ScalaWorld 2017 - A Pragmatic Introduction to Category Theory

Transcript

1. A PRAGMATIC
   INTRODUCTION TO
   CATEGORY THEORY
   @DANIELASFREGOLA
   SCALA WORLD 2017
   github.com/DanielaSfregola/tutorial-cat
   

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2. AGENDA
   > Intro
   > Monoid
   > Functor
   > Applicative
   > Monad
   github.com/DanielaSfregola/tutorial-cat
   

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3. HELLOOOOO
   > ex Java Developer
   > OOP background
   > I am not a mathematician !
   

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4. I AM NOT A MATHEMATICIAN
   

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5. YOU DO NOT NEED TO KNOW
   CATEGORY THEORY
   TO WRITE
   SCALA CODE
   

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6. YOU DO NOT NEED TO KNOW
   CATEGORY THEORY
   TO WRITE
   FUNCTIONAL CODE
   

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7. CATEGORY THEORY
   DEEPER UNDERSTANDING ON
   OUR CODE
   

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8. HOW DO WE
   REASON ?
   

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9. COMPOSITION
   ABSTRACTION
   

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10. OBJECT ORIENTED PROGRAMMING (OOP)
    > composition of objects
    > abstraction over the behaviour of each object
    

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11. FUNCTIONAL PROGRAMMING (FP)
    > composition of functions
    > abstraction over mathematical laws
    

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12. CATEGORY THEORY
    HOW THINGS COMPOSE
    

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13. ARROW THEORY
    CATEGORY THEORY
    HOW THINGS COMPOSE
    

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14. WHAT IS A CATEGORY?
    

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15. COMPOSITION LAW
    

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16. IDENTITY LAW
    

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17. COMPOSITION + ASSOCIATIVITY
    

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18. CATEGORY'S RULES
    > Identity
    > Composition
    > Associativity
    

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19. A PRACTICAL EXAMPLE
    

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20. CATEGORY WITH 1 OBJECT
    

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21. CATEGORY WITH 1 OBJECT
    =
    MONOID
    

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22. MONOID'S RULES
    Identity
    n o id == id o n == n
    Composition
    forall x, y => x o y
    Associativity
    x o (y o z) == (x o y) o z
    

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23. A PRACTICAL EXAMPLE
    

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24. MONOID
    trait Monoid[A] {
    def identity: A
    def compose(x: A, y: A): A
    }
    

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25. EXERCISES ON MONOID
    > Define a monoid for Int
    > Define a monoid for String
    sbt 'testOnly *Monoid*'
    github.com/DanielaSfregola/tutorial-cat
    

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26. CATEGORY WITH 1+ OBJECT
    

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27. CATEGORY IN A BOX
    

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28. CATEGORY IN A BOX
    > Objects are in a Box
    > All the arrows are copied
    

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29. LIFTING: CONTEXT VS CONTENT
    

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30. EXAMPLE OF BOXES
    > Option
    > Future
    > Try
    > List
    > Either
    

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31. CATEGORY IN A BOX
    =
    FUNCTOR
    

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32. FUNCTOR'S RULES
    Identity
    map(id) == id
    Composition
    map(g o f) == map(g) o map(f)
    Associativity
    map(h o g) o map(f) == map(h) o map(g o f)
    

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33. LIFTING: CONTEXT VS CONTENT
    

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34. CHANGE THE CONTENT OF A BOX
    > How to change the context
    

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35. FUNCTOR
    class Functor[Box[_]] {
    def map[A, B](boxA: Box[A])
    (f: A => B): Box[B]
    }
    

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36. EXERCISES ON FUNCTOR
    > Define a functor for Maybe
    > Define a functor for ZeroOrMore
    sbt 'testOnly *Functor*'
    github.com/DanielaSfregola/tutorial-cat
    

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37. BOX FUNCTION + BOX VALUES
    

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38. COMBINE MORE BOXES
    INTO ONE
    =
    APPLICATIVE
    

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39. APPLICATIVE'S RULES
    > Identity
    > Associativity
    > Homorphism
    > Interchange
    ...and more!
    

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40. COMBINE BOXES TOGETHER
    > How to create a new box
    > How to combine their values together
    

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41. APPLICATIVE
    class Applicative[Box[_]] extends Functor[Box] {
    def pure[A](a: A): Box[A]
    def ap[A, B](boxF: Box[A => B])(boxA: Box[A]): Box[B]
    def map[A, B](boxA: Box[A])(f: A => B): Box[B] = ???
    // spoiler alert
    }
    

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42. BOX FUNCTION + BOX VALUES
    

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43. APPLICATIVE
    class Applicative[Box[_]] extends Functor[Box] {
    def pure[A](a: A): Box[A]
    def ap[A, B](boxF: Box[A => B])(value: Box[A]): Box[B]
    def ap2[A1, A2, B](boxF: Box[(A1, A2) => B])
    (value1: Box[A1], value2: Box[A2]): Box[B]
    // up to 22 values!
    // same for map
    }
    

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44. EXERCISES ON APPLICATIVE
    > Define map in terms of ap and pure
    > Define an applicative for Maybe
    > Define an applicative for ZeroOrMore
    sbt 'testOnly *Applicative*'
    github.com/DanielaSfregola/tutorial-cat
    

