From 35eeaa4ea169b4de4b14159778e90415431aa693 Mon Sep 17 00:00:00 2001 From: "Yann Esposito (Yogsototh)" Date: Wed, 12 Dec 2012 11:09:15 +0100 Subject: [PATCH] added latest presentation blog post --- Rules | 11 +- content/css/main.sass | 58 +- .../en/blog/Category-Theory-Presentation.erb | 1000 ++++++++++++ .../fr/blog/Category-Theory-Presentation.erb | 1000 ++++++++++++ crash.log | 169 +- img_latest_blog_dir | 2 +- latest.ymd | 2 +- multi/blog/Category-Theory-Presentation.erb | 1001 ++++++++++++ output/Scratch/assets/css/main.css | 2 +- .../blog/Haskell-OpenGL-Mandelbrot/index.html | 4 + output/Scratch/en/blog/feed/feed.xml | 140 +- .../blog/Haskell-OpenGL-Mandelbrot/index.html | 4 + output/Scratch/fr/blog/feed/feed.xml | 137 +- output/Scratch/sitemap.xml | 1450 +++++++++-------- output/index.html | 6 +- output/keynote/categories.pdf | Bin 0 -> 13757697 bytes 16 files changed, 3965 insertions(+), 1021 deletions(-) create mode 100644 content/html/en/blog/Category-Theory-Presentation.erb create mode 100644 content/html/fr/blog/Category-Theory-Presentation.erb create mode 100644 multi/blog/Category-Theory-Presentation.erb create mode 100644 output/keynote/categories.pdf diff --git a/Rules b/Rules index 0e7e86a45..9b04e484d 100644 --- a/Rules +++ b/Rules @@ -46,17 +46,16 @@ compile '/html/*' do # item[:extension] returns 'html.erb' for multi-dotted filename ext = item[:extension].nil? ? nil : item[:extension].split('.').last - filter :ultraviolet # must be before kramdown - filter :graph - filter :description - filter :falacy - filter :blogimage - if ext == 'erb' filter :erb elsif ext == 'haml' || ext.nil? filter :haml elsif ext == 'md' || ext == 'markdown' + filter :ultraviolet # must be before kramdown + filter :graph + filter :description + filter :falacy + filter :blogimage filter :erb # I'll try not to use ruby anymore inside content filter :kramdown # filter :math_repair # after kramdown diff --git a/content/css/main.sass b/content/css/main.sass index d23392b1e..43e07e1c2 100644 --- a/content/css/main.sass +++ b/content/css/main.sass @@ -515,23 +515,16 @@ $hauteur_menu: 4em $contentMargin: 0em -.encadre, .black, .red, .intro, .resume, .shadow +.encadre, .black, .intro, .resume, .shadow padding: 2em margin: top: 2em bottom: 2em -.encadre, .black, .red, .shadow +.encadre, .black, .shadow +secondColor -pre .red - background: none - padding: 0 - margin: auto - border: none - box-shadow: none - .intro, .resume font-size: 0.9em +highlight @@ -600,6 +593,9 @@ figure text-align: center margin: .5em 0 +figure.left, figure.right + max-width: 30% + img.clean border: none background-color: none @@ -717,6 +713,11 @@ img background: none border: none +img.right + max-width: 30% + margin: + top: .6em + left: 2em img.left float: left max-width: 30% @@ -1070,3 +1071,42 @@ a.cut img max-width: 80% max-height: 6em + +section.slide + border-color: $borderColor + border: solid 1px + margin-bottom: 1em + padding: .5em + +.base03 + color: $base03 +.base02 + color: $base02 +.base01 + color: $base01 +.base00 + color: $base00 +.base0 + color: $base0 +.base1 + color: $base1 +.base2 + color: $base2 +.base3 + color: $base3 +.yellow + color: $yellow +.orange + color: $orange +.red + color: $red +.magenta + color: $magenta +.violet + color: $violet +.blue + color: $blue +.cyan + color: $cyan +.green + color: $green diff --git a/content/html/en/blog/Category-Theory-Presentation.erb b/content/html/en/blog/Category-Theory-Presentation.erb new file mode 100644 index 000000000..7fde2f554 --- /dev/null +++ b/content/html/en/blog/Category-Theory-Presentation.erb @@ -0,0 +1,1000 @@ +----- +isHidden: false +menupriority: 1 +kind: article +created_at: 2012-12-12T10:06:40+02:00 +title: Category Theory Presentation +author_name: Yann Esposito +author_uri: yannesposito.com +tags: + - category theory + - math + - functor +----- +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).

+ + + +

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

+ +
+\(\newcommand{\F}{\mathbf{F}}\) +\(\newcommand{\E}{\mathbf{E}}\) +\(\newcommand{\C}{\mathcal{C}}\) +\(\newcommand{\D}{\mathcal{D}}\) +\(\newcommand{\id}{\mathrm{id}}\) +\(\newcommand{\ob}[1]{\mathrm{ob}(#1)}\) +\(\newcommand{\hom}[1]{\mathrm{hom}(#1)}\) +\(\newcommand{\Set}{\mathbf{Set}}\) +\(\newcommand{\Mon}{\mathbf{Mon}}\) +\(\newcommand{\Vec}{\mathbf{Vec}}\) +\(\newcommand{\Grp}{\mathbf{Grp}}\) +\(\newcommand{\Rng}{\mathbf{Rng}}\) +\(\newcommand{\ML}{\mathbf{ML}}\) +\(\newcommand{\Hask}{\mathbf{Hask}}\) +\(\newcommand{\Cat}{\mathbf{Cat}}\) +\(\newcommand{\fmap}{\mathtt{fmap}}\) +
+ +
+

Category Theory & Programming

+
for Rivieria Scala Clojure (Note this presentation uses Haskell)
+by Yann Esposito +
+ + @yogsototh, + + + +yogsototh + +
+
+
ENTER FULLSCREEN
+ HTML presentation: use arrows, space, swipe to navigate. +
+
+
+

Plan

+ +
+
+

Not really about: Cat & glory

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

General Overview

+
+Samuel Eilenberg Saunders Mac Lane +
+ +

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

+

Certainly one of the more abstract branches of math

+ +

+★: When is one thing equal to some other thing?, Barry Mazur, 2007
☆: Physics, Topology, Logic and Computation: A Rosetta Stone, John C. Baez, Mike Stay, 2009 +

+ + +
+
+

From a Programmer perspective

+
+

Category Theory is a new language/framework for Math

+
+ +
+
+

Math Programming relation

+Buddha Fractal +

Programming is doing Math

+

Strong relations between type theory and category theory.

+

Not convinced?
Certainly a vocabulary problem.

+

One of the goal of Category Theory is to create a homogeneous vocabulary between different disciplines.

+
+
+

Vocabulary

+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, ...

+
+
+
+

Programmer Translation

+lolcat + + + + + + + + +
+Mathematician + +Programmer +
+Morphism + +Arrow +
+Monoid + +String-like +
+Preorder + +Acyclic graph +
+Isomorph + +The same +
+Natural transformation + +rearrangement function +
+Funny Category + +LOLCat +
+ + +
+
+

Plan

+ +
+
+

Category

+ +

A way of representing things and ways to go between things.

+ +

A Category \(\mathcal{C}\) is defined by:

+ +
+
+

Category: Objects

+ +objects + +

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

+
+
+

Category: Morphisms

+ +morphisms + +

\(A\) and \(B\) objects of \(\C\)
+\(\hom{A,B}\) is a collection of morphisms
+\(f:A→B\) denote the fact \(f\) belongs to \(\hom{A,B}\)

+

\(\hom{\C}\) the collection of all morphisms of \(\C\)

+
+
+

Category: Composition

+

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

+composition +
+
+

Category laws: neutral element

+

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

+identity +
+
+

Category laws: Associativity

+

Composition is associative:

+associative composition +
+
+

Commutative diagrams

+ +

Two path with the same source and destination are equal.

