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
2. AGENDA > Intro > Monoid > Functor > Applicative > Monad github.com/DanielaSfregola/tutorial-cat
3. HELLOOOOO > ex Java Developer > OOP background > I am not a mathematician !
4. I AM NOT A MATHEMATICIAN
5. YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE SCALA CODE
6. YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE FUNCTIONAL CODE
7. CATEGORY THEORY DEEPER UNDERSTANDING ON OUR CODE
8. HOW DO WE REASON ?
9. COMPOSITION ABSTRACTION
10. OBJECT ORIENTED PROGRAMMING (OOP) > composition of objects > abstraction over the behaviour of each object
11. FUNCTIONAL PROGRAMMING (FP) > composition of functions > abstraction over mathematical laws
12. CATEGORY THEORY HOW THINGS COMPOSE
13. ARROW THEORY CATEGORY THEORY HOW THINGS COMPOSE
14. WHAT IS A CATEGORY?
15. COMPOSITION LAW
16. IDENTITY LAW
17. COMPOSITION + ASSOCIATIVITY
18. CATEGORY'S RULES > Identity > Composition > Associativity
19. A PRACTICAL EXAMPLE
20. CATEGORY WITH 1 OBJECT
21. CATEGORY WITH 1 OBJECT = MONOID
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
23. A PRACTICAL EXAMPLE
24. MONOID trait Monoid[A] { def identity: A def compose(x: A, y: A): A }
25. EXERCISES ON MONOID > Define a monoid for Int > Define a monoid for String sbt 'testOnly *Monoid*' github.com/DanielaSfregola/tutorial-cat
26. CATEGORY WITH 1+ OBJECT
27. CATEGORY IN A BOX
28. CATEGORY IN A BOX > Objects are in a Box > All the arrows are copied
29. LIFTING: CONTEXT VS CONTENT
30. EXAMPLE OF BOXES > Option > Future > Try > List > Either
31. CATEGORY IN A BOX = FUNCTOR
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)
33. LIFTING: CONTEXT VS CONTENT
34. CHANGE THE CONTENT OF A BOX > How to change the context
35. FUNCTOR class Functor[Box[_]] { def map[A, B](boxA: Box[A]) (f: A => B): Box[B] }
36. EXERCISES ON FUNCTOR > Define a functor for Maybe > Define a functor for ZeroOrMore sbt 'testOnly *Functor*' github.com/DanielaSfregola/tutorial-cat
37. BOX FUNCTION + BOX VALUES
38. COMBINE MORE BOXES INTO ONE = APPLICATIVE
39. APPLICATIVE'S RULES > Identity > Associativity > Homorphism > Interchange ...and more!
40. COMBINE BOXES TOGETHER > How to create a new box > How to combine their values together
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 }
42. BOX FUNCTION + BOX VALUES
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 }
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
45. BOX IN A BOX
46. BOX IN A BOX
47. FUSE TWO BOXES TOGETHER = MONAD
48. MONAD'S RULES > Left Identity > Right Identity > Associativity
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 } }
50. BOXES IN A SEQUENCE
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
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] = ??? }
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
54. EXERCISES ON MONAD (2) > Define a monad for Maybe > Define a monad for ZeroOrMore sbt 'testOnly *Monad*' github.com/DanielaSfregola/tutorial-cat
55. MONAD IS A MONOID IN THE CATEGORY OF ENDOFUNCTORS
56. MONAD MONOID => pure + flatten ENDOFUNCTORS => map
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
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”.
59. FORGET ABOUT THE DETAILS FOCUS ON HOW THINGS COMPOSE
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
61. THANK YOU! > Twitter: @DanielaSfregola > Blog: danielasfregola.com github.com/DanielaSfregola/tutorial-cat