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45. BOX IN A BOX
    

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46. BOX IN A BOX
    

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47. FUSE TWO BOXES
    TOGETHER
    =
    MONAD
    

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48. MONAD'S RULES
    > Left Identity
    > Right Identity
    > Associativity
    

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49. MONAD (AS FUNCTOR)
    class Monad[Box[_]] extends Functor[Box] {
    def flatten[A](bb: Box[Box[A]]): Box[A]
    def flatMap[A, B](valueA: Box[A])
    (f: A => Box[B]): Box[B] = {
    val bb: Box[Box[B]] = map(valueA)(f)
    bb.flatten
    }
    }
    

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50. BOXES IN A SEQUENCE
    

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51. FOR-COMPREHENSION
    val boxA: Box[A]
    def toBoxB: A => Box[B]
    def toBoxC: B => Box[C]
    def toBoxD: C => Box[D]
    for {
    a <- boxA
    b <- toBoxB(a)
    c <- toBoxC(b)
    d <- toBoxD(c)
    } yield d
    

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52. MONAD (AS APPLICATIVE)
    trait Monad[Box[_]] extends Applicative[Box] {
    def flatMap[A, B](boxA: Box[A])(f: A => Box[B]): Box[B]
    // TODO - implement using flatMap
    def flatten[A](boxBoxA: Box[Box[A]]): Box[A] = ???
    // TODO - implement using flatMap and map
    override def ap[A, B](boxF: Box[A => B])(boxA: Box[A]): Box[B] = ???
    // TODO - implement using flatMap and pure
    override def map[A, B](boxA: Box[A])(f: A => B): Box[B] = ???
    }
    

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53. EXERCISES ON MONAD (1)
    > Define flatten using flatMap
    > Define map using flatMap and pure
    > Define ap using flatMap and map
    github.com/DanielaSfregola/tutorial-cat
    

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54. EXERCISES ON MONAD (2)
    > Define a monad for Maybe
    > Define a monad for ZeroOrMore
    sbt 'testOnly *Monad*'
    github.com/DanielaSfregola/tutorial-cat
    

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55. MONAD IS
    A MONOID
    IN THE CATEGORY OF
    ENDOFUNCTORS
    

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56. MONAD
    MONOID => pure + flatten
    ENDOFUNCTORS => map
    

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57. SUMMARY
    CATEGORY THEORY >> how things compose
    MONOID >> combining 2 values into 1
    FUNCTOR >> values lifted to a context
    APPLICATIVE >> independent values applied
    to a function in a context
    MONAD >> ops in sequence in a context
    

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58. Band
    BoundedSemilattice CommutativeGroup
    CommutativeMonoid
    CommutativeSemigroup
    Eq
    Group
    Semigroup
    Monoid
    Order
    PartialOrder
    Semilattice
    Alternative
    Applicative
    ApplicativeError
    Apply
    Bifoldable
    Bimonad
    Bitraverse
    Cartesian
    CoflatMap
    Comonad
    ContravariantCartesian
    FlatMap
    Foldable Functor
    Inject
    InvariantMonoidal
    Monad
    MonadError
    MonoidK
    NotNull
    Reducible
    SemigroupK
    Show
    ApplicativeAsk
    Bifunctor
    Contravariant
    Invariant
    Profunctor
    Strong
    Traverse
    Arrow
    Category
    Choice
    Compose
    Cats Type Classes
    kernel
    core/functor
    core/arrow
    core The highlighted type classes are the first ones
    you should learn. They’re well documented
    and well-known so it’s easy to get help. a |+| b
    a === b
    a =!= b
    a |@| b
    a *> b
    a <* b
    a <+> b
    a >>> b
    a <<< b
    a > b
    a >= b
    a < b
    a <= b
    Sync
    Async
    Effect
    LiftIO
    effect
    Some type classes introduce
    symbolic operators.
    NonEmptyTraverse
    InjectK
    CommutativeArrow
    CommutativeFlatMap
    CommutativeMonad
    ApplicativeLayer
    FunctorLayer
    ApplicativeLayerFunctor
    FunctorLayerFunctor
    ApplicativeLocal
    FunctorEmpty
    FunctorListen
    FunctorTell
    FunctorRaise
    MonadLayer
    MonadLayerFunctor
    MonadLayerControl
    MonadState
    TraverseEmpty
    Functor
    Applicative
    Monad
    Traverse
    mtl
    MTL type classes do not extend core type
    classes directly, but the effect is similar; the
    dashed line can be read “implies”.
    

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59. FORGET ABOUT THE DETAILS
    FOCUS ON
    HOW THINGS COMPOSE
    

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60. WANNA KNOW MORE?
    > Category Theory for the WH by @PhilipWadler
    > Category Theory by @BartoszMilewski
    > Cats-Infographics by Rob Norris - @tpolecat
    > Cats Documentation - Type Classes
    

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61. THANK YOU!
    > Twitter: @DanielaSfregola
    > Blog: danielasfregola.com
    github.com/DanielaSfregola/tutorial-cat
    

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