Functors and applicatives are very common in the functional programming landscape. They arise naturally in various computational contexts - optionality ( Option ), possible failure ( Either[A, ?] ), effect suspension ( IO ) and so forth.

All of these computational contexts are functors; the practical meaning is that given a function A => B (forall A , B ), we can lift it to operate within the context - Option[A] => Option[B] . That is the essential meaning of having an instance of Functor : the ability to lift ordinary functions to fancy functions that operate in a special context.

We’ll use these data types throughout the post:

case class Username (name: String) case class UserId (id: Int) case class User (id: UserId, name: Username)

Here’s a short example to illustrate this, using cats and cats-effect :

import cats. effect . IO trait DB { def retrieveInt (): IO[Int] } val testDB = new DB { def retrieveInt (): IO[Int] = IO ( 0 ) }

Our DB interface only provides an IO[Int] output; it has no knowledge of our UserId data type. Luckily, since IO has a Functor instance, we can lift the UserId.apply function, and compose it with DB.retrieveInt :

import cats. implicits ._, cats. effect . implicits ._ // import cats.implicits._ // import cats.effect.implicits._ import cats. Functor // import cats.Functor val original = UserId (_) // original: Int => UserId = <function1> val lifted = Functor[IO]. lift (original) // lifted: cats.effect.IO[Int] => cats.effect.IO[UserId] = <function1> val retrievedUserId = lifted (testDB. retrieveInt ()) // retrievedUserId: cats.effect.IO[UserId] = IO$1562772628

The essence of being able to lift ordinary functions means that having a Functor instance allows us to utilize our ordinary functions in special contexts.

Let’s turn to another data type: a JSON decoder. We’ll use play-json in this post. The decoder type, Reads[A] , is morally equivalent to a function:

type Reads[A] = String => JsResult[A] type JsResult[A] = Either[JsError, A]

A JSON decoder receives a String and returns a possible error value. We can write a functor for his data type:

import play. api . libs . json ._ implicit val decoderFunctor = new Functor[Reads] { def map[A, B](fa: Reads[A])(f: A => B): Reads[B] = Reads { fa. reads (_). map (f) } }

You’ll notice that we just reused the fact that JsResult[A] has a Functor instance. Now, going back to our original example of String and UserId , we can parse a UserId given a Reads[String] and a String => UserId :

val idDecoder = Reads. of [Int]. map ( UserId (_)) // idDecoder: play.api.libs.json.Reads[UserId] = [email protected]

What happens when we need to combine two parsed values?

val nameDecoder = Reads. of [String]. map ( Username (_)) // nameDecoder: play.api.libs.json.Reads[Username] = [email protected]

We have a function (UserId, Username) => User , and two instances of Reads . So we need a function of the form:

def lift2 (f: (UserId, Username) => User): (JsDecoder[UserId], JsDecoder[Username]) => JsDecoder[User]

Meaning, a function that lifts a function in two arguments, to a decoder that combines two decoders. This notion is available with the Applicative typeclass. cats doesn’t provide the lift2 function directly, but we have map2 at our disposal, which is similar.

The Applicative instance for Reads is available in play-json-cats so we’ll elide it for brevity, but let’s see how map2 is used:

import cats. Applicative // import cats.Applicative import com. iravid . playjsoncats . implicits ._ // import com.iravid.playjsoncats.implicits._ val userDecoder = Applicative[Reads]. map2 (idDecoder, nameDecoder)( User (_, _)) // userDecoder: play.api.libs.json.Reads[User] = [email protected]

Let’s now consider the dual of Reads : Writes . This, too, is just a glorified function, of the form A => String . Very similar to Reads , only this time the arrow is reversed and the possibility for failure is elided. The arrow being reversed is significant: it means that we can’t compose the encoder with functions of A => B as the encoder must be supplied with an A . Thus, we can’t have a regular Functor for this.

But if we have a function that produces an A , we can pre-compose it with the encoder. Let’s try to develop an intuition for what we can do with this through an example with a simplified Writes definition:

type PseudoWrites[A] = A => String val stringWrites: PseudoWrites[Int] = _. toString val userIdToString: UserId => Int = _. id

Here, we have a Writes[Int] = Int => String , and a function UserId => Int . Composing these functions together, we get UserId => String - which is a Writes[UserId] ! So if we know how to encode a simple type A as a String, and we know how to encode a complex type B as an A , we can encode B as String for free.

This notion is encoded in the contravariant functor typeclass:

trait Contravariant[F[_]] { def contramap[A, B](fa: F[A])(f: B => A): F[B] } // specialized for Reads: def contramap[A, B](fa: Reads[A])(f: B => A): Reads[B]

If we flip the argument order and curry the arguments, we can get something similar to lift :

def contralift[A, B](f: B => A): Writes[A] => Writes[B]

Notice how contralift flips the arrows in the lifted function; this is the notion of contravariance in a functor. As a side note, the “regular” functor is actually called a covariant functor.