+
+ Commutative Diagram (Associativity) +
+ \((h∘g)∘f = h∘(g∘f) \) +
+
+
+ Commutative Diagram (Identity law) +
+ \(id_B∘f = f = f∘id_A \) +
+
+
+
+

Question Time!

+ +
+ +
+- French-only joke - +
+
+
+
+

Can this be a category?

+

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

+
+ Category example 1 +
+ YES +
+
+
+ Category example 2 +
+ no candidate for \(g∘f\) +
NO +
+
+
+ Category example 3 +
+ YES +
+
+
+
+

Can this be a category?

+
+ Category example 4 +
+ no candidate for \(f:C→B\) +
NO +
+
+
+ Category example 5 +
+ \((h∘g)∘f=\id_B∘f=f\)
+ \(h∘(g∘f)=h∘\id_A=h\)
+ but \(h≠f\)
+ NO +
+
+
+
+

Categories Examples

+ +
+Basket of cats +
+- Basket of Cats - +
+
+
+
+

Category \(\Set\)

+ + + + +
+
+

Categories Everywhere?

+Cats everywhere + +
+
+

Smaller Examples

+ +

Strings

+Monoids are one object categories + + +
+
+

Finite Example?

+ +

Graph

+
+Each graph is a category +
+ + +
+
+

Number construction

+ +

Each Numbers as a whole category

+Each number as a category +
+
+

Degenerated Categories: Monoids

+ +Monoids are one object categories +

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

+

Only one object.

+

Examples:

+ +
+
+

Degenerated Categories: Preorders \((P,≤)\)

+ + + +

At most one morphism between two objects.

+ +preorder category +
+
+

Degenerated Categories: Discrete Categories

+ +Any set can be a category +

Any Set

+

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

+

Only identities

+
+
+

Choice

+

The same object can be seen in many different way as a category.

+

You can choose what are object, morphisms and composition.

+

ex: Str and discrete(Σ*)

+
+
+

Categorical Properties

+ +

Any property which can be expressed in term of category, objects, morphism and composition.

+ + +
+
+

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.

+

A≌B means A and B are essentially the same.
In Category Theory, = is in fact mostly ≌.
For example in commutative diagrams.

+
+
+

Functor

+ +

A functor is a mapping between two categories. +Let \(\C\) and \(\D\) be two categories. +A functor \(\F\) from \(\C\) to \(\D\):

+ +
+
+

Functor Example (ob → ob)

+ +Functor +
+
+

Functor Example (hom → hom)

+ +Functor +
+
+

Functor Example

+ +Functor +
+
+

Endofunctors

+ +

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

+Endofunctor +
+
+

Category of Categories

+ + + +

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

+ +
+
+

Plan

+ +
+
+

Hask

+ +

Category \(\Hask\):

+ +Haskell Category Representation + + + +

Forget glitches because of undefined.

+
+
+

Haskell Kinds

+

In Haskell some types can take type variable(s). Typically: [a].

+

Types have kinds; The kind is to type what type is to function. Kind are the types for types (so meta).

+
Int, Char :: *
+[], Maybe :: * -> *
+(,), (->) :: * -> * -> *
+[Int], Maybe Char, Maybe [Int] :: *
+
+
+

Haskell Types

+

Sometimes, the type determine a lot about the function★:

+
fst :: (a,b) -> a -- Only one choice
+snd :: (a,b) -> b -- Only one choice
+f :: a -> [a]     -- Many choices
+-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
+
+? :: [a] -> [a] -- Many choices
+-- can only rearrange: duplicate/remove/reorder elements
+-- for example: the type of addOne isn't [a] -> [a]
+addOne l = map (+1) l
+-- The (+1) force 'a' to be a Num.
+ +

+

★:Theorems for free!, Philip Wadler, 1989

+
+
+

Haskell Functor vs \(\Hask\) Functor

+ +

A Haskell Functor is a type F :: * -> * which belong to the type class Functor ; thus instantiate +fmap :: (a -> b) -> (F a -> F b). + +

& F: \(\ob{\Hask}→\ob{\Hask}\)
& fmap: \(\hom{\Hask}→\hom{\Hask}\) + +

The couple (F,fmap) is a \(\Hask\)'s functor if for any x :: F a:

+ +
+
+

Haskell Functors Example: Maybe

+ +
data Maybe a = Just a | Nothing
+instance Functor Maybe where
+    fmap :: (a -> b) -> (Maybe a -> Maybe b)
+    fmap f (Just a) = Just (f a)
+    fmap f Nothing = Nothing
+
fmap (+1) (Just 1) == Just 2
+fmap (+1) Nothing  == Nothing
+fmap head (Just [1,2,3]) == Just 1
+
+
+

Haskell Functors Example: List

+ +
instance Functor ([]) where
+	fmap :: (a -> b) -> [a] -> [b]
+	fmap = map
+
fmap (+1) [1,2,3]           == [2,3,4]
+fmap (+1) []                == []
+fmap head [[1,2,3],[4,5,6]] == [1,4]
+
+
+

Haskell Functors for the programmer

+

Functor is a type class used for types that can be mapped over.

+ +
+
+

Haskell Functor intuition

+ +

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

+ +Haskell Functor as a box play +

+
+

Haskell Functor properties

+ +

Haskell Functors are:

+ + +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

"Non Haskell" Hask's Functors

+

A simple basic example is the \(id_\Hask\) functor. It simply cannot be expressed as a couple (F,fmap) where

+ +

Another example:

+ +
+
+

Also Functor inside \(\Hask\)

+

\(\mathtt{[a]}∈\ob{\Hask}\) but is also a category. Idem for Int.

+

length is a Functor from the category [a] to the cateogry Int:

+ +

⇒

+ +
+ +
+
+

Category of \(\Hask\) Endofunctors

+Category of Hask endofunctors +
+
+

Category of Functors

+

If \(\C\) is small (\(\hom{\C}\) is a set). All functors from \(\C\) to some category \(\D\) form the category \(\mathrm{Func}(\C,\D)\).

+ +

\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).

+
+
+

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.

+

ex: between Haskell functors; F a -> G a
Rearragement functions only.

+
+
+

Natural Transformation Examples (1/4)

+
data Tree a = Empty | Node a [Tree a]
+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 +
+ + +
+
+

Natural Transformation Examples (2/4)

+
data Tree a = Empty | Node a [Tree a]
+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.
+
+ + +
+
+

Natural Transformation Examples (3/4)

+
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 +
+ + +
+
+

Natural Transformation Examples (4/4)

+
mToList :: Maybe a -> [a]
+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 +
+relation between [] and Maybe
There is no isomorphism.
Hint: Bool lists longer than 1.
+
+ + +
+
+

Composition problem

+

The Problem; example with lists:

+
f x = [x]       ⇒ f 1 = [1]   ⇒ (f.f) 1 = [[1]] ✗
+g x = [x+1]     ⇒ g 1 = [2]   ⇒ (g.g) 1 = ERROR [2]+1 ✗
+h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h.h) 1 = ERROR [2,3]+1 ✗
+ +

The same problem with most f :: a -> F a functions and functor F.

+
+
+

Composition Fixable?

+

How to fix that? We want to construct an operator which is able to compose:

+

f :: a -> F b & g :: b -> F c.

+

More specifically we want to create an operator ◎ of type

+

◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)

+

Note: if F = I, ◎ = (.).

+
+
+

Fix Composition (1/2)

+

Goal, find: ◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)
f :: a -> F b, g :: b -> F c:

+ +
+
+

Fix Composition (2/2)

+

Goal, find: ◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)
f :: a -> F b, g :: b -> F c, f x :: F b:

+ +
+
+

Necessary laws

+

For ◎ to work like composition, we need join to hold the following properties:

+ +
+
+

Klesli composition

+

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] ✓
+ + +
+
+

We reinvented Monads!

