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create mode 100644 themes/transition/vertical-slide.css create mode 100644 themes/transition/vertical-slide.scss diff --git a/categories.html b/categories.html new file mode 100644 index 0000000..e40a038 --- /dev/null +++ b/categories.html @@ -0,0 +1,1119 @@ + + + + + + + + Category Theory for Programming + + + + + + + + + + + + + + + + + + + + + + + + + +
+\(\newcommand{\F}{\mathbf{F}}\) +\(\newcommand{\E}{\mathbf{E}}\) +\(\newcommand{\C}{\mathcal{C}}\) +\(\newcommand{\D}{\mathcal{D}}\) +\(\newcommand{\id}{\mathrm{id}}\) +\(\newcommand{\ob}[1]{\mathrm{ob}(#1)}\) +\(\newcommand{\hom}[1]{\mathrm{hom}(#1)}\) +\(\newcommand{\Set}{\mathbf{Set}}\) +\(\newcommand{\Mon}{\mathbf{Mon}}\) +\(\newcommand{\Vec}{\mathbf{Vec}}\) +\(\newcommand{\Grp}{\mathbf{Grp}}\) +\(\newcommand{\Rng}{\mathbf{Rng}}\) +\(\newcommand{\ML}{\mathbf{ML}}\) +\(\newcommand{\Hask}{\mathbf{Hask}}\) +\(\newcommand{\Cat}{\mathbf{Cat}}\) +\(\newcommand{\fmap}{\mathtt{fmap}}\) +
+ + + +
+
+

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 category 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 List a = Nil | Cons a (List a)
+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 Examples (2/4)

+
data List a = Nil | Cons a (List a)
+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
toList . toHList = id & toHList . toList = id &
therefore [] & List 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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