The Free Monad with Multiple Algebras

In this post I will look at the machinery that makes it possible to use Free with more than one algebra at once: Coproducts and the Inject type class. This technique was first described in the paper Data types à la carte, and Cats and other libraries provide implementations. However we’re going to build our own implementation in this post, so we understand how it all works. I’m going to assume you understand Free at a basic level. If not, there are several good posts that introduce the concept.

The usual scenario with the Free monad is that of constructing DSLs: representing computations as immutable values and separating their execution into a separate structure known as an interpreter. Reifying the steps of a procedure or computation in order to represent them without actually executing them is a very common implementation strategy in functional programming. We would normally represent these steps, or actions, as an algebraic data type, without necessarily providing an implementation at the same time. Here is a very simple example of a set of operations represented in this way:

sealed trait Action[A] // parameterised on return type
final case class ReadData(port: Int) extends Action[String]
final case class TransformData(data: String) extends Action[String]
final case class WriteData(port: Int, data: String) extends Action[Unit]

Note how none of the types above have implementations, they are purely descriptive.

So, what good is it? It starts to look more useful when we imagine being able to sequence these things and compose them with other actions, using them as building blocks of bigger abstract ‘programs’. (The obvious example here would be sequencing ReadData and then TransformData and then WriteData into a combined action) As they are at the moment, though, we can’t compose these directly; right now we have defined no mechanism to allow it. We could go and write some methods on Action for this purpose, but there is a more general way: if this ‘instruction set’ was a monad, we could sequence instructions in a for comprehension, for example. And there is no need to make Action a monad ‘manually’, either: a practical way of achieving the same is by wrapping these types into the Free monad. (For a more general discussion of Free please see any of the references to this post. For now, we’ll just use it in a practical context to lead us to the next topic, multiple algebras.)

In Cats, the Free monad is already defined for us, and we can ‘lift’ a type constructor into the it with Free.liftF. Let us write a little bit of boilerplate to help us do that:

object ActionFree {
  def readData(port: Int): Free[Action, String] = Free.liftF(ReadData(port))
  def transformData(data: String): Free[Action, String] = Free.liftF(TransformData(data))
  def writeData(port: Int, data: String): Free[Action, Unit] = Free.liftF(WriteData(port, data))

The general pattern above is to turn an F[A] into a Free[F, A] so we have access to the monadic operations such as flatMap provided by Free.

Now we can do this:

import ActionFree._
val program = for {
  d <- readData(123)
  t <- transformData(d)
  _ <- writeData(789, t)
} yield ()
// program: Free[Action, Unit]

We can then take program and in turn compose it with other structures with the same signature.

(Note that in the convenience methods defined in ActionFree, we ensure that the resulting instances of Free are of type Free[Action, A] rather than allowing the compiler’s type inference to narrow them excessively and end up with a too-specific Free[ReadData, A] which wouldn’t compose with e.g. Free[WriteData, A].)

Multiple DSLs

The interesting part is yet to come. One of the main attractions of going down this avenue is the possibility of mixing ‘instruction sets’ and still getting them to compose.

Back to our example, we want to do more with data besides reading, writing and ‘transforming’ it. We want to add the following operations:

final case class EncryptData(key: String)
final case class DecryptData(key: String)

But let’s assume that we can’t just make them subclasses of Action, because Action is a sealed trait in a library over which we have no control (or for other good conceptual or architecture reasons). Whatever the reason, imagine that this is how we end up writing them:

sealed trait AdvancedAction[A]
final case class EncryptData(key: String) extends AdvancedAction[String]
final case class DecryptData(key: String) extends AdvancedAction[String]

Action and AdvancedAction are separate data types. They have no meaningful common supertype. How can we group them so we can build programs from commands of each type?

(Let’s remind ourselves that we’re not concerned with how any of these combined operations will be executed in practice. We simply want to find a suitable representation for them. Executing will be the job of the interpreter of our DSL.)

If we wanted to make use of operations from both Action[A] and AdvancedAction[A], Cats offers a Coproduct type we can use, which is roughly an Either but for higher kinded types. The idea is to use this to ‘merge’ the two algebras into a common type.

type ActionOrAdvanced[A] = Coproduct[Action, AdvancedAction, A]

We would like to use the newly-defined ActionOrAdvanced in place of either Action alone or AdvancedAction alone. At least in principle this should work and give us a way to do what we want. We would end up with Free instances of type Free[ActionOrAdvanced, A].