+

A monad is a triplet (M,⊙,η) where

+ +

Satisfying

+ +
+
+

Compare with Monoid

+

A Monoid is a triplet \((E,∙,e)\) s.t.

+ +

Satisfying

+ +
+
+

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.

+
+
+
+

Example: List

+ +
-- In Haskell ⊙ is "join" in "Control.Monad"
+join :: [[a]] -> [a]
+join = concat
+
+-- In Haskell the "return" function (unfortunate name)
+η :: a -> [a]
+η x = [x]
+ + +
+
+

Example: List (law verification)

+

Example: List is a functor (join is ⊙)

+ +
join [ join [[x,y,...,z]] ] = join [[x,y,...,z]]
+                            = join (join [[[x,y,...,z]]])
+join (η [x]) = [x] = join [η x]
+ +

Therefore ([],join,η) is a monad.

+
+
+

Monads useful?

+

A LOT of monad tutorial on the net. Just one example; the State Monad

+

DrawScene to State Screen DrawScene ; still pure.

+
main = drawImage (width,height)
+
+drawImage :: Screen -> DrawScene
+drawImage screen = do
+    drawPoint p screen
+    drawCircle c screen
+    drawRectangle r screen
+
+drawPoint point screen = ...
+drawCircle circle screen = ...
+drawRectangle rectangle screen = ...
+
main = do
+    put (Screen 1024 768)
+    drawImage
+
+drawImage :: State Screen DrawScene
+drawImage = do
+    drawPoint p
+    drawCircle c
+    drawRectangle r
+
+drawPoint :: Point -> State Screen DrawScene
+drawPoint p = do
+    Screen width height <- get
+    ...
+
+
+

fold

+fold +
+
+

κατα-morphism

+catamorphism +
+
+

κατα-morphism: fold generalization

+

acc type of the "accumulator":
fold :: (acc -> a -> acc) -> acc -> [a] -> acc

+

Idea: put the accumulated value inside the type.

+
-- Equivalent to fold (+1) 0 "cata"
+(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' Nil))))
+(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' 0))))
+(Cons 'c' (Cons 'a' (Cons 't' 1)))
+(Cons 'c' (Cons 'a' 2))
+(Cons 'c' 3)
+4
+ + +

But where are all the informations? (+1) and 0?

+
+
+

κατα-morphism: Missing Information

+

Where is the missing information?

+ +

First example, make length on [Char]

+
+
+

κατα-morphism: Type work

+

+data StrF a = Cons Char a | Nil
+data Str' = StrF Str'
+
+-- generalize the construction of Str to other datatype
+-- Mu: type fixed point
+-- Mu :: (* -> *) -> *
+
+data Mu f = InF { outF :: f (Mu f) }
+data Str = Mu StrF
+
+-- Example
+foo=InF { outF = Cons 'f'
+        (InF { outF = Cons 'o'
+            (InF { outF = Cons 'o'
+                (InF { outF = Nil })})})}
+ + +
+
+

κατα-morphism: missing information retrieved

+
type Algebra f a = f a -> a
+instance Functor (StrF a) =
+    fmap f (Cons c x) = Cons c (f x)
+    fmap _ Nil = Nil
+ +
cata :: Functor f => Algebra f a -> Mu f -> a
+cata f = f . fmap (cata f) . outF
+ + + +
+
+

κατα-morphism: Finally length

+

All needed information for making length.

+
instance Functor (StrF a) =
+    fmap f (Cons c x) = Cons c (f x)
+    fmap _ Nil = Nil
+
+length' :: Str -> Int
+length' = cata phi where
+    phi :: Algebra StrF Int -- StrF Int -> Int
+    phi (Cons a b) = 1 + b
+    phi Nil = 0
+
+main = do
+    l <- length' $ stringToStr "Toto"
+    ...
+
+
+

κατα-morphism: extension to Trees

+

Once you get the trick, it is easy to extent to most Functor.

+
type Tree = Mu TreeF
+data TreeF x = Node Int [x]
+
+instance Functor TreeF where
+  fmap f (Node e xs) = Node e (fmap f xs)
+
+depth = cata phi where
+  phi :: Algebra TreeF Int -- TreeF Int -> Int
+  phi (Node x sons) = 1 + foldr max 0 sons
+
+
+

Conclusion

+

Category Theory oriented Programming:

+ +

No cat were harmed in the making of this presentation.

+
diff --git a/content/html/fr/blog/Category-Theory-Presentation.erb b/content/html/fr/blog/Category-Theory-Presentation.erb new file mode 100644 index 000000000..7fde2f554 --- /dev/null +++ b/content/html/fr/blog/Category-Theory-Presentation.erb @@ -0,0 +1,1000 @@ +----- +isHidden: false +menupriority: 1 +kind: article +created_at: 2012-12-12T10:06:40+02:00 +title: Category Theory Presentation +author_name: Yann Esposito +author_uri: yannesposito.com +tags: + - category theory + - math + - functor +----- +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).

+ + + +

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

+ +
+\(\newcommand{\F}{\mathbf{F}}\) +\(\newcommand{\E}{\mathbf{E}}\) +\(\newcommand{\C}{\mathcal{C}}\) +\(\newcommand{\D}{\mathcal{D}}\) +\(\newcommand{\id}{\mathrm{id}}\) +\(\newcommand{\ob}[1]{\mathrm{ob}(#1)}\) +\(\newcommand{\hom}[1]{\mathrm{hom}(#1)}\) +\(\newcommand{\Set}{\mathbf{Set}}\) +\(\newcommand{\Mon}{\mathbf{Mon}}\) +\(\newcommand{\Vec}{\mathbf{Vec}}\) +\(\newcommand{\Grp}{\mathbf{Grp}}\) +\(\newcommand{\Rng}{\mathbf{Rng}}\) +\(\newcommand{\ML}{\mathbf{ML}}\) +\(\newcommand{\Hask}{\mathbf{Hask}}\) +\(\newcommand{\Cat}{\mathbf{Cat}}\) +\(\newcommand{\fmap}{\mathtt{fmap}}\) +
+ +
+

Category Theory & Programming

+
for Rivieria Scala Clojure (Note this presentation uses Haskell)
+by Yann Esposito +
+ + @yogsototh, + + + +yogsototh + +
+
+
ENTER FULLSCREEN
+ HTML presentation: use arrows, space, swipe to navigate. +
+
+
+

Plan

+ +
+
+

Not really about: Cat & glory

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

General Overview

+
+Samuel Eilenberg Saunders Mac Lane +
+ +

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

+

Certainly one of the more abstract branches of math

+ +

+★: When is one thing equal to some other thing?, Barry Mazur, 2007
☆: Physics, Topology, Logic and Computation: A Rosetta Stone, John C. Baez, Mike Stay, 2009 +

+ + +
+
+

From a Programmer perspective

+
+

Category Theory is a new language/framework for Math

+
+ +
+
+

Math Programming relation

+Buddha Fractal +

Programming is doing Math

+

Strong relations between type theory and category theory.

+

Not convinced?
Certainly a vocabulary problem.

+

One of the goal of Category Theory is to create a homogeneous vocabulary between different disciplines.

+
+
+

Vocabulary

+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, ...

+
+
+
+

Programmer Translation

+lolcat + + + + + + + + +
+Mathematician + +Programmer +
+Morphism + +Arrow +
+Monoid + +String-like +
+Preorder + +Acyclic graph +
+Isomorph + +The same +
+Natural transformation + +rearrangement function +
+Funny Category + +LOLCat +
+ + +
+
+

Plan

+ +
+
+

Category

+ +

A way of representing things and ways to go between things.

