If you don't want to read them through an HTML presentations framework or downloading a big PDF
@@ -4749,14 +4749,14 @@ just continue to read as a standard web page.
Not really about: Cat & glory
-credit to Tokuhiro Kawai (川井徳寛)
+credit to Tokuhiro Kawai (川井徳寛)
General Overview
-
+
Recent Math Field 1942-45, Samuel Eilenberg & Saunders Mac Lane
@@ -4782,7 +4782,7 @@ just continue to read as a standard web page.
Math Programming relation
-
+
Programming is doing Math
Strong relations between type theory and category theory.
Not convinced? Certainly a vocabulary problem.
@@ -4790,7 +4790,7 @@ just continue to read as a standard web page.
@@ -4798,7 +4798,7 @@ just continue to read as a standard web page.
Programmer Translation
-
+
Mathematician
@@ -4870,14 +4870,14 @@ LOLCat
Category: Objects
-
+
\(\ob{\mathcal{C}}\) is a collection
Category: Morphisms
-
+
\(A\) and \(B\) objects of \(\C\)
\(\hom{A,B}\) is a collection of morphisms
@@ -4889,18 +4889,18 @@ LOLCat
Composition (∘): associate to each couple \(f:A→B, g:B→C\)
$$g∘f:A\rightarrow C$$
-
+
Category laws: neutral element
for each object \(X\), there is an \(\id_X:X→X\),
such that for each \(f:A→B\):
-
+
Category laws: Associativity
Composition is associative:
-
+
Commutative diagrams
@@ -4908,7 +4908,7 @@ such that for each \(f:A→B\):
Two path with the same source and destination are equal.
\((h∘g)∘f = h∘(g∘f) \)
@@ -4916,7 +4916,7 @@ such that for each \(f:A→B\):
\(id_B∘f = f = f∘id_A \)
@@ -4927,7 +4927,7 @@ such that for each \(f:A→B\):
Question Time!
-
+- French-only joke -
@@ -4937,20 +4937,20 @@ such that for each \(f:A→B\):
Can this be a category?
\(\ob{\C},\hom{\C}\) fixed, is there a valid ∘?
-
+ YES
-
+
no candidate for \(g∘f\)
NO
-
+ YES
@@ -4959,14 +4959,14 @@ such that for each \(f:A→B\):
Can this be a category?
-
+
no candidate for \(f:C→B\)
NO
-
+
\((h∘g)∘f=\id_B∘f=f\)
\(h∘(g∘f)=h∘\id_A=h\)
@@ -4979,7 +4979,7 @@ such that for each \(f:A→B\):
Categories Examples
-
+- Basket of Cats -
@@ -5002,7 +5002,7 @@ such that for each \(f:A→B\):
Categories Everywhere?
-
+
\(\Mon\): (monoids, monoid morphisms,∘)
\(\Vec\): (Vectorial spaces, linear functions,∘)
@@ -5017,7 +5017,7 @@ such that for each \(f:A→B\):
Smaller Examples
Strings
-
+
\(\ob{Str}\) is a singleton
\(\hom{Str}\) each string
@@ -5033,7 +5033,7 @@ such that for each \(f:A→B\):
Graph
-
+
\(\ob{G}\) are vertices
@@ -5049,12 +5049,12 @@ such that for each \(f:A→B\):
Number construction
Each Numbers as a whole category
-
+
Degenerated Categories: Monoids
-
+
Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)
Only one object.
Examples:
@@ -5072,12 +5072,12 @@ such that for each \(f:A→B\):
At most one morphism between two objects.
-
+
Degenerated Categories: Discrete Categories
-
+
Any Set
Any set \(E: \ob{E}=E, \hom{x,y}=\{x\} ⇔ x=y \)
Only identities
@@ -5103,7 +5103,7 @@ such that for each \(f:A→B\):
Isomorph
-
isomorphism: \(f:A→B\) which can be "undone" i.e. \(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\) in this case, \(A\) & \(B\) are isomorphic.
