lec07.txt

Natural transformations
------------------------

Some of the constructions we have seen so far within a category also
apply to CAT.

For example, CAT has an initial object, the empty category, and a
terminal object, the singleton category.  CAT has products, C x D, and
coproducts, C + D.  Does CAT have exponentials?

We want D^C to have as objects functors C -> D, such that we have a
functor

  Apply : D^C x C -> D
  Apply (F, X) = F X

The question is: what should the arrows of D^C be so that Apply can be
extended to arrows?

Let alpha : F -> G be an arrow in D^C, and f : X -> Y an arrow in C.
We have

  Apply (alpha, f) : Apply (F, X) -> Apply (G, Y)
<=>  { action of Apply on objects }
  Apply (alpha, f) : F X -> G Y

For any f : X -> Y, alpha should be such that we obtain an arrow F X
-> G Y.  It helps to summarize the types of the elements we have so
far:

  In C                    In D

   X                F X          G X
   |                 |            |
 f |             F f |            | G f
   v                 v            v
   Y                F Y          G Y

To construct the desired diagonal arrow from F X to G Y it would help
to first connect the two "edges" F f and G f to make a square (and
then use a suitable composition of "edges").  This suggests that alpha
: F -> G should fill in _any_ such square in D.

Therefore alpha should be a collection of arrows (in D) of type F X ->
G X, one for every object X (in C).

The collection cannot be arbitrary.  Above, we would obtain _two_
arrows of the required type F X -> G Y.  Which should we pick?  To
avoid having to make an arbitrary choice, we impose instead a
condition on alpha: the two ways should be equal!

Definition: C, D categories, F, G : C -> D functors.  A
_transformation_ alpha : F -> G is a mapping from the objects of C to
the arrows of D such that for all X : Obj C

    alpha_X : F X -> G X

(it is a standard convention to denote the arrow "alpha X" by
alpha_X).

The arrows alpha_X are sometimes referred to as the _components_ of
the transformation alpha.

A transformation alpha is _natural_ if for all f : X -> Y in C
we have

    alpha_Y . F f = G f . alpha_X


                             alpha_X
    X                  F X  ---------> G Y
    |                   |               |
   f|                F f|               |G f
    |                   |               |
    v                   v    alpha_Y    v
    Y                  F Y  ---------> G Y

Natural transformations compose, and D^C is indeed a category.

Remark: sometimes (for example, in Hinze's articles) the type of a
natural transformation is denoted by an arrow with a dot on the top:

    alpha : F --°--> G

I find this irritating, since it obscures the fact that they are the
arrows of categories of functors. (end of remark)

In functional programming, natural transformations correspond to
polymorphic functions.

Examples:
--------

- reverse, flatten, length, append

- arrows given by universal constructions:

    1 : Id -> K 1

    f : X -> Y

    We have:

    1 natural transformation
  <=>  { definition }
    1_Y . Id f = K 1 f . 1_X
  <=>  { Id f = f, K 1 f = id_1 }
    1_Y . f = 1_X
  <=>  { fusion }
    true

- fst : (x) -> Fst

  (x) : C x C -> C, (x) (X, Y) = X x Y

  Fst : C x C -> C, Fst (X, Y) = X

  (f, g) : (X1, Y1) -> (X2, Y2)

    Fst (f, g) . fst_(X1, Y1) = fst_(X2, Y2) . (f x g)
  <=>  { definitions }
    f . fst_(X1, Y1) = fst_(X2, Y2) .
                       ((f . fst_(X1, Y1)) ^ (g . snd_(X1, Y1))
  <=>  { cancellation for split }
    f . fst_(X1, Y1) = f . fst_(X1, Y1)
  <=>  { duh }
    true

- In the above, we had Fst : C x D -> C, a "polymorphic functor", and
  fst : X x Y -> X a natural transformation.  Can we view Fst also as
  a natural transformation?  At this level of generality, no.  We
  would need Fst to be polymorphic in categories, i.e., to go between
  (x) : CAT x CAT -> CAT and FST : CAT x CAT -> CAT; but then (x) and
  FST would need to be functors _within_ CAT.  To avoid paradoxes, we
  would need to introduce various restrictions, such as universes,
  etc.

- In Haskell we have

    sort :: Ord a => [a] -> [a]

  Is this a natural transformation? That depends on the category.

  Let TORD be the category of _total_ orders and U the forgetful
  functor U : TORD -> Set, U (X, <=) = X.

  We have List . U : TORD -> SET, so we would need for any X, Y
  preorders and f : X -> Y monotonic function

     sort . (List . U) f = (List . U) f . sort
  <=>  { U f = f, List on arrows is map }
     sort . map f = map f . sort

  Is it always the case that mapping a _monotonic_ f to a sorted list
  gives the same result as sorting the list of the results?

  It is.  To see this, it suffices to note that sorting two lists
  containing the same elements leads to identical results.

- Let TPRE be the category of total preorders and consider the
  forgetful functor U : TPRE -> SET, which takes (A, [=) to A.  Assume
  we have a sorting function

     sortP : List X -> List X

  which can sort the elements of any total preorder X.  Is sort a
  natural transformation

     sortP : List . U -> List . U

  ?

