From dceac9e1e44b69593ef8d8585f9f411d970b8441 Mon Sep 17 00:00:00 2001 From: Yann Esposito Date: Fri, 7 Dec 2012 11:58:41 +0100 Subject: [PATCH] added natural transformation explanation --- categories.html | 63 ++++++++++++++++++ ...tion.html => 150_Number_construction.html} | 0 ...> 160_Degenerated_Categories_Monoids.html} | 0 ...rated_Categories_Discrete_Categories.html} | 0 ..._Haskell_Functors_for_the_programmer.html} | 0 ...80_Haskell_Functors_for_the_programmer.md} | 0 .../170_Natural_Transformation_Examples.html | 14 ++++ .../170_Natural_Transformation_Examples.md | 17 +++++ .../180_Natural_Transformation_Examples.html | 14 ++++ .../180_Natural_Transformation_Examples.md | 20 ++++++ .../30_How/190_Another_Nat_Trans_Example.html | 14 ++++ .../30_How/190_Another_Nat_Trans_Example.md | 20 ++++++ .../30_How/200_Another_Nat_Trans_Example.html | 13 ++++ .../30_How/200_Another_Nat_Trans_Example.md | 19 ++++++ .../{170_Monads.html => 210_Monads.html} | 0 .../30_How/{170_Monads.md => 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mode 100644 categories/img/mp/list-tree-endofunctor-morphism.mp create mode 100644 categories/img/mp/list-tree-endofunctor-morphism.png create mode 100644 categories/img/mp/nattrans-list-maybe.mp create mode 100644 categories/img/mp/nattrans-list-maybe.png create mode 100644 categories/img/mp/nattrans-list-tree.mp create mode 100644 categories/img/mp/nattrans-list-tree.png create mode 100644 categories/img/mp/nattrans-tree-list.mp create mode 100644 categories/img/mp/nattrans-tree-list.png create mode 100644 categories/img/mp/tree-list-endofunctor-morphism.mp create mode 100644 categories/img/mp/tree-list-endofunctor-morphism.png diff --git a/categories.html b/categories.html index 202ece7..a4c6512 100644 --- a/categories.html +++ b/categories.html @@ -659,6 +659,69 @@ Haskell types is fractal:

  • for Haskell functors: F a -> G a are the natural transformations.
    List to Trees, Tree to List, Tree to Maybe...
    Rearragement functions only.
  • + +
    +

    Natural Transformation Examples

    +
    data Tree a = Empty | Node a [Tree a]
    +              deriving (Show)
    +toTree :: [a] -> Tree a
    +toTree [] = Empty
    +toTree (x:xs) = Node x [toTree xs]
    + +

    toTree is a natural transformation. It is also a morphism from [] to Tree in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram +
    + + +
    +
    +

    Natural Transformation Examples

    +
    data Tree a = Empty | Node a [Tree a]
    +              deriving (Show)
    +toList :: Tree a -> [a]
    +toList Empty = []
    +toList (Node x l) = [x] ++ concat (map toList l)
    + +

    toList is a natural transformation. It is also a morphism from Tree to [] in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram
    toList.toTree=id & toTree.toList=id.
    Therefore [] & Tree are isomorph.
    +
    + + +
    +
    +

    Another Nat. Trans. Example

    +
    
    +toMaybe :: [a] -> Maybe a   ;   mToList :: Maybe a -> [a]
    +toMaybe [] = Nothing        ;   mToList Nothing = []
    +toMaybe (x:xs) = Just x     ;   mToList Just x  = [x]
    +
    + +

    toMaybe is a natural transformation. It is also a morphism from [] to Maybe in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram
    There is no isomorphism.
    Hint: Bool lists longer than 1.
    +
    + + +
    +
    +

    Another Nat. Trans. Example

    +
    
    +toMaybe :: [a] -> Maybe a
    +toMaybe [] = Nothing
    +toMaybe (x:xs) = Just x
    + +

    toMaybe is a natural transformation. It is also a morphism from [] to Maybe in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram
    There is no isomorphism.
    Hint: List with more than one element.
    +
    + +