+
isomorphism: \(f:A→B\) which can be "undone" i.e. \(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\) in this case, \(A\) & \(B\) are isomorphic.
A≌B means A and B are essentially the same. In Category Theory, = is in fact mostly ≌. For example in commutative diagrams.
@@ -5126,28 +5126,28 @@ A functor\(\F\) from \(
Functor Example (ob → ob)
-
+
Functor Example (hom → hom)
-
+
Functor Example
-
+
Endofunctors
An endofunctor for \(\C\) is a functor \(F:\C→\C\).
-
+
Category of Categories
-
+
Categories and functors form a category: \(\Cat\)
\(\ob{\Cat}\) are categories
@@ -5176,7 +5176,7 @@ A functor\(\F\) from \(
Put normal function inside a container. Ex: list, trees...
-
+
Haskell Functor properties
@@ -5283,7 +5283,7 @@ fmap head [[1,2,3],[4,5,6]] == [1,4]
Haskell functor can be seen as boxes containing all Haskell types and functions.
Haskell types look like a fractal:
-
+
Functor as boxes
@@ -5291,7 +5291,7 @@ Haskell types look like a fractal:
Haskell functor can be seen as boxes containing all Haskell types and functions.
Haskell types look like a fractal:
-
+
Functor as boxes
@@ -5299,7 +5299,7 @@ Haskell types look like a fractal:
Haskell functor can be seen as boxes containing all Haskell types and functions.
Haskell types look like a fractal:
-
+
"Non Haskell" Hask's Functors
@@ -5336,7 +5336,7 @@ Haskell types look like a fractal:
Category of \(\Hask\) Endofunctors
-
+
Category of Functors
@@ -5351,7 +5351,7 @@ Haskell types look like a fractal:
Natural Transformations
Let \(F\) and \(G\) be two functors from \(\C\) to \(\D\).
-
A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.
+
A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.
ex: between Haskell functors; F a -> G a Rearragement functions only.
@@ -5361,9 +5361,9 @@ toList :: [a] -> List a
toList [] = Nil
toList (x:xs) = Cons x (toList xs)
toList is a natural transformation. It is also a morphism from [] to List in the Category of \(\Hask\) endofunctors.
-
+
-
+
@@ -5374,9 +5374,9 @@ toHList :: List a -> [a]
toHList Nil = []
toHList (Cons x xs) = x:toHList xs
toHList is a natural transformation. It is also a morphism from List to [] in the Category of \(\Hask\) endofunctors.
-
+
-toList . toHList = id&toHList . toList = id& therefore []&List are isomorph.
+toList . toHList = id&toHList . toList = id& therefore []&List are isomorph.
@@ -5386,9 +5386,9 @@ toHList (Cons x xs) = x:toHList xs
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.
-
+
-
+
@@ -5398,9 +5398,9 @@ toMaybe (x:xs) = Just x
mToList Nothing = []
mToList Just x = [x]
toMaybe is a natural transformation. It is also a morphism from [] to Maybe in the Category of \(\Hask\) endofunctors.
-
+
-There is no isomorphism. Hint: Bool lists longer than 1.
+There is no isomorphism. Hint: Bool lists longer than 1.
@@ -5559,11 +5559,11 @@ drawPoint p = do
fold
-
+
κατα-morphism
-
+
κατα-morphism: fold generalization
diff --git a/src/Scratch/en/blog/index.html b/src/Scratch/en/blog/index.html
index 7e6e33d..70365b0 100644
--- a/src/Scratch/en/blog/index.html
+++ b/src/Scratch/en/blog/index.html
@@ -114,7 +114,7 @@ Learn Haskell Fast and Hard »2013-03-14
If you don't want to read them through an HTML presentations framework or downloading a big PDF
@@ -4744,14 +4744,14 @@ just continue to read as a standard web page.
Not really about: Cat & glory
-credit to Tokuhiro Kawai (川井徳寛)
+credit to Tokuhiro Kawai (川井徳寛)
General Overview
-
+
Recent Math Field 1942-45, Samuel Eilenberg & Saunders Mac Lane
@@ -4777,7 +4777,7 @@ just continue to read as a standard web page.
