+### Boxes: type expressions with one free type variable
+
+Recall that we've been using lower-case Greek letters
+<code>α, β, γ, ...</code> 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
+
+<u>Int</u>
+
+for the type of a boxed Int.
+
+At the most general level, we'll talk about *Kleisli arrows*:
+
+P -> <u>Q</u>
+
+A Kleisli arrow is the type of a function from objects of type P to
+objects of type box Q, for some choice of type expressions P and Q.
+For instance, the following are arrows:
+
+Int -> <u>Bool</u>
+
+Int List -> <u>Int List</u>
+
+Note that the left-hand schema can itself be a boxed type. That is,
+if `α List` is our box type, we can write the second arrow as
+
+<u>Int</u> -> <u><u>Int</u></u>
+
+We'll need a number of schematic functions to help us maneuver in the presence
+of box types. We will want to define a different instance of each of
+these for whichever box type we're dealing with:
+
+<code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
+
+<code>map (/maep/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
+
+<code>map2 (/maep/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
+
+<code>mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
+
+<code>mcompose (aka <=<): (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
+
+<code>mbind (aka >>=): ( <u>Q</u>) -> (Q -> <u>R</u>) -> ( <u>R</u>)</code>
+
+<code>mflipcompose (aka >=>): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
+
+<code>mflipbind (aka =<<) ( <u>Q</u>) -> (Q -> <u>R</u>) -> ( <u>R</u>)</code>
+
+<code>mjoin: <u><u>P</u></u> -> <u>P</u></code>
+
+Note that `mcompose` and `mbind` are interdefinable: <code>u >=> k ≡ \a. (ja >>= k)</code>.
+
+In most cases of interest, the specific instances of these types will
+provide certain useful guarantees.
+
+* ***Mappable*** ("functors") At the most general level, some box types are *Mappable*
+if there is a `map` function defined for that boxt type with the type given above.
+
+* ***MapNable*** ("applicatives") A Mappable box type is *MapNable*
+ if there are in addition `map2`, `mid`, and `mapply`.
+
+* ***Monad*** ("composable") A MapNable box type is a *Monad* if
+ there is in addition a `mcompose` and `join`. In addition, in
+ order to qualify as a monad, `mid` must be a left and right
+ identity for mcompose, and mcompose must be associative. That
+ is, the following "laws" must hold:
+
+ mcompose mid k = k
+ mcompose k mid = k
+ mcompose (mcompose j k) l = mcompose j (mcompose k l)
+
+To take a trivial example (but still useful, as we will see), consider
+the identity box type Id: `α -> α`. In terms of the box analogy, the
+Identity box type is an invisible box. With the following definitions
+
+ mid ≡ \p.p
+ mcompose ≡ \f\g\x.f(gx)
+
+Id is a monad. Here is a demonstration that the laws hold:
+
+ mcompose mid k == (\f\g\x.f(gx)) (\p.p) k
+ ~~> \x.(\p.p)(kx)
+ ~~> \x.kx
+ ~~> k
+ mcompose k mid == (\f\g\x.f(gx)) k (\p.p)
+ ~~> \x.k((\p.p)x)
+ ~~> \x.kx
+ ~~> k
+ mcompose (mcompose j k) l == mcompose ((\f\g\x.f(gx)) j k) l
+ ~~> mcompose (\x.j(kx)) l
+ == (\f\g\x.f(gx)) (\x.j(kx)) l
+ ~~> \x.(\x.j(kx))(lx)
+ ~~> \x.j(k(lx))
+ mcompose j (mcompose k l) == mcompose j ((\f\g\x.f(gx)) k l)
+ ~~> mcompose j (\x.k(lx))
+ == (\f\g\x.f(gx)) j (\x.k(lx))
+ ~~> \x.j((\x.k(lx)) x)
+ ~~> \x.j(k(lx))
+
+Id is the favorite monad of mimes everywhere.
+
+To take a slightly less trivial example, consider the box type `α
+List`, with the following operations:
+
+ mcompose f g p = [r | q <- g p, r <- f q]
+
+In words, if g maps a P to a list of Qs, and f maps a Q to a list of
+Rs, then mcompose f g maps a P to a list of Rs by first feeding the P
+to g, then feeding each of the Qs delivered by g to f. For example,
+
+ let f q = [q, q+1] in
+ let g p = [p*p, p+p] in
+ mcompose f g 7 = [49, 50, 14, 15]
+
+It is easy to see that these definitions obey the monad laws (see exercises).
+