Monads are just WTF in the category of huh?
April 02, 2016
Scala developers love to discuss Monads, their metaphors, and their many use cases.
burritos are just tacoids in the category of enchiladafunctors
— Richard Minerich (@rickasaurus) March 2, 2016
Parts of functional programming (FP) may be built on the mathematical principles from category theory, but at its core, FP is a style of programming. This post aims to prove you don’t need a PhD or be a Haskell programmer to understand these patterns. One disclaimer  the explanation does assume that you know some basics of Scala (like functions, 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/monadexamples
Monoid
A Monoid
is any type A
that carries the following properties:

Has some
append
method that can take two instances ofA
and produce another, singular, instance ofA
. 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 performingappend
withidentity
as one of the arguments returns the other argument.
In code:
trait Monoid[A] {
def append(a: A, b: A): A
def identity: A
}
Examples
Integer addition
object IntegerAddition extends Monoid[Int] {
def append(a: Int, b: Int): Int = a + b
def identity: Int = 0
}
Function composition
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
}
The extension here wouldn’t quite compile, but it’s a good example of using functions as types which will be important later. ^{0}
String concatenation
object StringConcat extends Monoid[String] {
def append(a: String, b: String): String = a + b
def identity: String = ""
}
List concatenation
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]
}
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 theidentity
function is a noop. 
Composition: calling
map
with a composition of functions is equivalent to composing separate calls tomap
on each function individually.
trait Functor[F[_]] {
def map[A, B](a: F[A])(fn: A => B): F[B]
}
If you have experience programming in 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 until the result is needed.
For all practical purposes, implementations of Functors
in Scala are also Endofunctors
(‘endo’ meaning “internal” or “within”) because it’s map
method goes from one category to itself  that category in Scala is Scala Types. ^{4}
Monads
The term monad is a bit vacuous if you are not a mathematician. An alternative term is computation builder. ^{5}
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
^{6} methods.
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 ^{7}).
Once you have the monad methods, you can work backwords to define the monoid and functor operations. Here you can see how we can define map
:
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. ^{0}
trait Monad[M[_]] extends Functor[M] /* with 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)
}
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 (think Future
and Option
). Building composable programs is extremely useful, it’s one of the things that functional programmers love the most about all their functionalprogrammingness. When we talk about composable architecture we often cite the benefits of modularity, statelessness, and managing 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 ^{9}
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 Monad
s are largely builtin to the Scala language whether you realize it or not. Using types like List
and Option
means using Monad
s, 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. The resources and references below are useful if you want to explore this more; I tried not to reference Haskell, but I do like this explanation using pictures.
Check out Part 2 where I 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
Slides from my Scala Days talk:
Notes and references:
 It’s really difficult to define a syntax in Scala that allows A and B to be any type. People have done it trying to copy something similar in Haskell https://stackoverflow.com/questions/7213676/forallinscala but that boilerplate isn’t necessary here to show the concepts.↩
 http://www.dictionary.com/browse/endo and http://hseeberger.github.io/blog/2010/11/25/introductiontocategorytheoryinscala/↩
 https://stackoverflow.com/questions/44965/whatisamonad↩
pure
is also known asreturn
orlift
.flatMap
is also known asbind
↩ https://byorgey.wordpress.com/2009/01/12/abstractionintuitionandthemonadtutorialfallacy/↩
 https://twitter.com/jessitron/status/713432439746654209↩