ScalaMatsuri 2018 - A pragmatic introduction to Category Theory

Category Theory has become one of the hot topics in the community. Why is this theory suddenly so interesting for developers? Why are the cool kids talking so much about it?

This talk will introduce the general principles of Category Theory in a pragmatic, non-mathematical way. We will show practical examples of how this theory has managed to simplify and solve common challenges that we encounter in our code daily, such as nullable values, error handling, parallel and sequential operations and data validation. Also, we will apply them to create our own category theory library from scratch using ScalaCheck as the only dependency.

Transcript

1. A PRAGMATIC INTRODUCTION TO
   CATEGORY THEORY
   @DANIELASFREGOLA
   SCALA MATSURI 2018
   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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4. I AM NOT A MATHEMATICIAN
   

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5. YOU DO NOT NEED TO KNOW
   CATEGORY THEORY
   TO WRITE
   GOOD 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. CATEGORY THEORY
    HOW THINGS COMPOSE
    

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11. ARROW THEORY
    CATEGORY THEORY
    HOW THINGS COMPOSE
    

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12. WHAT IS A CATEGORY?
    

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13. COMPOSITION LAW
    

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14. IDENTITY LAW
    

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15. COMPOSITION + ASSOCIATIVITY
    

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

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17. A PRACTICAL EXAMPLE
    

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18. CATEGORY WITH 1 OBJECT
    

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19. CATEGORY WITH 1
    OBJECT
    =
    MONOID
    

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20. 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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21. A PRACTICAL EXAMPLE
    

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

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23. 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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24. CATEGORY WITH 1+ OBJECT
    

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25. CATEGORY IN A BOX
    

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26. CATEGORY IN A BOX
    > Objects are in a Box
    > All the arrows are mapped
    

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27. LIFTING: CONTEXT VS CONTENT
    

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

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29. CATEGORY IN A BOX
    =
    FUNCTOR
    

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30. 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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31. LIFTING: CONTEXT VS CONTENT
    

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

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33. 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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34. BOX FUNCTION + BOX VALUES
    

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35. COMBINE MORE BOXES
    =
    APPLICATIVE
    

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

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

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38. 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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39. BOX FUNCTION + BOX VALUES
    

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40. 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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41. 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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42. BOX IN A BOX
    

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43. BOX IN A BOX
    

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44. FUSE TWO BOXES
    =
    MONAD
    

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45. MONAD'S RULES
    > Identity
    > Composition
    > Associativity
    

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46. 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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47. BOXES IN A SEQUENCE
    

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

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49. 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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50. 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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51. 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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52. FUNCTOR VS ENDOFUNCTOR
    

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53. MONAD IS
    A MONOID
    IN THE CATEGORY OF
    ENDOFUNCTORS
    

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

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55. 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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57. FORGET ABOUT THE DETAILS
    FOCUS ON
    HOW THINGS
    COMPOSE
    

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

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