From b00276eb7b4517eb4d9fde79092cce7e123e0337 Mon Sep 17 00:00:00 2001 From: "Yann Esposito (Yogsototh)" Date: Sun, 9 Dec 2012 21:49:02 +0100 Subject: [PATCH] added catamorphism intro --- categories.html | 53 +++++++++++------- .../30_How/300_Catamorphisms/010_fold.html | 6 ++ .../30_How/300_Catamorphisms/010_fold.md | 7 +++ .../300_Catamorphisms/020_morphism.html | 6 ++ .../30_How/300_Catamorphisms/020_morphism.md | 7 +++ categories/img/earth_catamorphed.gif | Bin 0 -> 507253 bytes categories/img/tower_folded.gif | Bin 0 -> 1039913 bytes 7 files changed, 60 insertions(+), 19 deletions(-) create mode 100644 categories/30_How/300_Catamorphisms/010_fold.html create mode 100644 categories/30_How/300_Catamorphisms/010_fold.md create mode 100644 categories/30_How/300_Catamorphisms/020_morphism.html create mode 100644 categories/30_How/300_Catamorphisms/020_morphism.md create mode 100644 categories/img/earth_catamorphed.gif create mode 100644 categories/img/tower_folded.gif diff --git a/categories.html b/categories.html index af1a2d5..e8ad1d6 100644 --- a/categories.html +++ b/categories.html @@ -472,7 +472,6 @@ A functor \(\F\) from \(
  • \(\Hask\) category
  • Functors
  • Monads -
  • Arrows
  • κατα-morphisms
  • @@ -767,6 +766,16 @@ h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h.h) 1 = ERROR [2,3]+1 ✗
  • abusing notations for join it is equivalent to
    (F ⊙ F) ⊙ F = F ⊙ (F ⊙ F)
  • The second rule fix the type of η :: a -> F a.

    + +
    +

    Klesli arrows

    +

    Now the composition works as expected. In Haskell ◎ is <=< in Control.Monad.

    +

    g <=< f = \x -> join ((fmap g) (f x))

    +
    f x = [x]       ⇒ f 1 = [1]   ⇒ (f <=< f) 1 = [1] ✓
    +g x = [x+1]     ⇒ g 1 = [2]   ⇒ (g <=< g) 1 = [3] ✓
    +h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h <=< h) 1 = [3,6,4,9] ✓
    + +

    Monads

    @@ -798,16 +807,6 @@ h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h.h) 1 = ERROR [2,3]+1 ✗
    -

    Monads are just Monoids

    -
    -

    A Monad is just a monoid in the category of endofunctors, what's the problem?

    -
    -

    The real sentence was:

    -
    -

    All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.

    -
    -
    -

    List Monad

    Example: List is a functor

      @@ -840,14 +839,14 @@ join (η [x]) = [x] = join [η x]

      Therefore ([],join,η) is a monad.

    -

    Klesli arrows

    -

    Now the composition works as expected. In Haskell ◎ is <=< in Control.Monad.

    -

    g <=< f = \x -> join ((fmap g) (f x))

    -
    f x = [x]       ⇒ f 1 = [1]   ⇒ (f <=< f) 1 = [1] ✓
    -g x = [x+1]     ⇒ g 1 = [2]   ⇒ (g <=< g) 1 = [3] ✓
    -h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h <=< h) 1 = [3,6,4,9] ✓
    - - +

    Monads are just Monoids

    +
    +

    A Monad is just a monoid in the category of endofunctors, what's the problem?

    +
    +

    The real sentence was:

    +
    +

    All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.

    +

    Monads utility

    @@ -878,6 +877,22 @@ drawPoint :: Point -> State Screen DrawScene drawPoint p = do Screen width height <- get ... +
    +
    +

    fold

    +
    +fold
    fold
    +
    + + +
    +
    +

    κατα-morphism

    +
    +catamorphism
    κατα-morphism
    +
    + +
    diff --git a/categories/30_How/300_Catamorphisms/010_fold.html b/categories/30_How/300_Catamorphisms/010_fold.html new file mode 100644 index 0000000..5110b4b --- /dev/null +++ b/categories/30_How/300_Catamorphisms/010_fold.html @@ -0,0 +1,6 @@ +

    fold

    +
    +fold
    fold
    +
    + + diff --git a/categories/30_How/300_Catamorphisms/010_fold.md b/categories/30_How/300_Catamorphisms/010_fold.md new file mode 100644 index 0000000..0b42fe7 --- /dev/null +++ b/categories/30_How/300_Catamorphisms/010_fold.md @@ -0,0 +1,7 @@ +`fold` +------ + +
    +fold +
    fold
    +
    diff --git a/categories/30_How/300_Catamorphisms/020_morphism.html b/categories/30_How/300_Catamorphisms/020_morphism.html new file mode 100644 index 0000000..2b373ff --- /dev/null +++ b/categories/30_How/300_Catamorphisms/020_morphism.html @@ -0,0 +1,6 @@ +

    κατα-morphism

    +
    +catamorphism
    κατα-morphism
    +
    + + diff --git a/categories/30_How/300_Catamorphisms/020_morphism.md b/categories/30_How/300_Catamorphisms/020_morphism.md new file mode 100644 index 0000000..7f15d55 --- /dev/null +++ b/categories/30_How/300_Catamorphisms/020_morphism.md @@ -0,0 +1,7 @@ +κατα-morphism +------------- + +
    +catamorphism +
    κατα-morphism
    +
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