I watched Dr. Emily Reihl’s Compose Conf talk1 last night and a new intuition emerged. The purpose of the talk was to explicate the categorical notion of a monad, and while watching it, I was struck with the realization of how a monad arises from an algebraic data type.

Prereqs

In her talk, she first discusses $T$ as a computation, and defines a $T$ as a monad: something that can take an $A$ and lift it to a $T(A)$, like this:

NB: There is another operation that comes with a monad, bind, but we’ll skip that for now.

The canonical example used through the first half of the talk is a $List$: A function from $A \rightarrow T(A)$ could be a function from an $A$ to a $List$ of $A$’s where $T$ is the computation which constructs a list of $A$’s. That is, $A \rightarrow T(A)$ is simply a more general version of $A \rightarrow List(A)$.

She goes on to use the notation of $\leadsto$ to denote a “program” which contains one of these lift operations, but with $\leadsto$, we elide the $T$:

This notation is meant to denote a “weak” map between $A$ and $B$, in that it’s not a complete $A \rightarrow B$, due to the fact that it requires the computation $T$. This lift from $A$ to $T$ of $B$ is called a Kleisli arrow.

An ADT Monad

Later in the talk, she defines a function from $A$ to $A + \bot$:

This should look familiar — it contains a $+$ after all! It’s an algebraic data type (ADT) — a sum type to be specfic. It can give us either an $A$ or ${\bot}$; $\bot$ means “bottom” or false in this context. It would look something like this in Haskell:

data Maybe a = Nothing | Just a

f :: a -> Maybe a

And as we know, Maybe admits a monad where if we have an a, we apply our lift to it to get a Just a. In Haskell, this lift is called return, and made available in the Monad typeclass:

return :: Monad m => a -> m a

Which, if we squint, looks an awful lot like A -> T(A). For edification purposes, our definition of return for Maybe and the other requisite pieces of a Monad in Haskell are below.

instance Monad Maybe where
  return = Just
  (Just x) >>= f = f x
  Nothing >>= _ = Nothing

If you’re still squinting, you can start to see how:

  • Our ADT becomes $T$, the computation which can give us either our a, or Nothing.
  • Using Dr. Reihl’s notation, we could denote f mathematically as $f : A \leadsto A$

References

  1. A Categorical View of Computational Effects
  2. What does the $\leadsto$ denote?
  3. Kleisli Category - Take a look at definition 2.4 in the Kleisli morphisms section.