Monads
======
The [[tradition in the functional programming
literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
introduce monads using a metaphor: monads are spacesuits, monads are
monsters, monads are burritos. We're part of the backlash that
prefers to say that monads are monads.
The closest we will come to metaphorical talk is to suggest that
monadic types place objects inside of *boxes*, and that monads wrap
and unwrap boxes to expose or enclose the objects inside of them. In
any case, the emphasis will be on starting with the abstract structure
of monads, followed by instances of monads from the philosophical and
linguistics literature.
### Boxes: type expressions with one free type variable
Recall that we've been using lower-case Greek letters
`α, β, γ, ...`

to represent types. We'll
use `P`, `Q`, `R`, and `S` as metavariables over type schemas, where a
type schema is a type expression that may or may not contain unbound
type variables. For instance, we might have
P ≡ Int
P ≡ α -> α
P ≡ ∀α. α -> α
P ≡ ∀α. α -> β
etc.
A box type will be a type expression that contains exactly one free
type variable. Some examples (using OCaml's type conventions):
α Maybe
α List
(α, P) Tree (assuming P contains no free type variables)
(α, α) Tree
The idea is that whatever type the free type variable α might be,
the boxed type will be a box that "contains" an object of type α.
For instance, if `α List` is our box type, and α is the basic type
Int, then in this context, `Int List` is the type of a boxed integer.
We'll often write box types as a box containing the value of the free
type variable. So if our box type is `α List`, and `α == Int`, we
would write