-<table border=2><tr><tl>x</tl></tr></table>
-
-
+<!-- λ Λ ∀ ≡ α β γ ρ ω Ω -->
+<!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
Monads
======
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.
+prefers to say that monads are (Just) 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
+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.
-<table border=2>x</table>
+## Box types: type expressions with one free type variable
+
+Recall that we've been using lower-case Greek letters
+<code>α, β, γ, ...</code> as variables over 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_1 ≡ Int
+ P_2 ≡ α -> α
+ P_3 ≡ ∀α. α -> α
+ P_4 ≡ ∀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.
+
+## Kleisli arrows
+
+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 classes of 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 (/maeptu/): (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>
+
+The managerie isn't quite as bewildering as you might suppose. For
+one thing, `mcompose` and `mbind` are interdefinable: <code>u >=> k ≡
+\a. (ja >>= k)</code>.
+
+In most cases of interest, instances of these types will provide
+certain useful guarantees.
+
+* ***Mappable*** ("functors") At the most general level, box types are *Mappable*
+if there is a `map` function defined for that box type with the type given above.
+
+* ***MapNable*** ("applicatives") A Mappable box type is *MapNable*
+ if there are in addition `map2`, `mid`, and `mapply`. (With
+ `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.)
+
+* ***Monad*** ("composable") A MapNable box type is a *Monad* if there
+ is in addition an `mcompose` and a `join` such that `mid` is be
+ a left and right identity for `mcompose`, and `mcompose` is
+ 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 (but, as we will see, still useful) example,
+consider the identity box type Id: `α -> α`. So if α is type Bool,
+then a boxed α is ... a Bool. In terms of the box analogy, the
+Identity box type is a completly 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 (and even more useful) example,
+consider the box type `α List`, with the following operations:
+
+ mid: α -> [α]
+ mid a = [a]
+
+ mcompose-crossy: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
+ mcompose-crossy f g a = [c | b <- g a, c <- f b]
+
+In words, `mcompose f g a` feeds the a (which has type α) to g, which
+returns a list of βs; each β in that list is fed to f, which returns a
+list of γs. The final result is the concatenation of those lists of γs.
+For example,
+
+ let f b = [b, b+1] in
+ let g a = [a*a, a+a] in
+ mcompose-crossy f g 7 = [49, 50, 14, 15]
+
+It is easy to see that these definitions obey the monad laws (see exercises).
+
+There can be multiple monads for any given box type. For isntance,
+using the same box type and the same mid, we can define
+
+ mcompose-zippy f g a = match (f,g) with
+ ([],_) -> []
+ (_,[]) -> []
+ (f:ftail, g:gtail) -> f(ga) && mcompoze-zippy ftail gtail a
+
+