But how would it work in practice?

If you’ve been reading other posts on this subject, this is the point where, usually, the word ‘inject’ starts to appear, requiring an act of faith on the part of any readers who haven’t encountered it before, and quite possibly at the same point their ability to follow what’s going on may begin to waver. So, let’s look at Coproduct and Inject in more detail to find out how they work, as they rely on a very interesting type-level mechanism.

Coproduct Basics

Before getting too far ahead, let’s have a look at how Coproduct works, to get familiar with it. Let’s take this example:

type MyCoproduct[A] = Coproduct[List, Option, A]

A reminder that the above is akin to saying Either[List[A], Option[A]] for any A.

Now, assuming I have a List, how do I ‘put it’ inside a coproduct? Like this:

val c1: MyCoproduct[Int] = Coproduct.leftc(List(1,2,3))
// c1 = Coproduct(Left(List(1, 2, 3)))

Similarly if I have an Option:

val c2: MyCoproduct[Int] = Coproduct.rightc(Some(5))
// c2 = Coproduct(Right(Some(5)))

So, as per the original definition of MyCoproduct, List is left, Option is right, and everything makes sense. If I try it any other way, it predictably fails:

// trying to put an Option on the left
val c3: MyCoproduct[Int] = Coproduct.leftc(Some(7))
// Error: type mismatch
// found   : Some[Int]
// required: List[Int]

Let’s apply this to our DSL example. One possibility would be to directly use leftc and rightc in the convenience methods that create the Free instances we need:

object NaiveActionFree {
  def readData(port: Int): Free[ActionOrAdvanced, String] = Free.liftF[ActionOrAdvanced, String](Coproduct.leftc(ReadData(port)))

We would then do a similar thing for the case classes of AdvancedAction Note, though, that in so doing we are tied to the particular coproduct we are choosing to use at the moment: first of all, we mention it by name (in the type arguments to liftF) and we rely on the knowledge that in it ReadData is on the left side (otherwise it wouldn’t compile). This in turn means that if we want to use another type of coproduct, i.e. another mix of DSLs (it’s easy to imagine wanting to add more instructions later!), the code snippet above (and all of the convenience methods thus defined) will have to be scrapped and rewritten. Nobody likes doing that, so is there a more automatic solution?


Let’s envision a typeclass that, given a type and a coproduct, ‘knows’ how to correctly lift it to the coproduct level. Introducing Inject, defined thusly:

trait Inject[F[_], G[_]] {
  def inj[A](fa: F[A]): G[A]

(N.B. The code above is a simplification of the actual library code available in cats.)

Inject is a type class that allows us to embed one algebra within another. An instance of Inject[Action, ActionOrAdvanced], for example, allows us to ‘lift’ instances of Action to type ActionOrAdvanced.

(As an aside, you may notice that Inject has the same type structure as a natural transformation. This is no coincidence: in a sense, an instance Inject[F, G] allows us to ‘interpret’ instances of F as G. However, the similarity is effectively academic. In practice we use Inject to embed algebras in coproducts and ~> to interpret them to meaningful results.)

Is it possible to come up with a way to automatically derive an instance of Inject so that stuffing our Coproduct types happened by magic?

Left Hand

Let’s start by examining the case where F is our DSL and G is a coproduct with our desired type F on the left-hand side. That is, G[A] is Coproduct[F,X,A], that is to say, a coproduct with our desired type on the left hand side. Then, writing an instance for this case is as simple as calling leftc as we did in the examples earlier:

implicit def injectCoproductLeft[F[_], X[_]]: Inject[F, Coproduct[F, X, ?]] =
  new Inject[F, Coproduct[F, X, ?]] {
    def inj[A](fa: F[A]): Coproduct[F, X, A] = Coproduct.leftc(fa)

The above will work for any X and so the left-hand problem is solved.

Right Hand

We could do the same with the right-hand side of the coproduct and end up with enough tools in our box to automatically (i.e. by implicit provision of Inject instances) handle simple two-way coproducts appropriately. But we can do better. If we wanted ‘three-way coproducts’ or more, it is possible to define them by nesting them:

Coproduct[F1, Coproduct[F2, F3, ?], ?] // 3-way
Coproduct[F1, Coproduct[F2, Coproduct[F3, F4, ?], ?], ?] // 4-way
// ...

