-<!-- λ Λ ∀ ≡ α β ρ ω Ω -->
+<!-- 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
of monads, followed by instances of monads from the philosophical and
linguistics literature.
-### Boxes: type expressions with one free type variable
+## Box types: 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
+<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 ≡ Int
- P ≡ α -> α
- P ≡ ∀α. α -> α
- P ≡ ∀α. α -> β
+ 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
+A *box type* will be a type expression that contains exactly one free
type variable. Some examples (using OCaml's type conventions):
α Maybe
for the type of a boxed Int.
+## Kleisli arrows
+
At the most general level, we'll talk about *Kleisli arrows*:
P -> <u>Q</u>
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. (This will
+become clearly shortly.)
+
+<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 (/m&ash;ptu/): (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*** ("composables") A MapNable box type is a *Monad* if there
+ is in addition an `mcompose` and a `join` such that `mid` is
+ 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 ≡ \fgx.f(gx)
+
+Id is a monad. Here is a demonstration that the laws hold:
+
+ mcompose mid k == (\fgx.f(gx)) (\p.p) k
+ ~~> \x.(\p.p)(kx)
+ ~~> \x.kx
+ ~~> k
+ mcompose k mid == (\fgx.f(gx)) k (\p.p)
+ ~~> \x.k((\p.p)x)
+ ~~> \x.kx
+ ~~> k
+ mcompose (mcompose j k) l == mcompose ((\fgx.f(gx)) j k) l
+ ~~> mcompose (\x.j(kx)) l
+ == (\fgx.f(gx)) (\x.j(kx)) l
+ ~~> \x.(\x.j(kx))(lx)
+ ~~> \x.j(k(lx))
+ mcompose j (mcompose k l) == mcompose j ((\fgx.f(gx)) k l)
+ ~~> mcompose j (\x.k(lx))
+ == (\fgx.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: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
+ mcompose f g a = concat (map f (g a))
+ = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
+ = [c | b <- g a, c <- f b]
+
+These three definitions are all equivalent. 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 f g 7 = [49, 50, 14, 15]
+
+It is easy to see that these definitions obey the monad laws (see exercises).
+