+ +

A Category \(\mathcal{C}\) is defined by:

+ +
+
+

Category: Objects

+ +objects + +

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

+
+
+

Category: Morphisms

+ +morphisms + +

\(A\) and \(B\) objects of \(\C\)
+\(\hom{A,B}\) is a collection of morphisms
+\(f:A→B\) denote the fact \(f\) belongs to \(\hom{A,B}\)

+

\(\hom{\C}\) the collection of all morphisms of \(\C\)

+
+
+

Category: Composition

+

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

+composition +
+
+

Category laws: neutral element

+

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

+identity +
+
+

Category laws: Associativity

+

Composition is associative:

+associative composition +
+
+

Commutative diagrams

+ +

Two path with the same source and destination are equal.

+
+ Commutative Diagram (Associativity) +
+ \((h∘g)∘f = h∘(g∘f) \) +
+
+
+ Commutative Diagram (Identity law) +
+ \(id_B∘f = f = f∘id_A \) +
+
+
+
+

Question Time!

+ +
+ +
+- French-only joke - +
+
+
+
+

Can this be a category?

+

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

+
+ Category example 1 +
+ YES +
+
+
+ Category example 2 +
+ no candidate for \(g∘f\) +
NO +
+
+
+ Category example 3 +
+ YES +
+
+
+
+

Can this be a category?

+
+ Category example 4 +
+ no candidate for \(f:C→B\) +
NO +
+
+
+ Category example 5 +
+ \((h∘g)∘f=\id_B∘f=f\)
+ \(h∘(g∘f)=h∘\id_A=h\)
+ but \(h≠f\)
+ NO +
+
+
+
+

Categories Examples

+ +
+Basket of cats +
+- Basket of Cats - +
+
+
+
+

Category \(\Set\)

+ + + + +
+
+

Categories Everywhere?

+Cats everywhere + +
+
+

Smaller Examples

+ +

Strings

+Monoids are one object categories + + +
+
+

Finite Example?

+ +

Graph

+
+Each graph is a category +
+ + +
+
+

Number construction

+ +

Each Numbers as a whole category

+Each number as a category +
+
+

Degenerated Categories: Monoids

+ +Monoids are one object categories +

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

+

Only one object.

+

Examples:

+ +
+
+

Degenerated Categories: Preorders \((P,≤)\)

+ + + +

At most one morphism between two objects.

+ +preorder category +
+
+

Degenerated Categories: Discrete Categories

+ +Any set can be a category +

Any Set

+

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

+

Only identities

+
+
+

Choice

+

The same object can be seen in many different way as a category.

+

You can choose what are object, morphisms and composition.

+

ex: Str and discrete(Σ*)

+
+
+

Categorical Properties

+ +

Any property which can be expressed in term of category, objects, morphism and composition.

+ + +
+
+

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.

+

A≌B means A and B are essentially the same.
In Category Theory, = is in fact mostly ≌.
For example in commutative diagrams.

+
+
+

Functor

+ +

A functor is a mapping between two categories. +Let \(\C\) and \(\D\) be two categories. +A functor \(\F\) from \(\C\) to \(\D\):

+ +
+
+

Functor Example (ob → ob)

+ +Functor +
+
+

Functor Example (hom → hom)

+ +Functor +
+
+

Functor Example

+ +Functor +
+
+

Endofunctors

+ +

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

+Endofunctor +
+
+

Category of Categories

+ + + +

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

+ +
+
+

Plan

+ +
+
+

Hask

+ +

Category \(\Hask\):

+ +Haskell Category Representation + + + +

Forget glitches because of undefined.

+
+
+

Haskell Kinds

+

In Haskell some types can take type variable(s). Typically: [a].

+

Types have kinds; The kind is to type what type is to function. Kind are the types for types (so meta).

+
Int, Char :: *
+[], Maybe :: * -> *
+(,), (->) :: * -> * -> *
+[Int], Maybe Char, Maybe [Int] :: *
+
+
+

Haskell Types

+

Sometimes, the type determine a lot about the function★:

+
fst :: (a,b) -> a -- Only one choice
+snd :: (a,b) -> b -- Only one choice
+f :: a -> [a]     -- Many choices
+-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
+
+? :: [a] -> [a] -- Many choices
+-- can only rearrange: duplicate/remove/reorder elements
+-- for example: the type of addOne isn't [a] -> [a]
+addOne l = map (+1) l
+-- The (+1) force 'a' to be a Num.
+ +

+

★:Theorems for free!, Philip Wadler, 1989

+
+
+

Haskell Functor vs \(\Hask\) Functor

+ +

A Haskell Functor is a type F :: * -> * which belong to the type class Functor ; thus instantiate +fmap :: (a -> b) -> (F a -> F b). + +

& F: \(\ob{\Hask}→\ob{\Hask}\)
& fmap: \(\hom{\Hask}→\hom{\Hask}\) + +

The couple (F,fmap) is a \(\Hask\)'s functor if for any x :: F a:

+ +
+
+

Haskell Functors Example: Maybe

+ +
data Maybe a = Just a | Nothing
+instance Functor Maybe where
+    fmap :: (a -> b) -> (Maybe a -> Maybe b)
+    fmap f (Just a) = Just (f a)
+    fmap f Nothing = Nothing
+
fmap (+1) (Just 1) == Just 2
+fmap (+1) Nothing  == Nothing
+fmap head (Just [1,2,3]) == Just 1
+
+
+

Haskell Functors Example: List

+ +
instance Functor ([]) where
+	fmap :: (a -> b) -> [a] -> [b]
+	fmap = map
+
fmap (+1) [1,2,3]           == [2,3,4]
+fmap (+1) []                == []
+fmap head [[1,2,3],[4,5,6]] == [1,4]
+
+
+

Haskell Functors for the programmer

+

Functor is a type class used for types that can be mapped over.

+ +
+
+

Haskell Functor intuition

+ +

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

+ +Haskell Functor as a box play +

+
+

Haskell Functor properties

+ +

Haskell Functors are:

+ + +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

"Non Haskell" Hask's Functors

+

A simple basic example is the \(id_\Hask\) functor. It simply cannot be expressed as a couple (F,fmap) where

+ +

Another example:

+ +
+
+

Also Functor inside \(\Hask\)

+

\(\mathtt{[a]}∈\ob{\Hask}\) but is also a category. Idem for Int.

+

length is a Functor from the category [a] to the cateogry Int:

+ +

⇒

+ +
+ +
+
+

Category of \(\Hask\) Endofunctors

+Category of Hask endofunctors +
+
+

Category of Functors

+

If \(\C\) is small (\(\hom{\C}\) is a set). All functors from \(\C\) to some category \(\D\) form the category \(\mathrm{Func}(\C,\D)\).

+ +

\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).

+
+
+

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.

+

ex: between Haskell functors; F a -> G a
Rearragement functions only.

+
+
+

Natural Transformation Examples (1/4)

+
data Tree a = Empty | Node a [Tree a]
+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 +
+ + +
+
+

Natural Transformation Examples (2/4)

+
data Tree a = Empty | Node a [Tree a]
+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.
+
+ + +
+
+

Natural Transformation Examples (3/4)

+
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 +
+ + +
+
+

Natural Transformation Examples (4/4)

+
mToList :: Maybe a -> [a]
+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 +
+relation between [] and Maybe
There is no isomorphism.
Hint: Bool lists longer than 1.
+
+ + +
+
+

Composition problem

+

The Problem; example with lists:

+
f x = [x]       ⇒ f 1 = [1]   ⇒ (f.f) 1 = [[1]] ✗
+g x = [x+1]     ⇒ g 1 = [2]   ⇒ (g.g) 1 = ERROR [2]+1 ✗
+h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h.h) 1 = ERROR [2,3]+1 ✗
+ +

The same problem with most f :: a -> F a functions and functor F.

+
+
+

Composition Fixable?

+

How to fix that? We want to construct an operator which is able to compose:

+

f :: a -> F b & g :: b -> F c.

+

More specifically we want to create an operator ◎ of type

+

◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)

+

Note: if F = I, ◎ = (.).