    Monads

    diff --git a/categories/20_What/145_Number_construction.html b/categories/20_What/150_Number_construction.html similarity index 100% rename from categories/20_What/145_Number_construction.html rename to categories/20_What/150_Number_construction.html diff --git a/categories/20_What/150_Degenerated_Categories.html b/categories/20_What/160_Degenerated_Categories_Monoids.html similarity index 100% rename from categories/20_What/150_Degenerated_Categories.html rename to categories/20_What/160_Degenerated_Categories_Monoids.html diff --git a/categories/20_What/180_Degenerated_Categories.html b/categories/20_What/180_Degenerated_Categories_Discrete_Categories.html similarity index 100% rename from categories/20_What/180_Degenerated_Categories.html rename to categories/20_What/180_Degenerated_Categories_Discrete_Categories.html diff --git a/categories/30_How/080_Haskell_Functors.html b/categories/30_How/080_Haskell_Functors_for_the_programmer.html similarity index 100% rename from categories/30_How/080_Haskell_Functors.html rename to categories/30_How/080_Haskell_Functors_for_the_programmer.html diff --git a/categories/30_How/080_Haskell_Functors.md b/categories/30_How/080_Haskell_Functors_for_the_programmer.md similarity index 100% rename from categories/30_How/080_Haskell_Functors.md rename to categories/30_How/080_Haskell_Functors_for_the_programmer.md diff --git a/categories/30_How/170_Natural_Transformation_Examples.html b/categories/30_How/170_Natural_Transformation_Examples.html new file mode 100644 index 0000000..86a90c4 --- /dev/null +++ b/categories/30_How/170_Natural_Transformation_Examples.html @@ -0,0 +1,14 @@ +

    Natural Transformation Examples

    +
    data Tree a = Empty | Node a [Tree a]
    +              deriving (Show)
    +toTree :: [a] -> Tree a
    +toTree [] = Empty
    +toTree (x:xs) = Node x [toTree xs]
    + +

    toTree is a natural transformation. It is also a morphism from [] to Tree in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram +
    + + diff --git a/categories/30_How/170_Natural_Transformation_Examples.md b/categories/30_How/170_Natural_Transformation_Examples.md new file mode 100644 index 0000000..40402f5 --- /dev/null +++ b/categories/30_How/170_Natural_Transformation_Examples.md @@ -0,0 +1,17 @@ +Natural Transformation Examples +------------------------------- + +
    data Tree a = Empty | Node a [Tree a]
    +              deriving (Show)
    +toTree :: [a] -> Tree a
    +toTree [] = Empty
    +toTree (x:xs) = Node x [toTree xs]
    + + +`toTree` is a natural transformation. +It is also a morphism from `[]` to `Tree` in the Category of \\(\Hask\\) endofunctors. + +natural transformation commutative diagram +
    +natural transformation commutative diagram +
    diff --git a/categories/30_How/180_Natural_Transformation_Examples.html b/categories/30_How/180_Natural_Transformation_Examples.html new file mode 100644 index 0000000..90b6c69 --- /dev/null +++ b/categories/30_How/180_Natural_Transformation_Examples.html @@ -0,0 +1,14 @@ +

    Natural Transformation Examples

    +
    data Tree a = Empty | Node a [Tree a]
    +              deriving (Show)
    +toList :: Tree a -> [a]
    +toList Empty = []
    +toList (Node x l) = [x] ++ concat (map toList l)
    + +

    toList is a natural transformation. It is also a morphism from Tree to [] in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram
    toList.toTree=id & toTree.toList=id.
    Therefore [] & Tree are isomorph.
    +
    + + diff --git a/categories/30_How/180_Natural_Transformation_Examples.md b/categories/30_How/180_Natural_Transformation_Examples.md new file mode 100644 index 0000000..1253fa2 --- /dev/null +++ b/categories/30_How/180_Natural_Transformation_Examples.md @@ -0,0 +1,20 @@ +Natural Transformation Examples +------------------------------- + +
    data Tree a = Empty | Node a [Tree a]
    +              deriving (Show)
    +toList :: Tree a -> [a]
    +toList Empty = []
    +toList (Node x l) = [x] ++ concat (map toList l)
    + + +`toList` is a natural transformation. +It is also a morphism from `Tree` to `[]` in the Category of \\(\Hask\\) endofunctors. + +natural transformation commutative diagram +
    +natural transformation commutative diagram +
    toList.toTree=id & toTree.toList=id.
    +Therefore [] & Tree are isomorph. +
    +
    diff --git a/categories/30_How/190_Another_Nat_Trans_Example.html b/categories/30_How/190_Another_Nat_Trans_Example.html new file mode 100644 index 0000000..193e7aa --- /dev/null +++ b/categories/30_How/190_Another_Nat_Trans_Example.html @@ -0,0 +1,14 @@ +

