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
2. AGENDA - Intro - Monoid - Functor - Applicative - Monad github.com/DanielaSfregola/tutorial-cat
4. I AM NOT A MATHEMATICIAN
5. YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE GOOD 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. CATEGORY THEORY HOW THINGS COMPOSE
11. ARROW THEORY CATEGORY THEORY HOW THINGS COMPOSE
12. WHAT IS A CATEGORY?
13. COMPOSITION LAW
14. IDENTITY LAW
15. COMPOSITION + ASSOCIATIVITY
16. CATEGORY'S RULES > Identity > Composition > Associativity
17. A PRACTICAL EXAMPLE
18. CATEGORY WITH 1 OBJECT
19. CATEGORY WITH 1 OBJECT = MONOID
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
21. A PRACTICAL EXAMPLE
22. MONOID trait Monoid[A] { def identity: A def compose(x: A, y: A): A }
23. EXERCISES ON MONOID > Define a monoid for Int > Define a monoid for String sbt 'testOnly *Monoid*' github.com/DanielaSfregola/tutorial-cat
24. CATEGORY WITH 1+ OBJECT
25. CATEGORY IN A BOX
26. CATEGORY IN A BOX > Objects are in a Box > All the arrows are mapped
27. LIFTING: CONTEXT VS CONTENT
28. EXAMPLE OF BOXES > Option > Future > Try > List > Either
29. CATEGORY IN A BOX = FUNCTOR
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)
31. LIFTING: CONTEXT VS CONTENT
32. FUNCTOR class Functor[Box[_]] { def map[A, B](boxA: Box[A]) (f: A => B): Box[B] }
33. EXERCISES ON FUNCTOR > Define a functor for Maybe > Define a functor for ZeroOrMore sbt 'testOnly *Functor*' github.com/DanielaSfregola/tutorial-cat
34. BOX FUNCTION + BOX VALUES
35. COMBINE MORE BOXES = APPLICATIVE
36. APPLICATIVE'S RULES > Identity > Associativity > Homorphism > Interchange ...and more!
37. COMBINE BOXES TOGETHER > How to create a new box > How to combine their values together
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 }
39. BOX FUNCTION + BOX VALUES
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 }
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
42. BOX IN A BOX
43. BOX IN A BOX
44. FUSE TWO BOXES = MONAD
45. MONAD'S RULES > Identity > Composition > Associativity
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 } }
47. BOXES IN A SEQUENCE
48. 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
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] = ??? }
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
51. EXERCISES ON MONAD (2) > Define a monad for Maybe > Define a monad for ZeroOrMore sbt 'testOnly *Monad*' github.com/DanielaSfregola/tutorial-cat
52. FUNCTOR VS ENDOFUNCTOR
53. MONAD IS A MONOID IN THE CATEGORY OF ENDOFUNCTORS
54. MONAD MONOID => pure + flatten ENDOFUNCTORS => map
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
57. FORGET ABOUT THE DETAILS FOCUS ON HOW THINGS COMPOSE
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
59. THANK YOU! > Twitter: @DanielaSfregola > Blog: danielasfregola.com github.com/DanielaSfregola/tutorial-cat