+
+
+

Fix Composition (1/2)

+

Goal, find: ◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)
f :: a -> F b, g :: b -> F c:

+ +
+
+

Fix Composition (2/2)

+

Goal, find: ◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)
f :: a -> F b, g :: b -> F c, f x :: F b:

+ +
+
+

Necessary laws

+

For ◎ to work like composition, we need join to hold the following properties:

+ +
+
+

Klesli composition

+

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] ✓
+ + +
+
+

We reinvented Monads!

+

A monad is a triplet (M,⊙,η) where

+ +

Satisfying

+ +
+
+

Compare with Monoid

+

A Monoid is a triplet \((E,∙,e)\) s.t.

+ +

Satisfying

+ +
+
+

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.

+
+
+
+

Example: List

+ +
-- In Haskell ⊙ is "join" in "Control.Monad"
+join :: [[a]] -> [a]
+join = concat
+
+-- In Haskell the "return" function (unfortunate name)
+η :: a -> [a]
+η x = [x]
+ + +
+
+

Example: List (law verification)

+

Example: List is a functor (join is ⊙)

+ +
join [ join [[x,y,...,z]] ] = join [[x,y,...,z]]
+                            = join (join [[[x,y,...,z]]])
+join (η [x]) = [x] = join [η x]
+ +

Therefore ([],join,η) is a monad.

+
+
+

Monads useful?

+

A LOT of monad tutorial on the net. Just one example; the State Monad

+

DrawScene to State Screen DrawScene ; still pure.

+
main = drawImage (width,height)
+
+drawImage :: Screen -> DrawScene
+drawImage screen = do
+    drawPoint p screen
+    drawCircle c screen
+    drawRectangle r screen
+
+drawPoint point screen = ...
+drawCircle circle screen = ...
+drawRectangle rectangle screen = ...
+
main = do
+    put (Screen 1024 768)
+    drawImage
+
+drawImage :: State Screen DrawScene
+drawImage = do
+    drawPoint p
+    drawCircle c
+    drawRectangle r
+
+drawPoint :: Point -> State Screen DrawScene
+drawPoint p = do
+    Screen width height <- get
+    ...
+
+
+

fold

+fold +
+
+

κατα-morphism

+catamorphism +
+
+

κατα-morphism: fold generalization

+

acc type of the "accumulator":
fold :: (acc -> a -> acc) -> acc -> [a] -> acc

+

Idea: put the accumulated value inside the type.

+
-- Equivalent to fold (+1) 0 "cata"
+(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' Nil))))
+(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' 0))))
+(Cons 'c' (Cons 'a' (Cons 't' 1)))
+(Cons 'c' (Cons 'a' 2))
+(Cons 'c' 3)
+4
+ + +

But where are all the informations? (+1) and 0?

+
+
+

κατα-morphism: Missing Information

+

Where is the missing information?

+ +

First example, make length on [Char]

+
+
+

κατα-morphism: Type work

+

+data StrF a = Cons Char a | Nil
+data Str' = StrF Str'
+
+-- generalize the construction of Str to other datatype
+-- Mu: type fixed point
+-- Mu :: (* -> *) -> *
+
+data Mu f = InF { outF :: f (Mu f) }
+data Str = Mu StrF
+
+-- Example
+foo=InF { outF = Cons 'f'
+        (InF { outF = Cons 'o'
+            (InF { outF = Cons 'o'
+                (InF { outF = Nil })})})}
+ + +
+
+

κατα-morphism: missing information retrieved

+
type Algebra f a = f a -> a
+instance Functor (StrF a) =
+    fmap f (Cons c x) = Cons c (f x)
+    fmap _ Nil = Nil
+ +
cata :: Functor f => Algebra f a -> Mu f -> a
+cata f = f . fmap (cata f) . outF
+ + + +
+
+

κατα-morphism: Finally length

+

All needed information for making length.

+
instance Functor (StrF a) =
+    fmap f (Cons c x) = Cons c (f x)
+    fmap _ Nil = Nil
+
+length' :: Str -> Int
+length' = cata phi where
+    phi :: Algebra StrF Int -- StrF Int -> Int
+    phi (Cons a b) = 1 + b
+    phi Nil = 0
+
+main = do
+    l <- length' $ stringToStr "Toto"
+    ...
+
+
+

κατα-morphism: extension to Trees

+

Once you get the trick, it is easy to extent to most Functor.

+
type Tree = Mu TreeF
+data TreeF x = Node Int [x]
+
+instance Functor TreeF where
+  fmap f (Node e xs) = Node e (fmap f xs)
+
+depth = cata phi where
+  phi :: Algebra TreeF Int -- TreeF Int -> Int
+  phi (Node x sons) = 1 + foldr max 0 sons
+
+
+

Conclusion

+

Category Theory oriented Programming:

+ +

No cat were harmed in the making of this presentation.

+
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-=== $LOAD_PATH: - - 0. /var/lib/gems/1.9.1/gems/nanoc-3.4.0/lib - 1. /var/lib/gems/1.9.1/gems/nanoc-3.4.0/lib - 2. /var/lib/gems/1.9.1/gems/colored-1.2/lib - 3. /var/lib/gems/1.9.1/gems/cri-2.3.0/lib - 4. /var/lib/gems/1.9.1/gems/builder-3.0.0/lib - 5. /var/lib/gems/1.9.1/gems/kramdown-0.13.7/lib - 6. /var/lib/gems/1.9.1/gems/sass-3.1.19/lib - 7. /usr/local/lib/site_ruby/1.9.1 - 8. /usr/local/lib/site_ruby/1.9.1/i686-linux - 9. /usr/local/lib/site_ruby - 10. /usr/lib/ruby/vendor_ruby/1.9.1 - 11. /usr/lib/ruby/vendor_ruby/1.9.1/i686-linux - 12. /usr/lib/ruby/vendor_ruby - 13. /usr/lib/ruby/1.9.1 - 14. /usr/lib/ruby/1.9.1/i686-linux diff --git a/img_latest_blog_dir b/img_latest_blog_dir index 1c72e24d9..fa9bee8c2 120000 --- a/img_latest_blog_dir +++ b/img_latest_blog_dir @@ -1 +1 @@ -output/Scratch/img/blog/Haskell-OpenGL-Mandelbrot \ No newline at end of file +./output/Scratch/img/blog/Category-Theory-Presentation \ No newline at end of file diff --git a/latest.ymd b/latest.ymd index 1b8b8e4aa..5a7fd9383 120000 --- a/latest.ymd +++ b/latest.ymd @@ -1 +1 @@ -multi/blog/Haskell-OpenGL-Mandelbrot.md \ No newline at end of file +multi/blog/Category-Theory-Presentation.md \ No newline at end of file diff --git a/multi/blog/Category-Theory-Presentation.erb b/multi/blog/Category-Theory-Presentation.erb new file mode 100644 index 000000000..d0b5cb676 --- /dev/null +++ b/multi/blog/Category-Theory-Presentation.erb @@ -0,0 +1,1001 @@ +----- +isHidden: false +menupriority: 1 +kind: article +created_at: 2012-12-12T10:06:40+02:00 +en: title: Category Theory Presentation +fr: title: Category Theory Presentation +author_name: Yann Esposito +author_uri: yannesposito.com +tags: + - category theory + - math + - functor +----- +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).

