Monads are confusing. Let us help

Scala developers love to discuss Monads, their metaphors, and their many use cases. We joke that Monads are “just Monoids in the category of Endofunctors,” but what does that really mean?

Parts of functional programming (FP) may be built on the mathematical principles from category theory, but you don’t need a PhD – or to be a Haskell programmer – to understand these patterns. One disclaimer - the explanation does assume that you know some basics of Scala (like types, polymorphism, and traits).

We’ll start by defining some of the most referenced components in order to define Monads. We also explore why Monadic design is useful, why it’s dangerous, and discuss some tradeoffs of using these types.

Code examples used can be found here: https://github.com/robinske/monad-examples

Monoid

A Monoid is any type A that carries the following properties:

• Has some append method that can take two instances of A and produce another, singular, instance of A. This method is associative; if you use it to append multiple values together, the grouping of values doesn’t matter.

• Has some identity element such that performing append with identity as one of the arguments returns the other argument.

In code:

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trait Monoid[A] {
  def append(a: A, b: A): A
  def identity: A

  /*
   * Such that:
   * Associativity property: `append(a, append(b,c)) == append(append(a,b),c)`
   * Identity property: `append(a, identity) == append(identity, a) == a`
   */
}

Examples

Integer addition

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object IntegerAddition extends Monoid[Int] {
  def append(a: Int, b: Int): Int = a + b
  def identity: Int = 0
  // Associativity: 2 + (3 + 4) == (2 + 3) + 4
  // Identity: (1 + 0) == (0 + 1) == 1
}

Function composition

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object FunctionComposition /* extends Monoid[_ => _] */ {
  def append[A, B, C](a: A => B, b: B => C): A => C = a.andThen(b)
  def identity[A]: A => A = a => a
  // Associativity: (f.andThen(g.andThen(h)))(x) == ((f.andThen(g)).andThen(h))(x)
  // Identity: identitity(f(x)) == f(identity(x)) == f(x)
}

The extension here wouldn’t quite compile, but it’s a good example of using functions as types which will be important later.

String concatenation

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object StringConcat extends Monoid[String] {
  def append(a: String, b: String): String = a + b
  def identity: String = ""
  // Associativity: "foo" + ("bar" + "baz") == ("foo" + "bar") + "baz"
  // Identity: ("foo" + "") == ("" + "foo") == "foo"
}

List concatenation

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class ListConcat[A] extends Monoid[List[A]] {
  def append(a: List[A], b: List[A]): List[A] = a ++ b
  def identity: List[A] = List.empty[A]
  // Associativity: List(1,2,3) ++ (List(4,5,6) ++ List(7,8,9)) == (List(1,2,3) ++ List(4,5,6)) ++ List(7,8,9)
  // Identity: (List(1,2,3) ++ Nil) == (Nil ++ List(1,2,3)) == List(1,2,3)
}

Monoids are a useful construct in every language. While not always explicitly defined as this type, the four examples above are ubiquitous language features.

Functors

A Functor is concept that applies to a family of types F with a single generic type parameter. For example, List is a type family, because List[A] is a distinct type for each distinct type A. A type family F is a Functor if it can define a map method with the following properties:

• Identity: calling map with the identity function is a no-op.

• Composition: calling map with a composition of functions is equivalent to composing separate calls to map on each function individually.

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trait Functor[F[_]] {
  def map[A, B](a: F[A])(fn: A => B): F[B]
  // Identity: map(fa)(identity) == fa
  // Composition: map(fa)(f andThen g) == map(map(fa)(f))(g)
}

If you write Scala, you’ll know this encompasses a lot of types. map is a useful method because it allows you to chain operations together (composition). Since mapped functions don’t need to be executed immediately, you can also defer evaluation and side effects until the result is needed.

Implementations of Functors in Scala are also Endofunctors (‘endo’ meaning “internal” or “within”) because the input and output parameters are always Scala Types. ¹

Monads

The term monad is a bit vacuous if you are not a mathematician. An alternative term is computation builder. ²

We’ve established that we don’t have to be mathematicians to do this, so let’s take a look at the practical implementation details.

A Monad is a type that has implemented the pure and flatMap ³ methods.

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trait Monad[M[_]] {
  def pure[A](a: A): M[A]
  def flatMap[A, B](a: M[A])(fn: A => M[B]): M[B]
}

pure is a method that takes any type and creates the “computation builder”, wrapping it in the container type or “context”. (Why some people have described monads as burritos ⁴).

With these two methods, you can define map:

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trait Monad[M[_]] {
  def pure[A](a: A): M[A]
  def flatMap[A, B](a: M[A])(fn: A => M[B]): M[B]

  def map[A, B](a: M[A])(fn: A => B): M[B] = {
    flatMap(a){ b: A => pure(fn(b)) }
  }
}

You can also define the Monoid operations append and identity by using flatMap and pure. Above, we defined the trait Monoid with a generic type. Here, that type is a function: A => M[B] where A and B are not fixed and can be any type. ⁵

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trait Monad[M[_]] { // extends Monoid[_ => M[_]]
  def pure[A](a: A): M[A]
  def flatMap[A, B](a: M[A])(fn: A => M[B]): M[B]

  def map[A, B](a: M[A])(fn: A => B): M[B] = {
    flatMap(a){ b: A => pure(fn(b)) }
  }

  def append[A, B, C](f1: A => M[B], f2: B => M[C]): A => M[C] = { a: A =>
    val bs: M[B] = f1(a)
    val cs: M[C] = flatMap(bs) { b: B => f2(b) }
    cs
  }

  def identity[A]: A => M[A] = a => pure(a)

  // And the laws apply!
  // Associativity: flatMap(pure(a), x => flatMap(f(x), g)) == flatMap(flatMap(pure(a), f), g)
  // Identity: flatMap(pure(a), f) == flatMap(f(x), pure) == f(x)
}

Monoids already allow composition of functions as we saw above. Monads are useful because they allow you to compose functions for values in a context (M[_]), something that we see all over our programs (like Lists and Options). Building composable programs is extremely useful, it’s one of the things that functional programmers love the most about all their functional-programming-ness. When we talk about composable architecture we often cite the benefits of modularity, statelessness, and deferring side effects:

A functional style pushes side effects to the edges: “gather information, make decisions, act.” A good plan in most life situations too. - Jessica Kerr ⁶

Building systems in this manner can provide greater maintainability and code reuse, and increase understanding of complex logic by breaking it into smaller, simpler pieces. What’s better is that the benefits of Monads are largely builtin to the Scala language whether you realize it or not. Using types like List and Option means using Monads, without having to do any of the tedious setup or method definitions.

Takeaways

These are complicated concepts, but hopefully ( by applying the principles of FP! ) we have broken it into smaller, digestable explanations. If anything is still confusing, leave me a note in the comments. The resources and references below are useful if you want to explore this more; I promised not to reference Haskell, but I especially like this explanation using pictures.

Stay tuned for Part 2 where I’ll dive into the details of the Free Monad.

Sound interesting? Want to convince me of your metaphor? I’m talking more about this at Scala Days in May - or send me a note on Twitter @kelleyrobinson

Notes and references: