Category Theory Presentation
Yesterday I was happy to make a presentation about Category Theory at Riviera Scala Clojure Meetup (note I used only Haskell for my examples).
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Category Theory & Programming
Plan
 General overview
 Definitions
 Applications
Not really about: Cat & glory
General Overview
Recent Math Field
194245, 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
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
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, καταmorphism, ...
Programmer Translation
Mathematician  Programmer 

Morphism  Arrow 
Monoid  Stringlike 
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
\(\ob{\mathcal{C}}\) is a collection
Category: 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 \(f:A→B, g:B→C\) $$g∘f:A\rightarrow C$$
Category laws: neutral element
for each object \(X\), there is an \(\id_X:X→X\),
such that for each \(f:A→B\):
Category laws: Associativity
Composition is associative:
Commutative diagrams
Two path with the same source and destination are equal.
Question Time!
Can this be a category?
\(\ob{\C},\hom{\C}\) fixed, is there a valid ∘?
Can this be a category?
Categories Examples
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?
 \(\Mon\): (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
 \(\ob{Str}\) is a singleton
 \(\hom{Str}\) each string
 ∘ is concatenation
(++)

"" ++ u = u = u ++ ""

(u ++ v) ++ w = u ++ (v ++ w)
Finite Example?
Graph
 \(\ob{G}\) 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
Degenerated Categories: Monoids
Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)
Only one object.
Examples:
(Integer,0,+)
,(Integer,1,*)
,(Strings,"",++)
, for eacha
,([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.
Degenerated Categories: Discrete Categories
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
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 Example (hom → hom)
Functor Example
Endofunctors
An endofunctor for \(\C\) is a functor \(F:\C→\C\).
Category of Categories
Categories and functors form a category: \(\Cat\)
 \(\ob{\Cat}\) are categories
 \(\hom{\Cat}\) are functors
 ∘ is functor composition
Plan
 General overview
 Definitions
 Applications
 \(\Hask\) category
 Functors
 Natural transformations
 Monads
 καταmorphisms
Hask
Category \(\Hask\):
 \(\ob{\Hask} = \) 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.
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 ofa
.
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 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 look like a fractal:
Functor as boxes
Haskell functor can be seen as boxes containing all Haskell types and functions. Haskell types look like a fractal:
Functor as boxes
Haskell functor can be seen as boxes containing all Haskell types and functions. Haskell types look like a fractal:
"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 category 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 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 \(\D\).
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 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 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 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.
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
tox
⇒f x :: F b
 Then how to apply
g
properly to an element of typeF 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:<=<
(inControl.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
associateM 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
καταmorphism
καταmorphism: 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
?
καταmorphism: Missing Information
Where is the missing information?
 Functor operator
fmap
 Algebra representing the
(+1)
and also knowing about the0
.
First example, make length
on [Char]
καταmorphism: 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 })})})}
καταmorphism: 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
καταmorphism: 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"
...
καταmorphism: 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.