How is this useful, you ask? Well, libraries such as play-json usually come preloaded with encoders and decoders for the primitive types. Using the contravariant functor, we can succinctly derive an encoder for a wrapper type (again using play-json-cats):

val intWrites = Writes. of [Int] val userIdWrites: Writes[UserId] = intWrites. contramap (_. id ) val stringWrites = Writes. of [String] val usernameWrites: Writes[Username] = stringWrites. contramap (_. name )

Continuing along, can we generalize this to more than one Writes , as we did with Applicative ? Asking differently, if we know how to encode a UserId and a Username , do we know how to encode the User data type?

Let’s see what’s the signature we’re looking for:

def contramap2 (fa: JsEncoder[UserId], fb: JsEncoder[Username])(f: User => (UserId, Username)): JsEncoder[User]

The typeclass that describes this operation is called Divide , originating in Edward Kmett’s contravariant package; here’s how it looks like in Scala:

trait Divide[F[_]] extends Contravariant[F] { def divide[A, B, C](fb: F[A], fc: F[B])(f: C => (A, B)): F[C] }

The division wording comes from the fact that we are dividing a big problem (the notion of encoding a User ) into smaller problems that are solvable (encoding a UserId and a Username ).

This typeclass not available in cats yet, but we do have contramap available on the cartesian builders, which acts as divide ; we just need an instance of Cartesian for our Writes :

import cats. Cartesian // import cats.Cartesian implicit val cartesian: Cartesian[Writes] = new Cartesian[Writes] { def product[A, B](fa: Writes[A], fb: Writes[B]): Writes[(A, B)] = Writes { case (a, b) => Json. arr (fa. writes (a), fb. writes (b)) } } // cartesian: cats.Cartesian[play.api.libs.json.Writes] = [email protected] val userWrites: Writes[User] = (userIdWrites |@| usernameWrites). contramap (u => (u. id , u. name )) // userWrites: play.api.libs.json.Writes[User] = [email protected]

Do note that our choice of combining the output of the individual Writes instances as a JsArray is quite arbitrary. If we combine more than one Writes instances, our types line up, but we get nested arrays:

implicit val tripleWrites: Writes[(Int, Int, Int)] = (Writes. of [Int] |@| Writes. of [Int] |@| Writes. of [Int]). tupled // tripleWrites: play.api.libs.json.Writes[(Int, Int, Int)] = [email protected] writes (( 1 , 1 , 1 )). toString // res6: String = [1,[1,1]]

This typeclass instance is, in fact, unlawful. Instances of the Cartesian typeclass are required to uphold the associativity law - (a product b) product c must be equal to a product (b product c) . In our case, the two formulations would result in encoders which produce different JSON arrays:

implicit val intWriter = Writes. of [Int] // intWriter: play.api.libs.json.Writes[Int] = [email protected] implicit val sideA: Writes[(Int, Int, Int)] = (intWriter product (intWriter product intWriter)). contramap { case (a, b, c) => (a, (b, c)) } // sideA: play.api.libs.json.Writes[(Int, Int, Int)] = [email protected] implicit val sideB: Writes[(Int, Int, Int)] = ((intWriter product intWriter) product intWriter). contramap { case (a, b, c) => ((a, b), c) } // sideB: play.api.libs.json.Writes[(Int, Int, Int)] = [email protected] writes (( 1 , 1 , 1 )). toString // res7: String = [1,[1,1]] sideB. writes (( 1 , 1 , 1 )). toString // res8: String = [[1,1],1]

A more principled way of combining encoders is required. We could introspect the resulting JsValue in the composed encoder and flatten the resulting JSON array, but what would happen though if we would like to preserve nesting in products? This is not an easy problem. Sam Halliday has been tackling the same issues in his scalaz-deriving project. We’ll follow along to see what solutions he’ll discover :-)

We can summarize by saying that the laws can be successfully upheld when the target datatype for the typeclass is a lawful semigroup (e.g. (a |+| b) |+| c <-> a |+| (b |+| c) ). JsValue under the JSON array concatenation as formulated here is not.

Other useful, lawful instances often show up when dealing with type constructors that are isomorphic to functions that consume the data. For example:

Equal[A] , which is isomorphic to (A, A) => Boolean ;

, which is isomorphic to ; Order[A] , which is isomorphic to (A, A) => LT/EQ/GT ;

, which is isomorphic to ; Predicate[A] - which is just a fancy name for A => Boolean .

Admittedly, these aren’t as common as the covariant functors, but they’re quite useful for reducing boilerplate.

When starting out this post, I had hoped that I would figure out a good way to build up encoder instances for product types out of the instances of their fields, but we have not managed to achieve that lawfully. I intend on a following post to look into the relationship between encoders (contravariant functors), decoders (covariant functors) and profunctors - functors that are contravariant on one type parameter and covariant on the other. Should be interesting!