Math Programming relation
-
+
Programming is doing Math
Strong relations between type theory and category theory.
Not convinced? Certainly a vocabulary problem.
@@ -4785,7 +4785,7 @@ just continue to read as a standard web page.
@@ -4793,7 +4793,7 @@ just continue to read as a standard web page.
Programmer Translation
-
+
Mathematician
@@ -4865,14 +4865,14 @@ LOLCat
Category: Objects
-
+
\(\ob{\mathcal{C}}\) is a collection
Category: Morphisms
-
+
\(A\) and \(B\) objects of \(\C\)
\(\hom{A,B}\) is a collection of morphisms
@@ -4884,18 +4884,18 @@ LOLCat
Composition (∘): associate to each couple \(f:A→B, g:B→C\)
$$g∘f:A\rightarrow C$$
-
+
Category laws: neutral element
for each object \(X\), there is an \(\id_X:X→X\),
such that for each \(f:A→B\):
-
+
Category laws: Associativity
Composition is associative:
-
+
Commutative diagrams
@@ -4903,7 +4903,7 @@ such that for each \(f:A→B\):
Two path with the same source and destination are equal.
\((h∘g)∘f = h∘(g∘f) \)
@@ -4911,7 +4911,7 @@ such that for each \(f:A→B\):
\(id_B∘f = f = f∘id_A \)
@@ -4922,7 +4922,7 @@ such that for each \(f:A→B\):
Question Time!
-
+- French-only joke -
@@ -4932,20 +4932,20 @@ such that for each \(f:A→B\):
Can this be a category?
\(\ob{\C},\hom{\C}\) fixed, is there a valid ∘?
-
+ YES
-
+
no candidate for \(g∘f\)
NO
-
+ YES
@@ -4954,14 +4954,14 @@ such that for each \(f:A→B\):
Can this be a category?
-
+
no candidate for \(f:C→B\)
NO
-
+
\((h∘g)∘f=\id_B∘f=f\)
\(h∘(g∘f)=h∘\id_A=h\)
@@ -4974,7 +4974,7 @@ such that for each \(f:A→B\):
Categories Examples
-
+- Basket of Cats -
@@ -4997,7 +4997,7 @@ such that for each \(f:A→B\):
Categories Everywhere?
-
+
\(\Mon\): (monoids, monoid morphisms,∘)
\(\Vec\): (Vectorial spaces, linear functions,∘)
@@ -5012,7 +5012,7 @@ such that for each \(f:A→B\):
Smaller Examples
Strings
-
+
\(\ob{Str}\) is a singleton
\(\hom{Str}\) each string
@@ -5028,7 +5028,7 @@ such that for each \(f:A→B\):
Graph
-
+
\(\ob{G}\) are vertices
@@ -5044,12 +5044,12 @@ such that for each \(f:A→B\):
Number construction
Each Numbers as a whole category
-
+
Degenerated Categories: Monoids
-
+
Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)
Only one object.
Examples:
@@ -5067,12 +5067,12 @@ such that for each \(f:A→B\):
At most one morphism between two objects.
-
+
Degenerated Categories: Discrete Categories
-
+
Any Set
Any set \(E: \ob{E}=E, \hom{x,y}=\{x\} ⇔ x=y \)
Only identities
@@ -5098,7 +5098,7 @@ such that for each \(f:A→B\):
Isomorph
-
isomorphism: \(f:A→B\) which can be "undone" i.e. \(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\) in this case, \(A\) & \(B\) are isomorphic.
+
isomorphism: \(f:A→B\) which can be "undone" i.e. \(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\) in this case, \(A\) & \(B\) are isomorphic.
A≌B means A and B are essentially the same. In Category Theory, = is in fact mostly ≌. For example in commutative diagrams.
@@ -5121,28 +5121,28 @@ A functor\(\F\) from \(
Functor Example (ob → ob)
-
+
Functor Example (hom → hom)
-
+
Functor Example
-
+
Endofunctors
An endofunctor for \(\C\) is a functor \(F:\C→\C\).
-
+
Category of Categories
-
+
Categories and functors form a category: \(\Cat\)
\(\ob{\Cat}\) are categories
@@ -5171,7 +5171,7 @@ A functor\(\F\) from \(
Put normal function inside a container. Ex: list, trees...
-
+
Haskell Functor properties
@@ -5278,7 +5278,7 @@ fmap head [[1,2,3],[4,5,6]] == [1,4]
Haskell functor can be seen as boxes containing all Haskell types and functions.
Haskell types look like a fractal:
-
+
Functor as boxes
@@ -5286,7 +5286,7 @@ Haskell types look like a fractal:
Haskell functor can be seen as boxes containing all Haskell types and functions.
Haskell types look like a fractal:
-
+
Functor as boxes
@@ -5294,7 +5294,7 @@ Haskell types look like a fractal:
Haskell functor can be seen as boxes containing all Haskell types and functions.
Haskell types look like a fractal:
-
+
"Non Haskell" Hask's Functors
@@ -5331,7 +5331,7 @@ Haskell types look like a fractal:
Category of \(\Hask\) Endofunctors
-
+
Category of Functors
@@ -5346,7 +5346,7 @@ Haskell types look like a fractal:
Natural Transformations
Let \(F\) and \(G\) be two functors from \(\C\) to \(\D\).
-
A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.
+
A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.
ex: between Haskell functors; F a -> G a Rearragement functions only.
@@ -5356,9 +5356,9 @@ toList :: [a] -> List a
toList [] = Nil
toList (x:xs) = Cons x (toList xs)
toList is a natural transformation. It is also a morphism from [] to List in the Category of \(\Hask\) endofunctors.
-
+
-
+
@@ -5369,9 +5369,9 @@ toHList :: List a -> [a]
toHList Nil = []
toHList (Cons x xs) = x:toHList xs
toHList is a natural transformation. It is also a morphism from List to [] in the Category of \(\Hask\) endofunctors.
-
+
-toList . toHList = id&toHList . toList = id& therefore []&List are isomorph.
+toList . toHList = id&toHList . toList = id& therefore []&List are isomorph.
@@ -5381,9 +5381,9 @@ toHList (Cons x xs) = x:toHList xs
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.
-
+
-
+
@@ -5393,9 +5393,9 @@ toMaybe (x:xs) = Just x
mToList Nothing = []
mToList Just x = [x]
toMaybe is a natural transformation. It is also a morphism from [] to Maybe in the Category of \(\Hask\) endofunctors.
-
+
-There is no isomorphism. Hint: Bool lists longer than 1.
+There is no isomorphism. Hint: Bool lists longer than 1.
@@ -5554,11 +5554,11 @@ drawPoint p = do
fold
-
+
κατα-morphism
-
+
κατα-morphism: fold generalization
diff --git a/src/Scratch/fr/blog/index.html b/src/Scratch/fr/blog/index.html
index 82716dc..a5c70ea 100644
--- a/src/Scratch/fr/blog/index.html
+++ b/src/Scratch/fr/blog/index.html
@@ -114,7 +114,7 @@ Haskell Vite et Direct »2013-03-14
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+ 
+ 
+ 
+
+
+
+
+
+
+
+
isomorphism: \(f:A→B\) which can be "undone" i.e.
isomorphism: \(f:A→B\) which can be "undone" i.e.
+
+
+
+
+
+
+
+
+
+
+
A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.
A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.
+
+
+
+
+
+
![relation between [] and Maybe](http://yogsototh.github.com/Category-Theory-Presentation/categories/img/mp/maybe-list-endofunctor-morphsm.png)
![relation between [] and Maybe](http://yogsototh.github.io/Category-Theory-Presentation/categories/img/mp/maybe-list-endofunctor-morphsm.png)
+
+
