From 72f79ad842ef682e9a68e5068817ae60f5769d4d Mon Sep 17 00:00:00 2001 From: "Yann Esposito (Yogsototh)" Date: Fri, 8 Oct 2021 16:16:02 +0200 Subject: [PATCH] Fix github pages links --- .../Category-Theory-Presentation/index.html | 104 +++++++++--------- .../Category-Theory-Presentation/index.html | 104 +++++++++--------- 2 files changed, 104 insertions(+), 104 deletions(-) diff --git a/src/Scratch/en/blog/Category-Theory-Presentation/index.html b/src/Scratch/en/blog/Category-Theory-Presentation/index.html index 28841fe..9d57a67 100644 --- a/src/Scratch/en/blog/Category-Theory-Presentation/index.html +++ b/src/Scratch/en/blog/Category-Theory-Presentation/index.html @@ -39,12 +39,12 @@
- Cateogry of Hask's endofunctors + Cateogry of Hask's endofunctors

Yesterday I was happy to make a presentation about Category Theory at Riviera Scala Clojure Meetup (note I used only Haskell for my examples).

-
  • Click here to go to the HTML presentation. -
  • Click Here to download the PDF slides (LaTeX not rendered properly) +

    If you don't want to read them through an HTML presentations framework or downloading a big PDF @@ -94,14 +94,14 @@ just continue to read as a standard web page.

    Not really about: Cat & glory

    -Cat n glory
    credit to Tokuhiro Kawai (川井徳寛)
    +Cat n glory
    credit to Tokuhiro Kawai (川井徳寛)

    General Overview

    -Samuel Eilenberg Saunders Mac Lane +Samuel Eilenberg Saunders Mac Lane

    Recent Math Field
    1942-45, Samuel Eilenberg & Saunders Mac Lane

    @@ -127,7 +127,7 @@ just continue to read as a standard web page.

    Math Programming relation

    -Buddha Fractal +Buddha Fractal

    Programming is doing Math

    Strong relations between type theory and category theory.

    Not convinced?
    Certainly a vocabulary problem.

    @@ -135,7 +135,7 @@ just continue to read as a standard web page.

    Vocabulary

    -mind blown +mind blown

    Math vocabulary used in this presentation:

    Category, Morphism, Associativity, Preorder, Functor, Endofunctor, Categorial property, Commutative diagram, Isomorph, Initial, Dual, Monoid, Natural transformation, Monad, Klesli arrows, κατα-morphism, ...

    @@ -143,7 +143,7 @@ just continue to read as a standard web page.

    Programmer Translation

    -lolcat +lolcat
    Mathematician @@ -215,14 +215,14 @@ LOLCat

    Category: Objects

    -objects +objects

    \(\ob{\mathcal{C}}\) is a collection

    Category: Morphisms

    -morphisms +morphisms

    \(A\) and \(B\) objects of \(\C\)
    \(\hom{A,B}\) is a collection of morphisms
    @@ -234,31 +234,31 @@ LOLCat

    Composition (∘): associate to each couple \(f:A→B, g:B→C\) $$g∘f:A\rightarrow C$$

    -composition +composition

    Category laws: neutral element

    for each object \(X\), there is an \(\id_X:X→X\),
    such that for each \(f:A→B\):

    -identity +identity

    Category laws: Associativity

    Composition is associative:

    -associative composition +associative composition

    Commutative diagrams

    Two path with the same source and destination are equal.

    - Commutative Diagram (Associativity) + Commutative Diagram (Associativity)
    \((h∘g)∘f = h∘(g∘f) \)
    - Commutative Diagram (Identity law) + Commutative Diagram (Identity law)
    \(id_B∘f = f = f∘id_A \)
    @@ -268,7 +268,7 @@ such that for each \(f:A→B\):

    Question Time!

    - +
    - French-only joke -
    @@ -278,20 +278,20 @@ such that for each \(f:A→B\):

    Can this be a category?

    \(\ob{\C},\hom{\C}\) fixed, is there a valid ∘?

    - Category example 1 + Category example 1
    YES
    - Category example 2 + Category example 2
    no candidate for \(g∘f\)
    NO
    - Category example 3 + Category example 3
    YES
    @@ -300,14 +300,14 @@ such that for each \(f:A→B\):

    Can this be a category?

    - Category example 4 + Category example 4
    no candidate for \(f:C→B\)
    NO
    - Category example 5 + Category example 5
    \((h∘g)∘f=\id_B∘f=f\)
    \(h∘(g∘f)=h∘\id_A=h\)
    @@ -320,7 +320,7 @@ such that for each \(f:A→B\):

    Categories Examples

    -Basket of cats +Basket of cats
    - Basket of Cats -
    @@ -343,7 +343,7 @@ such that for each \(f:A→B\):

    Categories Everywhere?

    -Cats everywhere +Cats everywhere
    • \(\Mon\): (monoids, monoid morphisms,∘)
    • \(\Vec\): (Vectorial spaces, linear functions,∘)
      • @@ -358,7 +358,7 @@ such that for each \(f:A→B\):

      Smaller Examples

      Strings

      -Monoids are one object categories +Monoids are one object categories
      • \(\ob{Str}\) is a singleton
      • \(\hom{Str}\) each string
        • @@ -374,7 +374,7 @@ such that for each \(f:A→B\):

        Graph

        -Each graph is a category +Each graph is a category
        • \(\ob{G}\) are vertices
          • @@ -390,12 +390,12 @@ such that for each \(f:A→B\):

          Number construction

          Each Numbers as a whole category

          -Each number as a category +Each number as a category

    Degenerated Categories: Monoids

    -Monoids are one object categories +Monoids are one object categories

    Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)

    Only one object.

    Examples:

    @@ -413,12 +413,12 @@ such that for each \(f:A→B\):

    At most one morphism between two objects.

    -preorder category +preorder category

    Degenerated Categories: Discrete Categories

    -Any set can be a category +Any set can be a category

    Any Set

    Any set \(E: \ob{E}=E, \hom{x,y}=\{x\} ⇔ x=y \)

    Only identities

    @@ -444,7 +444,7 @@ such that for each \(f:A→B\):

    Isomorph

    -

    isomorph cats 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.

    +

    isomorph cats 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.

    @@ -467,28 +467,28 @@ A functor \(\F\) from \(

    Functor Example (ob → ob)

    -Functor +Functor

    Functor Example (hom → hom)

    -Functor +Functor

    Functor Example

    -Functor +Functor

    Endofunctors

    An endofunctor for \(\C\) is a functor \(F:\C→\C\).

    -Endofunctor +Endofunctor

    Category of Categories

    - +

    Categories and functors form a category: \(\Cat\)

    • \(\ob{\Cat}\) are categories @@ -517,7 +517,7 @@ A functor \(\F\) from \(

      Category \(\Hask\):

      -Haskell Category Representation +Haskell Category Representation
      • \(\ob{\Hask} = \) Haskell types @@ -607,7 +607,7 @@ fmap head [[1,2,3],[4,5,6]] == [1,4]

        Put normal function inside a container. Ex: list, trees...

        -Haskell Functor as a box play +Haskell Functor as a box play

    Haskell Functor properties

    @@ -624,7 +624,7 @@ 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:

    -Haskell functor representation +Haskell functor representation

    Functor as boxes

    @@ -632,7 +632,7 @@ 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:

    -Haskell functor representation +Haskell functor representation

    Functor as boxes

    @@ -640,7 +640,7 @@ 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:

    -Haskell functor representation +Haskell functor representation

    "Non Haskell" Hask's Functors

    @@ -677,7 +677,7 @@ Haskell types look like a fractal:

    Category of \(\Hask\) Endofunctors

    -Category of Hask endofunctors +Category of Hask endofunctors

    Category of Functors

    @@ -692,7 +692,7 @@ Haskell types look like a fractal:

    Natural Transformations

    Let \(F\) and \(G\) be two functors from \(\C\) to \(\D\).

    -

    Natural transformation commutative diagram A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.

    +

    Natural transformation commutative diagram 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.

    @@ -702,9 +702,9 @@ 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.

    -natural transformation commutative diagram +natural transformation commutative diagram
    -natural transformation commutative diagram +natural transformation commutative diagram
    @@ -715,9 +715,9 @@ 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.

    -natural transformation commutative diagram +natural transformation commutative diagram
    -natural transformation commutative diagram
    toList . toHList = id & toHList . toList = id &
    therefore [] & List are isomorph.
    +natural transformation commutative diagram
    toList . toHList = id & toHList . toList = id &
    therefore [] & List are isomorph.
    @@ -727,9 +727,9 @@ 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.

    -natural transformation commutative diagram +natural transformation commutative diagram
    -natural transformation commutative diagram +natural transformation commutative diagram
    @@ -739,9 +739,9 @@ 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.

    -natural transformation commutative diagram +natural transformation commutative diagram
    -relation between [] and Maybe
    There is no isomorphism.
    Hint: Bool lists longer than 1.
    +relation between [] and Maybe
    There is no isomorphism.
    Hint: Bool lists longer than 1.
    @@ -900,11 +900,11 @@ drawPoint p = do

    fold

    -fold +fold

    κατα-morphism

    -catamorphism +catamorphism

    κατα-morphism: fold generalization

    diff --git a/src/Scratch/fr/blog/Category-Theory-Presentation/index.html b/src/Scratch/fr/blog/Category-Theory-Presentation/index.html index 5ea237d..f07a7f4 100644 --- a/src/Scratch/fr/blog/Category-Theory-Presentation/index.html +++ b/src/Scratch/fr/blog/Category-Theory-Presentation/index.html @@ -39,12 +39,12 @@
    - Cateogry of Hask's endofunctors + Cateogry of Hask's endofunctors

    Yesterday I was happy to make a presentation about Category Theory at Riviera Scala Clojure Meetup (note I used only Haskell for my examples).

    -
    • Click here to go to the HTML presentation. -
    • Click Here to download the PDF slides (LaTeX not rendered properly) +

      If you don't want to read them through an HTML presentations framework or downloading a big PDF @@ -94,14 +94,14 @@ just continue to read as a standard web page.

      Not really about: Cat & glory

      -Cat n glory
      credit to Tokuhiro Kawai (川井徳寛)
      +Cat n glory
      credit to Tokuhiro Kawai (川井徳寛)

      General Overview

      -Samuel Eilenberg Saunders Mac Lane +Samuel Eilenberg Saunders Mac Lane

      Recent Math Field
      1942-45, Samuel Eilenberg & Saunders Mac Lane

      @@ -127,7 +127,7 @@ just continue to read as a standard web page.

      Math Programming relation

      -Buddha Fractal +Buddha Fractal

      Programming is doing Math

      Strong relations between type theory and category theory.

      Not convinced?
      Certainly a vocabulary problem.

      @@ -135,7 +135,7 @@ just continue to read as a standard web page.

      Vocabulary

      -mind blown +mind blown

      Math vocabulary used in this presentation:

      Category, Morphism, Associativity, Preorder, Functor, Endofunctor, Categorial property, Commutative diagram, Isomorph, Initial, Dual, Monoid, Natural transformation, Monad, Klesli arrows, κατα-morphism, ...

      @@ -143,7 +143,7 @@ just continue to read as a standard web page.

      Programmer Translation

      -lolcat +lolcat
      Mathematician @@ -215,14 +215,14 @@ LOLCat

      Category: Objects

      -objects +objects

      \(\ob{\mathcal{C}}\) is a collection

      Category: Morphisms

      -morphisms +morphisms

      \(A\) and \(B\) objects of \(\C\)
      \(\hom{A,B}\) is a collection of morphisms
      @@ -234,31 +234,31 @@ LOLCat

      Composition (∘): associate to each couple \(f:A→B, g:B→C\) $$g∘f:A\rightarrow C$$

      -composition +composition

      Category laws: neutral element

      for each object \(X\), there is an \(\id_X:X→X\),
      such that for each \(f:A→B\):

      -identity +identity

      Category laws: Associativity

      Composition is associative:

      -associative composition +associative composition

      Commutative diagrams

      Two path with the same source and destination are equal.

      - Commutative Diagram (Associativity) + Commutative Diagram (Associativity)
      \((h∘g)∘f = h∘(g∘f) \)
      - Commutative Diagram (Identity law) + Commutative Diagram (Identity law)
      \(id_B∘f = f = f∘id_A \)
      @@ -268,7 +268,7 @@ such that for each \(f:A→B\):

      Question Time!

      - +
      - French-only joke -
      @@ -278,20 +278,20 @@ such that for each \(f:A→B\):

      Can this be a category?

      \(\ob{\C},\hom{\C}\) fixed, is there a valid ∘?

      - Category example 1 + Category example 1
      YES
      - Category example 2 + Category example 2
      no candidate for \(g∘f\)
      NO
      - Category example 3 + Category example 3
      YES
      @@ -300,14 +300,14 @@ such that for each \(f:A→B\):

      Can this be a category?

      - Category example 4 + Category example 4
      no candidate for \(f:C→B\)
      NO
      - Category example 5 + Category example 5
      \((h∘g)∘f=\id_B∘f=f\)
      \(h∘(g∘f)=h∘\id_A=h\)
      @@ -320,7 +320,7 @@ such that for each \(f:A→B\):

      Categories Examples

      -Basket of cats +Basket of cats
      - Basket of Cats -
      @@ -343,7 +343,7 @@ such that for each \(f:A→B\):

      Categories Everywhere?

      -Cats everywhere +Cats everywhere
      • \(\Mon\): (monoids, monoid morphisms,∘)
      • \(\Vec\): (Vectorial spaces, linear functions,∘)
        • @@ -358,7 +358,7 @@ such that for each \(f:A→B\):

        Smaller Examples

        Strings

        -Monoids are one object categories +Monoids are one object categories
        • \(\ob{Str}\) is a singleton
        • \(\hom{Str}\) each string
          • @@ -374,7 +374,7 @@ such that for each \(f:A→B\):

          Graph

          -Each graph is a category +Each graph is a category
          • \(\ob{G}\) are vertices
            • @@ -390,12 +390,12 @@ such that for each \(f:A→B\):

            Number construction

            Each Numbers as a whole category

            -Each number as a category +Each number as a category

      Degenerated Categories: Monoids

      -Monoids are one object categories +Monoids are one object categories

      Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)

      Only one object.

      Examples:

      @@ -413,12 +413,12 @@ such that for each \(f:A→B\):

      At most one morphism between two objects.

      -preorder category +preorder category

      Degenerated Categories: Discrete Categories

      -Any set can be a category +Any set can be a category

      Any Set

      Any set \(E: \ob{E}=E, \hom{x,y}=\{x\} ⇔ x=y \)

      Only identities

      @@ -444,7 +444,7 @@ such that for each \(f:A→B\):

      Isomorph

      -

      isomorph cats 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.

      +

      isomorph cats 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.

      @@ -467,28 +467,28 @@ A functor \(\F\) from \(

      Functor Example (ob → ob)

      -Functor +Functor

      Functor Example (hom → hom)

      -Functor +Functor

      Functor Example

      -Functor +Functor

      Endofunctors

      An endofunctor for \(\C\) is a functor \(F:\C→\C\).

      -Endofunctor +Endofunctor

      Category of Categories

      - +

      Categories and functors form a category: \(\Cat\)

      • \(\ob{\Cat}\) are categories @@ -517,7 +517,7 @@ A functor \(\F\) from \(

        Category \(\Hask\):

        -Haskell Category Representation +Haskell Category Representation
        • \(\ob{\Hask} = \) Haskell types @@ -607,7 +607,7 @@ fmap head [[1,2,3],[4,5,6]] == [1,4]

          Put normal function inside a container. Ex: list, trees...

          -Haskell Functor as a box play +Haskell Functor as a box play

      Haskell Functor properties

      @@ -624,7 +624,7 @@ 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:

      -Haskell functor representation +Haskell functor representation

      Functor as boxes

      @@ -632,7 +632,7 @@ 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:

      -Haskell functor representation +Haskell functor representation

      Functor as boxes

      @@ -640,7 +640,7 @@ 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:

      -Haskell functor representation +Haskell functor representation

      "Non Haskell" Hask's Functors

      @@ -677,7 +677,7 @@ Haskell types look like a fractal:

      Category of \(\Hask\) Endofunctors

      -Category of Hask endofunctors +Category of Hask endofunctors

      Category of Functors

      @@ -692,7 +692,7 @@ Haskell types look like a fractal:

      Natural Transformations

      Let \(F\) and \(G\) be two functors from \(\C\) to \(\D\).

      -

      Natural transformation commutative diagram A natural transformation: familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.

      +

      Natural transformation commutative diagram 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.

      @@ -702,9 +702,9 @@ 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.

      -natural transformation commutative diagram +natural transformation commutative diagram
      -natural transformation commutative diagram +natural transformation commutative diagram
      @@ -715,9 +715,9 @@ 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.

      -natural transformation commutative diagram +natural transformation commutative diagram
      -natural transformation commutative diagram
      toList . toHList = id & toHList . toList = id &
      therefore [] & List are isomorph.
      +natural transformation commutative diagram
      toList . toHList = id & toHList . toList = id &
      therefore [] & List are isomorph.
      @@ -727,9 +727,9 @@ 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.

      -natural transformation commutative diagram +natural transformation commutative diagram
      -natural transformation commutative diagram +natural transformation commutative diagram
      @@ -739,9 +739,9 @@ 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.

      -natural transformation commutative diagram +natural transformation commutative diagram
      -relation between [] and Maybe
      There is no isomorphism.
      Hint: Bool lists longer than 1.
      +relation between [] and Maybe
      There is no isomorphism.
      Hint: Bool lists longer than 1.
      @@ -900,11 +900,11 @@ drawPoint p = do

      fold

      -fold +fold

      κατα-morphism

      -catamorphism +catamorphism

      κατα-morphism: fold generalization