Scala IO 2017 - 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 Scala and ScalaCheck as the only dependency.

Transcript

1. A PRAGMATIC INTRODUCTION TO CATEGORY THEORY @DANIELASFREGOLA SCALA IO 2017 github.com/DanielaSfregola/tutorial-cat
2. Every good idea is discovered twice: once by a mathematician once by a computer scientist — Philip Wadler
3. A Pragmatic Intro to Category Theory by Daniela Sfregola THURSDAY 10.00 AM vs A crash course on Category Theory by Bartosz Milewski FRIDAY 15.30 PM
4. HELLOOOOO > ex Java Developer > OOP background > I am not a mathematician !
5. I AM NOT A MATHEMATICIAN
6. YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE SCALA CODE
7. YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE FUNCTIONAL CODE
8. CATEGORY THEORY DEEPER UNDERSTANDING ON OUR CODE
9. HOW DO WE REASON ?
10. COMPOSITION ABSTRACTION
11. CATEGORY THEORY HOW THINGS COMPOSE
12. ARROW THEORY CATEGORY THEORY HOW THINGS COMPOSE
13. WHAT IS A CATEGORY?
14. COMPOSITION LAW
15. IDENTITY LAW
16. COMPOSITION + ASSOCIATIVITY
17. CATEGORY'S RULES > Identity > Composition > Associativity
18. A PRACTICAL EXAMPLE
19. CATEGORY WITH 1 OBJECT
20. CATEGORY WITH 1 OBJECT = MONOID
21. 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
22. SCALACHECK FTW! MonoidSpec // n o id == id o n == n property("identity") = forAll { n: A => monoid.compose(n, id) == n && monoid.compose(id, n) == n } // forall x, y => x o y property("composition") = forAll { (x: A, y: A) => monoid.compose(x, y).isInstanceOf[A] } // x o (y o z) == (x o y) o z property("associativity") = forAll { (x: A, y: A, z: A) => val xY = monoid.compose(x,y) val yZ = monoid.compose(y,z) monoid.compose(xY, z) == monoid.compose(x, yZ) }
23. A PRACTICAL EXAMPLE
24. MONOID trait Monoid[A] { def identity: A def compose(x: A, y: A): A }
25. MONOID INSTANCES (1) implicit val intMonoid: Monoid[Int] = new Monoid[Int] { def compose(x: Int, y: Int): Int = x + y def identity: Int = 0 }
26. MONOID INSTANCES (2) implicit val stringMonoid: Monoid[String] = new Monoid[String] { def compose(x: String, y: String): String = x + y def identity: String = "" }
27. CATEGORY WITH 1+ OBJECT
28. CATEGORY IN A BOX
29. CATEGORY IN A BOX > Objects are in a Box > All the arrows are copied
30. LIFTING: CONTEXT VS CONTENT
31. EXAMPLE OF BOXES > Option > Future > Try > List > Either
32. CATEGORY IN A BOX = FUNCTOR
33. 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)
34. SCALACHECK FTW! FunctorSpec // map_id == id property("identity") = forAll { box: Box[A] => map(box)(identity) == box } // map_(g o f) == (map_g) o (map_f) property("composition") = forAll { boxA: Box[A] => val fG = f andThen g val mapFG: Box[A] => Box[C] = map(_)(fG) mapFG(boxA) == (mapF andThen mapG)(boxA) } // map_(h o g) o map_f == map_h o map_(g o f) property("associativity") = forAll { boxA: Box[A] => val fG = f andThen g val mapFG: Box[A] => Box[C] = map(_)(fG) val gH = g andThen h val mapGH: Box[B] => Box[D] = map(_)(gH) (mapF andThen mapGH)(boxA) == (mapFG andThen mapH)(boxA) }
35. LIFTING: CONTEXT VS CONTENT
36. FUNCTOR class Functor[Box[_]] { def map[A, B](boxA: Box[A]) (f: A => B): Box[B] }
37. MAYBE sealed abstract class Maybe[+A] final case class Just[A](a: A) extends Maybe[A] case object Empty extends Maybe[Nothing]
38. FUNCTOR FOR MAYBE implicit val maybeFunctor: Functor[Maybe] = new Functor[Maybe] { override def map[A, B](boxA: Maybe[A]) (f: A => B): Maybe[B] = boxA match { case Just(a) => Just(f(a)) case Empty => Empty } }