And we can still derive automatically Inject instances that work with these extended coproducts. As long as the ‘nested coproduct’ is in the right-hand position, at least. This is because we define the Inject instance for the right-hand side like this:

implicit def injectCoproductRight[F[_], R[_], X[_]](implicit I: Inject[F, R]): Inject[F, Coproduct[X, R, ?]] =
  new Inject[F, Coproduct[X, R, ?]] {
    def inj[A](fa: F[A]): Coproduct[X, R, A] = Coproduct.rightc(I.inj(fa))

Note that, unlike the left-hand case earlier, we don’t try to match the case where F itself is in the right-hand position. Rather, we identify the right-hand as another type constructor (R), about which we know nothing, except that we demand, as an implicit argument, proof that we are able to inject F into it. Which we then use to do exactly that. So, if this additional implicit I: Inject[F, R] can be found, we are done and the right-hand case also ties up. It boils down to how we get that proof, i.e. that additional Inject[F, R] instance we require.

  • Case 1: R is another coproduct with F on its left-hand side. We know how to handle this case: we’ve written it earlier (injectCoproductLeft).
  • Case 2: R is another coproduct with F on its right-hand side. In this case injectCoproductRight will be called recursively again and we’ll be back at this point!
  • Case 3: R is actually F itself. in other words, the type that we were trying to inject is the same type as what’s in the right-hand side of the coproduct. In this case we’re looking for an Inject[F, F], which will be trivial to write:
    implicit def injectReflexive[F[_]]: Inject[F, F] =
    new Inject[F, F] {
      def inj[A](fa: F[A]): F[A] = fa
  • And what if there is a Case 4, which is to say, R is something else entirely? Then the implicit resolution will fail, as is reasonable to expect, as we can’t put something into a coproduct that doesn’t have it as its options!

With these three implicit instances of Inject (the left, right and reflexive) we are able to automatically derive mechanisms to inject any desired type into any coproduct that contains it. Don’t worry though, all of these implicits are already in Cats and we won’t actually have to write them ourselves.

Finishing up

Back to our multiple DSLs. We have seen how to lift a single type constructor to the Free monad:

def readData(port: Int) = Free.liftF[Action, String](ReadData(port))

What if instead we had a coproduct Cop[A], which is a coproduct of Action and other algebras? We don’t need to know the exact structure of Cop. We can simply call:

def readData(port: Int): Free[Cop, String] = Free.inject[Action, Cop](ReadData(port))

Free.inject[F, Cop] is simply shorthand for:

  1. implicitly resolving an instance of Inject[F, Cop],
  2. using it to inject F[A] into Cop[A], and finally
  3. lifting Cop[A] into the Free monad.

So, let’s bring our mind back to those convenience methods we defined at the start, to lift those Actions into Free:

object ActionFree {
  def readData(port: Int): Free[Action, Int] = Free.liftF(ReadData(port))
  def transformData(data: String): Free[Action, Int] = Free.liftF(TransformData(data))
  def writeData(port: Int, data: String): Free[Action, Unit] = Free.liftF(WriteData(port, data))

Let’s generalise them with what we have learnt. We’ll no longer have a static object but a class with a type parameter:

class ActionFree[C[_]](implicit inject: Inject[Action, C]) {
  def readData(port: Int): Free[C, String] = Free.inject[Action, C](ReadData(port))
  def transformData(data: String): Free[C, String] = Free.inject[Action, C](TransformData(data))
  def writeData(port: Int, data: String): Free[C, Unit] = Free.inject[Action, C](WriteData(port, data))

The C type parameter is the coproduct definition we use, and we can change it easily if we need to. We could swap the terms, from Coproduct[Action, AdvancedAction, ?] to Coproduct[AdvancedAction, Action, ?] or we could add more, like Coproduct[Action, Coproduct[AdvancedAction, AdminAction, ?], ?], and the implicit resolution of inject would still find the correct Inject instance to deal with it.


In this post we discussed the implementation of Free in Cats. We went into a lot of depth about the implementation of Coproduct and Inject, and their application to mixing free algebras. I found this detail useful when trying to understand Free. I hope you found it useful as well.


On Free:

On injector classes:

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