An Algebraic Data Type's Monad

I watched Dr. Emily Riehl’s Compose Conf talkA Categorical View of Computational Effects

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:

\[ A \rightarrow T(A) \]

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 \(\text{List}\): 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 \(T\):

\[ A \rightarrow T(B) = A \leadsto B \]

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 \(A + \bot\):

\[ A \rightarrow A + \{\bot\} \]

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. Riehl’s notation, we could denote f mathematically as \(f : A \leadsto A\)