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