+ + + +

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

+ +
+\(\newcommand{\F}{\mathbf{F}}\) +\(\newcommand{\E}{\mathbf{E}}\) +\(\newcommand{\C}{\mathcal{C}}\) +\(\newcommand{\D}{\mathcal{D}}\) +\(\newcommand{\id}{\mathrm{id}}\) +\(\newcommand{\ob}[1]{\mathrm{ob}(#1)}\) +\(\newcommand{\hom}[1]{\mathrm{hom}(#1)}\) +\(\newcommand{\Set}{\mathbf{Set}}\) +\(\newcommand{\Mon}{\mathbf{Mon}}\) +\(\newcommand{\Vec}{\mathbf{Vec}}\) +\(\newcommand{\Grp}{\mathbf{Grp}}\) +\(\newcommand{\Rng}{\mathbf{Rng}}\) +\(\newcommand{\ML}{\mathbf{ML}}\) +\(\newcommand{\Hask}{\mathbf{Hask}}\) +\(\newcommand{\Cat}{\mathbf{Cat}}\) +\(\newcommand{\fmap}{\mathtt{fmap}}\) +
+ +
+

Category Theory & Programming

+
for Rivieria Scala Clojure (Note this presentation uses Haskell)
+by Yann Esposito +
+ + @yogsototh, + + + +yogsototh + +
+
+
ENTER FULLSCREEN
+ HTML presentation: use arrows, space, swipe to navigate. +
+
+
+

Plan

+
    +
  • General overview
    • +
  • Definitions
    • +
  • Applications
    • +
+
+
+

Not really about: Cat & glory

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

General Overview

+
+Samuel Eilenberg Saunders Mac Lane +
+ +

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

+

Certainly one of the more abstract branches of math

+
    +
  • New math foundation
    formalism abstraction, package entire theory★
    • +
  • Bridge between disciplines
    Physics, Quantum Physics, Topology, Logic, Computer Science☆
    • +
+

+★: When is one thing equal to some other thing?, Barry Mazur, 2007
☆: Physics, Topology, Logic and Computation: A Rosetta Stone, John C. Baez, Mike Stay, 2009 +

+ + +
+
+

From a Programmer perspective

+
+

Category Theory is a new language/framework for Math

+
+
    +
  • Another way of thinking
    • +
  • Extremely efficient for generalization
    • +
+
+
+

Math Programming relation

+Buddha Fractal +

Programming is doing Math

+

Strong relations between type theory and category theory.

+

Not convinced?
Certainly a vocabulary problem.

+

One of the goal of Category Theory is to create a homogeneous vocabulary between different disciplines.

+
+
+

Vocabulary

+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, ...

+
+
+
+

Programmer Translation

+lolcat + + + + + + + + +
+Mathematician + +Programmer +
+Morphism + +Arrow +
+Monoid + +String-like +
+Preorder + +Acyclic graph +
+Isomorph + +The same +
+Natural transformation + +rearrangement function +
+Funny Category + +LOLCat +
+ + +
+
+

Plan

+
    +
  • General overview
    • +
  • Definitions +
      +
    • Category
      • +
    • Intuition
      • +
    • Examples
      • +
    • Functor
      • +
    • Examples
      • +
    +
    • +
  • Applications
    • +
+
+
+

Category

+ +

A way of representing things and ways to go between things.

+ +

A Category \(\mathcal{C}\) is defined by:

+
    +
  • Objects \(\ob{C}\),
    • +
  • Morphisms \(\hom{C}\),
    • +
  • a Composition law (∘)
    • +
  • obeying some Properties.
    • +
+
+
+

Category: Objects

+ +objects + +

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

+
+
+

Category: Morphisms

+ +morphisms + +

\(A\) and \(B\) objects of \(\C\)
+\(\hom{A,B}\) is a collection of morphisms
+\(f:A→B\) denote the fact \(f\) belongs to \(\hom{A,B}\)

+

\(\hom{\C}\) the collection of all morphisms of \(\C\)

+
+
+

Category: Composition

+

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

+composition +
+
+

Category laws: neutral element

+

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

+identity +
+
+

Category laws: Associativity

+

Composition is associative:

+associative composition +
+
+

Commutative diagrams

+ +

Two path with the same source and destination are equal.

+
+ Commutative Diagram (Associativity) +
+ \((h∘g)∘f = h∘(g∘f) \) +
+
+
+ Commutative Diagram (Identity law) +
+ \(id_B∘f = f = f∘id_A \) +
+
+
+
+

Question Time!

+ +
+ +
+- French-only joke - +
+
+
+
+

Can this be a category?

+

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

+
+ Category example 1 +
+ YES +
+
+
+ Category example 2 +
+ no candidate for \(g∘f\) +
NO +
+
+
+ Category example 3 +
+ YES +
+
+
+
+

Can this be a category?

+
+ Category example 4 +
+ no candidate for \(f:C→B\) +
NO +
+
+
+ Category example 5 +
+ \((h∘g)∘f=\id_B∘f=f\)
+ \(h∘(g∘f)=h∘\id_A=h\)
+ but \(h≠f\)
+ NO +
+
+
+
+

Categories Examples

+ +
+Basket of cats +
+- Basket of Cats - +
+
+
+
+

Category \(\Set\)

+ +
    +
  • \(\ob{\Set}\) are all the sets
    • +
  • \(\hom{E,F}\) are all functions from \(E\) to \(F\)
    • +
  • ∘ is functions composition
    • +
+ +
    +
  • \(\ob{\Set}\) is a proper class ; not a set
    • +
  • \(\hom{E,F}\) is a set
    • +
  • \(\Set\) is then a locally small category
    • +
+
+
+

Categories Everywhere?

+Cats everywhere +
    +
  • \(\Mon\): (monoids, monoid morphisms,∘)
    • +
  • \(\Vec\): (Vectorial spaces, linear functions,∘)
    • +
  • \(\Grp\): (groups, group morphisms,∘)
    • +
  • \(\Rng\): (rings, ring morphisms,∘)
    • +
  • Any deductive system T: (theorems, proofs, proof concatenation)
    • +
  • \( \Hask\): (Haskell types, functions, (.) )
    • +
  • ...
    • +
+
+
+

Smaller Examples

+ +

Strings

+Monoids are one object categories +
    +
  • \(\ob{Str}\) is a singleton
    • +
  • \(\hom{Str}\) each string
    • +
  • ∘ is concatenation (++)
    • +
+
    +
  • "" ++ u = u = u ++ ""
    • +
  • (u ++ v) ++ w = u ++ (v ++ w)
    • +
+
+
+

Finite Example?

+ +

Graph

+
+Each graph is a category +
+
    +
  • \(\ob{G}\) are vertices
    • +
  • \(\hom{G}\) each path
    • +
  • ∘ is path concatenation
    • +
+
  • \(\ob{G}=\{X,Y,Z\}\), +
  • \(\hom{G}=\{ε,α,β,γ,αβ,βγ,...\}\) +
  • \(αβ∘γ=αβγ\) +
+
+
+

Number construction

+ +

Each Numbers as a whole category

+Each number as a category +
+
+

Degenerated Categories: Monoids

+ +Monoids are one object categories +

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

+

Only one object.

+

Examples:

+
  • (Integer,0,+), (Integer,1,*), +
  • (Strings,"",++), for each a, ([a],[],++) +
+
+
+

Degenerated Categories: Preorders \((P,≤)\)

+ +
  • \(\ob{P}={P}\), +
  • \(\hom{x,y}=\{x≤y\} ⇔ x≤y\), +
  • \((y≤z) \circ (x≤y) = (x≤z) \) +
+ +

At most one morphism between two objects.

+ +preorder category +
+
+

Degenerated Categories: Discrete Categories

+ +Any set can be a category +

Any Set

+

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

+

Only identities

+
+
+

Choice

+

The same object can be seen in many different way as a category.

+

You can choose what are object, morphisms and composition.

+

ex: Str and discrete(Σ*)

+
+
+

Categorical Properties

+ +

Any property which can be expressed in term of category, objects, morphism and composition.

+ +
  • Dual: \(\D\) is \(\C\) with reversed morphisms. +
  • Initial: \(Z\in\ob{\C}\) s.t. \(∀Y∈\ob{\C}, \#\hom{Z,Y}=1\) +
    Unique ("up to isormophism") +
  • Terminal: \(T\in\ob{\C}\) s.t. \(T\) is initial in the dual of \(\C\) +
  • Functor: structure preserving mapping between categories +
  • ... +
+
+
+

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.

+

A≌B means A and B are essentially the same.
In Category Theory, = is in fact mostly ≌.
For example in commutative diagrams.

+
+
+

Functor

+ +

A functor is a mapping between two categories. +Let \(\C\) and \(\D\) be two categories. +A functor \(\F\) from \(\C\) to \(\D\):

+
    +
  • Associate objects: \(A\in\ob{\C}\) to \(\F(A)\in\ob{\D}\)
    • +
  • Associate morphisms: \(f:A\to B\) to \(\F(f) : \F(A) \to \F(B)\) + such that +
      +
    • \( \F (\)\(\id_X\)\()= \)\(\id\)\(\vphantom{\id}_{\F(}\)\(\vphantom{\id}_X\)\(\vphantom{\id}_{)} \),
      • +
    • \( \F (\)\(g∘f\)\()= \)\( \F(\)\(g\)\() \)\(\circ\)\( \F(\)\(f\)\() \)
      • +
    +
    • +
+
+
+

Functor Example (ob → ob)

+ +Functor +
+
+

Functor Example (hom → hom)

+ +Functor +
+
+

Functor Example

+ +Functor +
+
+

Endofunctors

+ +

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

+Endofunctor +
+
+

Category of Categories

+ + + +

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

+
  • \(\ob{\Cat}\) are categories +
  • \(\hom{\Cat}\) are functors +
  • ∘ is functor composition +
+
+
+

Plan

+
    +
  • General overview
    • +
  • Definitions
    • +
  • Applications +
      +
    • \(\Hask\) category +
    • Functors +
    • Natural transformations +
    • Monads +
    • κατα-morphisms +
    +
    • +
+
+
+

Hask

+ +

Category \(\Hask\):

+ +Haskell Category Representation + +
  • +\(\ob{\Hask} = \) Haskell types +
  • +\(\hom{\Hask} = \) Haskell functions +
  • +∘ = (.) Haskell function composition +
+ +

Forget glitches because of undefined.

+
+
+

Haskell Kinds

+

In Haskell some types can take type variable(s). Typically: [a].

+

Types have kinds; The kind is to type what type is to function. Kind are the types for types (so meta).

+
Int, Char :: *
+[], Maybe :: * -> *
+(,), (->) :: * -> * -> *
+[Int], Maybe Char, Maybe [Int] :: *
+
+
+

Haskell Types

+

Sometimes, the type determine a lot about the function★:

+
fst :: (a,b) -> a -- Only one choice
+snd :: (a,b) -> b -- Only one choice
+f :: a -> [a]     -- Many choices
+-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
+
+? :: [a] -> [a] -- Many choices
+-- can only rearrange: duplicate/remove/reorder elements
+-- for example: the type of addOne isn't [a] -> [a]
+addOne l = map (+1) l
+-- The (+1) force 'a' to be a Num.
+ +

+

★:Theorems for free!, Philip Wadler, 1989

+
+
+

Haskell Functor vs \(\Hask\) Functor

+ +

A Haskell Functor is a type F :: * -> * which belong to the type class Functor ; thus instantiate +fmap :: (a -> b) -> (F a -> F b). + +

& F: \(\ob{\Hask}→\ob{\Hask}\)
& fmap: \(\hom{\Hask}→\hom{\Hask}\) + +

The couple (F,fmap) is a \(\Hask\)'s functor if for any x :: F a:

+
  • fmap id x = x +
  • fmap (f.g) x= (fmap f . fmap g) x +
+
+
+

Haskell Functors Example: Maybe

+ +
data Maybe a = Just a | Nothing
+instance Functor Maybe where
+    fmap :: (a -> b) -> (Maybe a -> Maybe b)
+    fmap f (Just a) = Just (f a)
+    fmap f Nothing = Nothing
+
fmap (+1) (Just 1) == Just 2
+fmap (+1) Nothing  == Nothing
+fmap head (Just [1,2,3]) == Just 1
+
+
+

Haskell Functors Example: List

+ +
instance Functor ([]) where
+	fmap :: (a -> b) -> [a] -> [b]
+	fmap = map
+
fmap (+1) [1,2,3]           == [2,3,4]
+fmap (+1) []                == []
+fmap head [[1,2,3],[4,5,6]] == [1,4]
+
+
+

Haskell Functors for the programmer

+

Functor is a type class used for types that can be mapped over.

+
    +
  • Containers: [], Trees, Map, HashMap...
    • +
  • "Feature Type": +
      +
    • Maybe a: help to handle absence of a.
      Ex: safeDiv x 0 ⇒ Nothing
      • +
    • Either String a: help to handle errors
      Ex: reportDiv x 0 ⇒ Left "Division by 0!"
      • +
    • +
+
+
+

Haskell Functor intuition

+ +

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

+ +Haskell Functor as a box play +

+
+

Haskell Functor properties

+ +

Haskell Functors are:

+ +
  • endofunctors ; \(F:\C→\C\) here \(\C = \Hask\), +
  • a couple (Object,Morphism) in \(\Hask\). +
+
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

Functor as boxes

+ +

Haskell functor can be seen as boxes containing all Haskell types and functions. +Haskell types is fractal:

+ +Haskell functor representation +
+
+

"Non Haskell" Hask's Functors

+

A simple basic example is the \(id_\Hask\) functor. It simply cannot be expressed as a couple (F,fmap) where

+
    +
  • F::* -> *
    • +
  • fmap :: (a -> b) -> (F a) -> (F b)
    • +
+

Another example:

+
    +
  • F(T)=Int
    • +
  • F(f)=\_->0
    • +
+
+
+

Also Functor inside \(\Hask\)

+

\(\mathtt{[a]}∈\ob{\Hask}\) but is also a category. Idem for Int.

+

length is a Functor from the category [a] to the cateogry Int:

+
    +
  • \(\ob{\mathtt{[a]}}=\{∙\}\)
    • +
  • \(\hom{\mathtt{[a]}}=\mathtt{[a]}\)
    • +
  • \(∘=\mathtt{(++)}\)
    • +
+

⇒

+
    +
  • \(\ob{\mathtt{Int}}=\{∙\}\)
    • +
  • \(\hom{\mathtt{Int}}=\mathtt{Int}\)
    • +
  • \(∘=\mathtt{(+)}\)
    • +
+
+
  • id: length [] = 0 +
  • comp: length (l ++ l') = (length l) + (length l') +
+
+
+

Category of \(\Hask\) Endofunctors

+Category of Hask endofunctors +
+
+

Category of Functors

+

If \(\C\) is small (\(\hom{\C}\) is a set). All functors from \(\C\) to some category \(\D\) form the category \(\mathrm{Func}(\C,\D)\).

+
    +
  • \(\ob{\mathrm{Func}(\C,\D)}\): Functors \(F:\C→\D\)
    • +
  • \(\hom{\mathrm{Func}(\C,\D)}\): natural transformations
    • +
  • ∘: Functor composition
    • +
+

\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).

+
+
+

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.

+

ex: between Haskell functors; F a -> G a
Rearragement functions only.

+
+
+

Natural Transformation Examples (1/4)

+
data Tree a = Empty | Node a [Tree a]
+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 +
+ + +
+
+

Natural Transformation Examples (2/4)

+
data Tree a = Empty | Node a [Tree a]
+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.
+
+ + +
+
+

Natural Transformation Examples (3/4)

+
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 +
+ + +
+
+

Natural Transformation Examples (4/4)

+
mToList :: Maybe a -> [a]
+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 +
+relation between [] and Maybe
There is no isomorphism.
Hint: Bool lists longer than 1.
+
+ + +
+
+

Composition problem

+

The Problem; example with lists:

+
f x = [x]       ⇒ f 1 = [1]   ⇒ (f.f) 1 = [[1]] ✗
+g x = [x+1]     ⇒ g 1 = [2]   ⇒ (g.g) 1 = ERROR [2]+1 ✗
+h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h.h) 1 = ERROR [2,3]+1 ✗
+ +

The same problem with most f :: a -> F a functions and functor F.

