Invariant Functors Unlocked

Afsal Thaj
2 min readJun 29, 2018

We tend to forget this quite often. An invariant functor or an exponential functor is, given A => B and B => A , it converts type A to typeB in the same context F[_]. We call this xmap .

Covariant Functor

That’s the famous Functor ! Covariant functor implements xmap by discarding the function B => A .

Contravariant Functor

As expected, it discards f: A => B and makes use of contramap to implement xmap

Example for Contravariant Functor


Example for covariant Functor

As expected, it is DecodeJson, where the type parameter in the type class comes at covariant position (method result)


If type parameters are at covariant position, that means the method return contains the type.

If type parameters are at contravariant position, that means the method parameters contain the type.

When is invariant functor?

We may have types at covariant (output) or contravariant (input) position. However, we may sometime deal with both covariance and contravariance in the same type class.

Let’s bring in EncodeJson and DecodeJson into one type class.

EncodeJson and DecodeJson

Functor but invariant

So an individual map or contramap to upcast (or downcast) an A to B in the context of F[_] is not possible if F has types both in covariant and contravariant positions. It means, F has to have an invariant functor for it!

Apply, Applicative to Divide, Divisible

We saw the contra-variant version of functor. We will see the contra-variant version of Apply , which is called Divide and how that can be more powerful if EncodeJson had an instance of Divide in the next blog. Along with it, we will discuss more on Apply and Applicative and how Divide and Divisible form the hierarchy!

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Afsal Thaj

A software engineer, traveller, fitness trainer, hiker, skydiver and I write at times.