1. Category Theory for Beginners
Your data structures
are made of maths!
Melbourne Scala User Group Mar 2015
@KenScambler

2. Data structures
More and more “logical”
Less low-level memory/hardware connection
More and more support for immutability
We can use maths to reason about them!
Category Theory can reveal even deeper symmetries

3. Algebraic Data Types
Constants, sums, products, exponents
We can directly manipulate them with algebra!

4. Products
A × B

5. NinjaTurtles

6. NinjaTurtles
×
BluesBrothers×

7. NinjaTurtles BluesBrothers
=
NTs and
BBs× =
×

8. =4 × 2 8

9. Integers as types?
We can actually use integers to represent our types!
The integers correspond to the size of the type

10. Products in code
(NinjaTurtle, BluesBrother)

11. (NinjaTurtle, BluesBrother)
4

12. (NinjaTurtle, BluesBrother)
4 2

13. NinjaTurtle (,) BluesBrother
4 2×

14. (NinjaTurtle, BluesBrother)
8

15. case class TurtleAndBrother(
foo1: NinjaTurtle,
foo2: BluesBrother)

16. case class TurtleAndBrother(
foo1: NinjaTurtle,
foo2: BluesBrother)
4

17. case class TurtleAndBrother(
foo1: NinjaTurtle,
foo2: BluesBrother)
4
2

18. case class TurtleAndBrother(…)
NinjaTurtle
BluesBrother
4
2
×

19. case class TurtleAndBrother(
foo1: NinjaTurtle,
foo2: BluesBrother)
8

20. trait Confluster {
def defrabulate(): BluesBrother
def empontigle(): NinjaTurtle
}

21. trait Confluster {
def defrabulate(): BluesBrother
def empontigle(): NinjaTurtle
}
2

22. trait Confluster {
def defrabulate(): BluesBrother
def empontigle(): NinjaTurtle
}
4
2

23. trait Confluster { def; def; }
BluesBrother
NinjaTurtle
2
4
×

24. trait Confluster {
def defrabulate(): BluesBrother
def empontigle(): NinjaTurtle
}
8

25. Sums
A + B

26. true
false
Boolean

27. true
false
+
Boolean Shapes+

28. true
false
+ =
Boolean Shapes
Boolean or
Shapes
true
false
+ =

29. true
false true
false
=2 + 4 6

30. sealed trait Holiday
case object Christmas extends Holiday
case object Easter extends Holiday
case object AnzacDay extends Holiday
Sums in code

31. sealed trait Holiday
case object Christmas extends Holiday
case object Easter extends Holiday
case object AnzacDay extends Holiday
1

32. sealed trait Holiday
case object Christmas extends Holiday
case object Easter extends Holiday
case object AnzacDay extends Holiday
11

33. sealed trait Holiday
case object Christmas extends Holiday
case object Easter extends Holiday
case object AnzacDay extends Holiday
11
1

34. sealed trait Holiday, case, extends
Christmas
Easter
AnzacDay
11
1
+

35. sealed trait Holiday
case object Christmas extends Holiday
case object Easter extends Holiday
case object AnzacDay extends Holiday
3

36. sealed trait Opt[A]
case class Some[A](a: A) extends Opt[A]
case class None[A] extends Opt[A]

37. sealed trait Opt[A]
case class Some[A](a: A) extends Opt[A]
case class None[A] extends Opt[A]
A

38. sealed trait Opt[A]
case class Some[A](a: A) extends Opt[A]
case class None[A] extends Opt[A]
A
1

39. sealed trait Opt[A], case class, extends
Some[A](a: A)
None[A]
A
1
+

40. sealed trait Opt[A]
case class Some[A](a: A) extends Opt[A]
case class None[A] extends Opt[A]A + 1

41. Either[Holiday, Opt[Boolean]]

42. Either[Holiday, Opt[Boolean]]
3

43. Either[Holiday, Opt[Boolean]]
3 2

44. Either[Holiday, Opt[Boolean]]
3 2 + 1

45. Either[Holiday, Opt[Boolean]]
3 3

46. Either[Holiday, Opt[Boolean]]
3 3+

47. Either[Holiday, Opt[Boolean]]
6

48. Exponents
BA

49. Exponents
BA
A type to the power of another type?? Huh?

50. true
false
Boolean

51. true
false
Boolean
Shapes

52. true
false
Boolean
Shapes = Shapes  Boolean
true
false
true
false
true
false
true
false
true
false
true
false
true
false
true
false

53. true
false
23 =
true
false
true
false
true
false
true
false
true
false
true
false
true
false
true
false
8

54. Function types =
exponents!
A  B = BA

55. Exponents in code
def getTurtle(): NinjaTurtles

56. def getTurtle: () => NinjaTurtles

57. def getTurtle: () => NinjaTurtles
1

58. def getTurtle: () => NinjaTurtles
1 4

59. def getTurtle: () => NinjaTurtles
1  4

60. def getTurtle: () => NinjaTurtles
41

61. def getTurtle: () => NinjaTurtles
4

62. Functions “with no arguments”
are tacitly from a singleton type
such as Unit
Singleton types carry no
information.

63. trait State[S, A] {
def run(state: S): (A, S)
}

64. trait State[S, A] {
def run(state: S): (A, S)
}
A×S

65. trait State[S, A] {
def run(state: S): (A, S)
}
S  A×S

66. trait State[S, A] {
def run(state: S): (A, S)
}
(A×S)S

67. {•}
Zero
On
e
Product
s
Sum
sExponent
s
Sets Scala Algebra
{}
A×B
A∪B
AB
(A, B)
0
1
AB
A+B
A -> B
A / B
Nothing
Unit
BA

68. Currying
def tupled[A, B, C](a: A, b: B): C
def curried[A, B, C]: A => (B => C)

69. Currying
def tupled[A, B, C](a: A, b: B): C
def curried[A, B, C]: A => (B => C)
A×B

70. Currying
def tupled[A, B, C](a: A, b: B): C
def curried[A, B, C]: A => (B => C)
A×B  C

71. Currying
def tupled[A, B,