39. BOX FUNCTION + BOX VALUES
40. COMBINE MORE BOXES INTO ONE = APPLICATIVE
41. APPLICATIVE'S RULES > Identity > Composition > Associativity > Homorphism > Interchange ...and more!
42. SCALACHECK FTW! ApplicativeSpec extends FunctorSpec // ap(id)(a) == a property("identity") = forAll { box: Box[A] => ap(pureIdentity)(box) == box } // ap(pure(f))(pure(a)) == pure(f(a)) property("homorphism") = forAll { a: A => ap(pureF)(pure(a)) == pure(f(a)) } // {x => pure(x)}(a) == pure(a) property("interchange") = forAll { a: A => toPureA(a) == pure(a) } // pure(h o g o f) == ap(pure(h o g))(pure(f(a))) property("composition") = forAll { a: A => val gH = g andThen h val fGH = f andThen gH val pureGH = pure(gH) val pureFA = pure(f(a)) pure(fGH(a)) == ap(pureGH)(pureFA) }
43. COMBINE BOXES TOGETHER > How to create a new box > How to combine their values together
44. 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] = ap[A, B](pure(f))(boxA) }
45. BOX FUNCTION + BOX VALUES
46. 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 }
47. APPLICATIVE FOR MAYBE implicit val maybeApplicative: Applicative[Maybe] = new Applicative[Maybe] { def pure[A](a: A): Maybe[A] = Just(a) def ap[A, B](boxF: Maybe[A => B])(boxA: Maybe[A]): Maybe[B] = (boxF, boxA) match { case (Just(f), Just(a)) => pure(f(a)) case _ => Empty } }
48. BOX IN A BOX
49. BOX IN A BOX
50. FUSE TWO BOXES TOGETHER = MONAD
51. MONAD'S RULES > Identity > Composition > Associativity
52. SCALACHECK FTW! MonadSpec extends ApplicativeSpec // flatMap(pure(a))(f(a)) == f(a) property("left identity") = forAll { a: A => flatMap(pure(a))(toPureFa) == toPureFa(a) } // flatMap(pure(a))(f(a)) == f(a) property("right identity") = forAll { a: A => flatMap(toPureFa(a))(pure) == toPureFa(a) } // pure(h o g o f) == ap(pure(h o g))(pure(f(a))) property("associativity") = forAll { boxA: Box[A] => val left: Box[C] = flatMap(flatMap(boxA)(toPureFa))(toPureGb) val right: Box[C] = flatMap(boxA)(a => flatMap(toPureFa(a))(toPureGb)) left == right }
53. MONAD (AS FUNCTOR) class Monad[Box[_]] extends Functor[Box] { // map from Functor 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 } }
54. BOXES IN A SEQUENCE
55. 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
56. MONAD (AS APPLICATIVE) trait Monad[Box[_]] extends Applicative[Box] { // pure from Applicative def flatMap[A, B](boxA: Box[A])(f: A => Box[B]): Box[B] /******/ def flatten[A](boxBoxA: Box[Box[A]]): Box[A] = flatMap(boxBoxA)(identity) def ap[A, B](boxF: Box[A => B])(boxA: Box[A]): Box[B] = flatMap(boxF)(f => map(boxA)(f)) def map[A, B](boxA: Box[A])(f: A => B): Box[B] = flatMap(boxA)(a => pure(f(a))) }
57. MONAD FOR MAYBE implicit val maybeMonad: Monad[Maybe] = new Monad[Maybe] { def flatMap[A, B](boxA: Maybe[A]) (f: (A) => Maybe[B]): Maybe[B] = boxA match { case Just(a) => f(a) case Empty => Empty } def pure[A](a: A): Maybe[A] = maybeApplicative.pure(a) }
58. MONAD IS A MONOID IN THE CATEGORY OF ENDOFUNCTORS
59. MONAD MONOID => pure + flatten ENDOFUNCTORS => map
60. 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
61. 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”.
62. FORGET ABOUT THE DETAILS FOCUS ON HOW THINGS COMPOSE
63. WANNA KNOW MORE? > Category Theory for the WH by @PhilipWadler > Category Theory by @BartoszMilewski > Cats-Infographics by Rob Norris - @tpolecat > Cats Documentation - Type Classes
64. THANK YOU! > Twitter: @DanielaSfregola > Blog: danielasfregola.com github.com/DanielaSfregola/tutorial-cat