  The answer is no.  Consider a sorting method that preserves the
  order of equivalent elements.  Consider the preorder (String+, [=)
  where non-empty strings are compared only by their first character
  in alphabetical order, i.e., (c : cs) [= (c' : cs) iff c comes
  before c' in alphabetical order.

  The function

      f : String+ -> String+
      f cs = 'X' : cs

  is monotonic w.r.t. [=, since f cs [= f cs' for all cs, cs' (the
  monotonicity condition is vacuously satisfied).

  Then

    sortP (map f ["b", "a"])
  =  { definition f }
    sortP ["Xb", "Xa"]
  =  { "Xb" ~ "Xa", order is preserved by assumption }
    ["Xb", "Xa"]

  but

    map f (sortP ["b", "a"])
  =  { 'a' comes before 'b' }
    map f ["a", "b"]
  =  { definition f }
    ["Xa", "Xb"]

Functors and natural transformations
------------------------------------

Let F, G : C -> D and alpha : F -> G.

For any functor H : B -> C we have

    alpha_H : F . H -> G . H
    (alpha_H)_X = alpha_(H X)

is a natural transformation (_precomposing_ alpha with H).

Similarly, for any functor I : D -> E

    I alpha : I . F -> I . G
    (I alpha)_X = I (alpha X)

is a natural transformation (_postcomposing_ alpha with I).

Given functors F, G : A -> B and H, I : B -> C, and natural
transformations alpha : F -> G and beta : H -> I, we can define a new
natural transformation

    beta * alpha : H . F -> I . G

Writing the types of the elements involved we have

    H alpha : H . F -> H . G
    I alpha : I . F -> I . G
    beta_F  : H . F -> I . F
    beta_G  : H . G -> I . G

We can see we have two choices (for a type correct dep. of *):

    beta * alpha = beta_G . H alpha

or

    beta * alpha = I alpha . beta_F


Which should we pick?

    beta_(G X) . H alpha_X
  =  { naturality of beta }
    I (alpha_X) . beta_(F X)
  =  { precomposition }
    I (alpha_X) . (beta_F)_X

We have used alpha_H for the precomposition, and I alpha for the
postcomposition.  In many textbooks, the composition notation is
overloaded: alpha . H and I . alpha would be used, respectively.

This suggests that composition-like properties hold, in particular
associativity:

    alpha . G . F = (alpha_G)_F = alpha_(G . F)
    I . H . alpha = I (H alpha) = (I . H) alpha
    I . alpha . H = (I alpha)_H = I (alpha_H)

and the less obvious

    (beta . alpha) . H = (beta . alpha)_H = beta_H . alpha_H

We also have an "abide" law relating . and *:

    alpha : F -> G, beta : G -> H, delta : I -> J, gamma : J -> K

    (gamma . delta) * (beta . alpha) =
    (gamma * beta) . (delta * alpha)

Remark: "abide" comes from "above beside" as in multiplication of
fractions (with · as multiplication):

    gamma · delta     gamma   delta
    -------------  =  ----- · -----
     beta · alpha     beta    alpha

With this comparison, "*" for natural transformations is like division
and "." is multiplication.

(end remark).


Mendler-style folds
-------------------

F endofunctor, initial F-algebra  in : F uF -> uF

The universal property of folds states that the equation

    x . in = f . F x

has the unique solution x = fold f.

Mendler proposed the following generalisation of folds:

    x . in = psi x

where

    x : uF -> X
    psi : Hom(uF, X) -> Hom(F uF, X)

with the condition that psi be natural in the contravariant argument,
i.e., the X is fixed and

    psi : H1 -> H2

    H1, H2 : C^op -> Set
    H1 Y = Hom(Y, X)
    H2 Y = Hom(F Y, X)

    for f : Y2 -> Y1        (in C)
    H1 f : H1 Y1 -> H1 Y2   (in Set)
    H1 f g = g . f
    H2 f : H2 Y1 -> H2 Y2   (in Set)
    H2 f g = g . F f

The naturality condition gives us for every f : Y2 -> Y1

    H2 f . psi_Y1 = psi_Y2 . H1 f
  <=>  { extensionality (in Set) }
    (H2 f . psi_Y1) g = (psi_Y2 . H1 f) g  for all g : Y1 -> X
  <=>  { Composition }
    H2 f (psi_Y1 g) = psi_Y2 (H1 f g)      for all g : Y1 -> X
  <=>  { Expand def. of H2 and H1 }
    psi_Y1 g . F f = psi_Y2 (g . f)        for all g : Y1 -> X

In the case of f with target X, i.e. Y1 = X, we can choose the
identity for g:

    f : Y2 -> X
  =>  { id : X -> X }
    psi_X id . F f = psi_Y2 f

Therefore, for f with target X, the action of psi is determined by
psi_X id.

But this is the case of x : uF -> X above!  Therefore we have

    x . in = psi x
  <=>  { naturality of psi above }
    x . in = psi id . F x
  <=>  { universal property of fold }
    x = fold (psi id)

Therefore, for any natural transformation psi : H1 -> H2, the Mendler
equation has a unique solution, fold (psi id), and therefore Mendler
style folds are not more general than standard folds.

Conversely, for any F-algebra a : F X -> X we can define a natural
transformation

    psi : H1 -> H2
    psi g = a . F g

and therefore standard folds are also not more general than Mendler
folds.