    Another Nat. Trans. Example

    +
    
    +toMaybe :: [a] -> Maybe a   ;   mToList :: Maybe a -> [a]
    +toMaybe [] = Nothing        ;   mToList Nothing = []
    +toMaybe (x:xs) = Just x     ;   mToList Just x  = [x]
    +
    + +

    toMaybe is a natural transformation. It is also a morphism from [] to Maybe in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram
    There is no isomorphism.
    Hint: Bool lists longer than 1.
    +
    + + diff --git a/categories/30_How/190_Another_Nat_Trans_Example.md b/categories/30_How/190_Another_Nat_Trans_Example.md new file mode 100644 index 0000000..854cf7b --- /dev/null +++ b/categories/30_How/190_Another_Nat_Trans_Example.md @@ -0,0 +1,20 @@ +Another Nat. Trans. Example +--------------------------- + +
    
    +toMaybe :: [a] -> Maybe a   ;   mToList :: Maybe a -> [a]
    +toMaybe [] = Nothing        ;   mToList Nothing = []
    +toMaybe (x:xs) = Just x     ;   mToList Just x  = [x]
    +
    + + +`toMaybe` is a natural transformation. +It is also a morphism from `[]` to `Maybe` in the Category of \\(\Hask\\) endofunctors. + +natural transformation commutative diagram +
    +natural transformation commutative diagram +
    There is no isomorphism.
    +Hint: Bool lists longer than 1. +
    +
    diff --git a/categories/30_How/200_Another_Nat_Trans_Example.html b/categories/30_How/200_Another_Nat_Trans_Example.html new file mode 100644 index 0000000..7e54ab4 --- /dev/null +++ b/categories/30_How/200_Another_Nat_Trans_Example.html @@ -0,0 +1,13 @@ +

    Another Nat. Trans. Example

    +
    
    +toMaybe :: [a] -> Maybe a
    +toMaybe [] = Nothing
    +toMaybe (x:xs) = Just x
    + +

    toMaybe is a natural transformation. It is also a morphism from [] to Maybe in the Category of \(\) endofunctors.

    +natural transformation commutative diagram +
    +natural transformation commutative diagram
    There is no isomorphism.
    Hint: List with more than one element.
    +
    + + diff --git a/categories/30_How/200_Another_Nat_Trans_Example.md b/categories/30_How/200_Another_Nat_Trans_Example.md new file mode 100644 index 0000000..f7b4920 --- /dev/null +++ b/categories/30_How/200_Another_Nat_Trans_Example.md @@ -0,0 +1,19 @@ +Another Nat. Trans. Example +--------------------------- + +
    
    +toMaybe :: [a] -> Maybe a
    +toMaybe [] = Nothing
    +toMaybe (x:xs) = Just x
    + + +`toMaybe` is a natural transformation. +It is also a morphism from `[]` to `Maybe` in the Category of \\(\Hask\\) endofunctors. + +natural transformation commutative diagram +
    +natural transformation commutative diagram +
    There is no isomorphism.
    +Hint: List with more than one element. +
    +
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a/categories/30_How/220_Monads_generalize_composition.md b/categories/30_How/260_Monads_generalize_composition.md similarity index 100% rename from categories/30_How/220_Monads_generalize_composition.md rename to categories/30_How/260_Monads_generalize_composition.md diff --git a/categories/30_How/230_Monads_utility.html b/categories/30_How/270_Monads_utility.html similarity index 100% rename from categories/30_How/230_Monads_utility.html rename to categories/30_How/270_Monads_utility.html diff --git a/categories/img/mp/list-maybe-endofunctor-morphism.mp b/categories/img/mp/list-maybe-endofunctor-morphism.mp new file mode 100644 index 0000000..af3fb88 --- /dev/null +++ b/categories/img/mp/list-maybe-endofunctor-morphism.mp @@ -0,0 +1,8 @@ +pair tree,list; +resize(.8cm); +list=origin; +tree=list shifted (gu,0); +drawState(list,"\mathtt{[]}"); +drawState(tree,"\mathtt{Maybe}"); +drawEdgeAngle(list,tree,"\mathtt{toMaybe}",30); +drawEdgeAngle(tree,list,"\mathtt{mToList}",30); diff --git 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