+
+
+

Composition Fixable?

+

How to fix that? We want to construct an operator which is able to compose:

+

f :: a -> F b & g :: b -> F c.

+

More specifically we want to create an operator ◎ of type

+

◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)

+

Note: if F = I, ◎ = (.).

+
+
+

Fix Composition (1/2)

+

Goal, find: ◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)
f :: a -> F b, g :: b -> F c:

+
    +
  • (g ◎ f) x ???
    • +
  • First apply f to x ⇒ f x :: F b
    • +
  • Then how to apply g properly to an element of type F b?
    • +
+
+
+

Fix Composition (2/2)

+

Goal, find: ◎ :: (b -> F c) -> (a -> F b) -> (a -> F c)
f :: a -> F b, g :: b -> F c, f x :: F b:

+
    +
  • Use fmap :: (t -> u) -> (F t -> F u)!
    • +
  • (fmap g) :: F b -> F (F c) ; (t=b, u=F c)
    • +
  • (fmap g) (f x) :: F (F c) it almost WORKS!
    • +
  • We lack an important component, join :: F (F c) -> F c
    • +
  • (g ◎ f) x = join ((fmap g) (f x)) ☺
    ◎ is the Kleisli composition; in Haskell: <=< (in Control.Monad).
    • +
+
+
+

Necessary laws

+

For ◎ to work like composition, we need join to hold the following properties:

+
    +
  • join (join (F (F (F a))))=join (F (join (F (F a))))
    • +
  • abusing notations denoting join by ⊙; this is equivalent to
    (F ⊙ F) ⊙ F = F ⊙ (F ⊙ F)
    • +
  • There exists η :: a -> F a s.t.
    η⊙F=F=F⊙η
    • +
+
+
+

Klesli composition

+

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] ✓
+ + +
+
+

We reinvented Monads!

+

A monad is a triplet (M,⊙,η) where

+
    +
  • \(M\) an Endofunctor (to type a associate M a)
    • +
  • \(⊙:M×M→M\) a nat. trans. (i.e. ⊙::M (M a) → M a ; join)
    • +
  • \(η:I→M\) a nat. trans. (\(I\) identity functor ; η::a → M a)
    • +
+

Satisfying

+
    +
  • \(M ⊙ (M ⊙ M) = (M ⊙ M) ⊙ M\)
    • +
  • \(η ⊙ M = M = M ⊙ η\)
    • +
+
+
+

Compare with Monoid

+

A Monoid is a triplet \((E,∙,e)\) s.t.

+
    +
  • \(E\) a set
    • +
  • \(∙:E×E→E\)
    • +
  • \(e:1→E\)
    • +
+

Satisfying

+
    +
  • \(x∙(y∙z) = (x∙y)∙z, ∀x,y,z∈E\)
    • +
  • \(e∙x = x = x∙e, ∀x∈E\)
    • +
+
+
+

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.

+
+
+
+

Example: List

+
    +
  • [] :: * -> * an Endofunctor
    • +
  • \(⊙:M×M→M\) a nat. trans. (join :: M (M a) -> M a)
    • +
  • \(η:I→M\) a nat. trans.
    • +
+
-- In Haskell ⊙ is "join" in "Control.Monad"
+join :: [[a]] -> [a]
+join = concat
+
+-- In Haskell the "return" function (unfortunate name)
+η :: a -> [a]
+η x = [x]
+ + +
+
+

Example: List (law verification)

+

Example: List is a functor (join is ⊙)

+
    +
  • \(M ⊙ (M ⊙ M) = (M ⊙ M) ⊙ M\)
    • +
  • \(η ⊙ M = M = M ⊙ η\)
    • +
+
join [ join [[x,y,...,z]] ] = join [[x,y,...,z]]
+                            = join (join [[[x,y,...,z]]])
+join (η [x]) = [x] = join [η x]
+ +

Therefore ([],join,η) is a monad.

+
+
+

Monads useful?

+

A LOT of monad tutorial on the net. Just one example; the State Monad

+

DrawScene to State Screen DrawScene ; still pure.

+
main = drawImage (width,height)
+
+drawImage :: Screen -> DrawScene
+drawImage screen = do
+    drawPoint p screen
+    drawCircle c screen
+    drawRectangle r screen
+
+drawPoint point screen = ...
+drawCircle circle screen = ...
+drawRectangle rectangle screen = ...
+
main = do
+    put (Screen 1024 768)
+    drawImage
+
+drawImage :: State Screen DrawScene
+drawImage = do
+    drawPoint p
+    drawCircle c
+    drawRectangle r
+
+drawPoint :: Point -> State Screen DrawScene
+drawPoint p = do
+    Screen width height <- get
+    ...
+
+
+

fold

+fold +
+
+

κατα-morphism

+catamorphism +
+
+

κατα-morphism: fold generalization

+

acc type of the "accumulator":
fold :: (acc -> a -> acc) -> acc -> [a] -> acc

+

Idea: put the accumulated value inside the type.

+
-- Equivalent to fold (+1) 0 "cata"
+(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' Nil))))
+(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' 0))))
+(Cons 'c' (Cons 'a' (Cons 't' 1)))
+(Cons 'c' (Cons 'a' 2))
+(Cons 'c' 3)
+4
+ + +

But where are all the informations? (+1) and 0?

+
+
+

κατα-morphism: Missing Information

+

Where is the missing information?

+
    +
  • Functor operator fmap
    • +
  • Algebra representing the (+1) and also knowing about the 0.
    • +
+

First example, make length on [Char]

+
+
+

κατα-morphism: Type work

+

+data StrF a = Cons Char a | Nil
+data Str' = StrF Str'
+
+-- generalize the construction of Str to other datatype
+-- Mu: type fixed point
+-- Mu :: (* -> *) -> *
+
+data Mu f = InF { outF :: f (Mu f) }
+data Str = Mu StrF
+
+-- Example
+foo=InF { outF = Cons 'f'
+        (InF { outF = Cons 'o'
+            (InF { outF = Cons 'o'
+                (InF { outF = Nil })})})}
+ + +
+
+

κατα-morphism: missing information retrieved

+
type Algebra f a = f a -> a
+instance Functor (StrF a) =
+    fmap f (Cons c x) = Cons c (f x)
+    fmap _ Nil = Nil
+ +
cata :: Functor f => Algebra f a -> Mu f -> a
+cata f = f . fmap (cata f) . outF
+ + + +
+
+

κατα-morphism: Finally length

+

All needed information for making length.

+
instance Functor (StrF a) =
+    fmap f (Cons c x) = Cons c (f x)
+    fmap _ Nil = Nil
+
+length' :: Str -> Int
+length' = cata phi where
+    phi :: Algebra StrF Int -- StrF Int -> Int
+    phi (Cons a b) = 1 + b
+    phi Nil = 0
+
+main = do
+    l <- length' $ stringToStr "Toto"
+    ...
+
+
+

κατα-morphism: extension to Trees

+

Once you get the trick, it is easy to extent to most Functor.

+
type Tree = Mu TreeF
+data TreeF x = Node Int [x]
+
+instance Functor TreeF where
+  fmap f (Node e xs) = Node e (fmap f xs)
+
+depth = cata phi where
+  phi :: Algebra TreeF Int -- TreeF Int -> Int
+  phi (Node x sons) = 1 + foldr max 0 sons
+
+
+

Conclusion

+

Category Theory oriented Programming:

+
    +
  • Focus on the type and operators
    • +
  • Extreme generalisation
    • +
  • Better modularity
    • +
  • Better control through properties of types
    • +
+

No cat were harmed in the making of this presentation.

+
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