to describe items in a given collection but, in truth, the type is describing the collection itself and not any of its contents. Thus it is more powerful to use a more concrete type so that we can reason about the individual items as well as describe the collection we are interested in.

Finally, a quick little mnemonic device! As we were getting educated on contra-/co-variants, it struck me that usually developers are more interested in subtypes, or covariants. You can find them in iterables, for instance. Contravariants are “contradictory to what you would expect” – all of the inherited types.

On top of all of this, it looks like the concept of contra-/co- variance does have some relevance in haskell as these concepts can be seen in vectors (via multilinear algebra and tensor analysis) and functors! I’ll have to investgate this more later – but to keep your interest for a brief moment more: covariant functors (or cofunctors) are ordinary functors, while functors which invert the morphism are the contravariants. If you want to play around with a fun library, it looks like Profunctors are fun to use and return a bifunctor of the contravariant and the covariant!