1. Category Theory for
beginners
Melbourne Scala User Group Feb 2015
@KenScambler
A B
C

2. Abstract maths… for us?
Dizzyingly abstract branch of maths
“Abstract nonsense”?
Programming = maths
Programming = abstraction
Really useful to programming!

3. The plan
Basic Category Theory concepts
New vocabulary (helpful for further reading)
How it relates to programming
Category Theory as seen by maths versus FP

4. A bit of background
1940s Eilenberg, Mac Lane invent Category Theory
1958 Monads discovered by Godement
In programming:
1990 Moggi, Wadler apply monads to programming
2006 “Applicative Programming with Effects” McBride &
Paterson
2006 “Essence of the Iterator Pattern” Gibbons & Oliveira

5. I. Categories

6. Category
Objects

7. Category
Objects
Arrows or
morphisms

8. Category
Objects
Arrows
Domain f
dom(f)

9. Category
Objects
Arrows
Domain/Codomain f
cod(f)
dom(f)

10. Category
Objects
Arrows
Domain/Codomain
dom(g)
cod(g)
g

11. Category
Objects
Arrows
Domain/Codomain

12. Category
Objects
Arrows
Domain/Codomain
Composition
f

13. Category
Objects
Arrows
Domain/Codomain
Composition
f
g

14. Category
Objects
Arrows
Domain/Codomain
Composition
f
g
g ∘ f

15. Category
Objects
Arrows
Domain/Codomain
Composition
f

16. Category
Objects
Arrows
Domain/Codomain
Composition
f
h

17. Category
Objects
Arrows
Domain/Codomain
Composition
f
h
h ∘ f

18. Category
Objects
Arrows
Domain/Codomain
Composition
Identity

19. Category
Compose
∘ : (B  C)  (A  B)  (A 
C)
Identity
id : A  A

20. Category Laws
Associative Law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
Identity Laws
f ∘ id = id ∘ f = f

21. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

22. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

23. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

24. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

25. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

26. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

27. Associative law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
f
g
h
(g ∘ h)
(f ∘ g)

28. Identity laws
f ∘ id = id ∘ f = f
f
id id

29. Identity laws
f ∘ id = id ∘ f = f
f
id id

30. Identity laws
f ∘ id = id ∘ f = f
f
id id

31. Identity laws
f ∘ id = id ∘ f = f
f
id id

32. Examples
Infinite categories
Finite categories
Objects can represent anything
Arrows can represent anything
As long as we have composition and identity!

33. Sets & functions
Person
String Integer
bestFriend
length
name age
+1

34. Sets & functions
Infinite arrows from composition
+1∘ length ∘ name
bestFriend ∘ bestFriend
bestFriend ∘ bestFriend ∘ bestFriend
+1∘ age∘ bestFriend

35. Sets & functions
Objects
Arrows
Composition
Identity

36. Sets & functions
Objects = sets (or types)
Arrows = functions
Composition = function composition
Identity = identity function

37. Zero

38. One

39. Two

40. Three

41. Class hierarchy
IFruit
IBanana
AbstractBanana
BananaImpl MockBanana
Tool
Spanner

42. Class hierarchy
Objects
Arrows
Composition
Identity

43. Class hierarchy
Objects = classes
Arrows = “extends”
Composition = “extends” is transitive
Identity = trivial

44. Class hierarchy
Partially ordered sets (posets)
Objects = elements in the set
Arrows = ordering relation ≤
Composition = ≤ is transitive
Identity = trivial

45. World Wide Web
www.naaawcats.com
No dogs allowed!
www.robodogs.com
See here for more
robots
www.coolrobots.com
BUY NOW!!!!

46. World Wide Web
Objects = webpages
Arrows = hyperlinks
Composition = Links don’t compose
Identity

47. World Wide Web
Graphs
Objects = nodes
Arrows = edges
Composition = Edges don’t compose
Identity

48. “Free Category” from
graphs!
Objects = nodes
Arrows = paths (0 to many edges)
Composition = aligning paths end to
end
Identity = you’re already there

49. Categories in code
trait Category[Arrow[_,_]] {
def compose[A,B,C](
c: Arrow[B,C],
d: Arrow[A,B]): Arrow[A,C]
def id[A]: Arrow[A,A]
}

50. Category of Types & Functions
object FnCat
extends Category[Function1] {
def compose[A,B,C](
c: B => C,
d: A => B): A => C = {
a => c(d(a))
}
def id[A]: A => A = (a => a)
}

51. Category of Garden Hoses
sealed trait Hose[In, Out] {
def leaks: Int
def kinked: Boolean
def >>[A](in: Hose[A, In]):
Hose[A, Out]
def <<[A](out: Hose[Out, A]):
Hose[In, A]
}

52. Category of Garden Hoses
[live code example]

53. Categories embody the
principle of
strongly-typed
composability

54. II. Functors

55. Functors
Functors map between categories
Objects  objects
Arrows  arrows
Preserves composition & identity

56. Functor laws
Composition Law
F(g ∘ f) = F(g) ∘ F(f)
Identity Law
F(idA) = idF(A)

57. C
F
DCategory Category
Functor

58. C
F
DCategory Category
Functor
CatCategory of categories

59. C
F
DCategory Category
Functor
CatCategory of categories
Objects = categories
Arrows = functors
Composition = functor composition
Identity = Identity functor

60. C
F
D
A
B
C
g ∘ f
f
g

61. C
F
D
A
B
C
g ∘ f
f
g
F(A)

62. C
F
D
A
B
C
g ∘ f
f
g
F(A)
F(B)

63. C
F
D
A
B
C
g ∘ f
f
g
F(A)
F(B)
F(C)

64. C
F
D
A
B
C
g ∘ f
f
g
F(A)
F(B)
F(C)
F(f)

65. C
F
D
A
B
C
g ∘ f
f
g
F(A)
F(B)
F(C)
F(f)
F(g
)

66. C
F
D
A
B
C
g ∘ f
f
g
F(A)
F(B)
F(C)
F(f)
F(g
)
F(g ∘ f)

67. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

68. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

69. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

70. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

71. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

72. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

73. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

74. g ∘ f
f
g
F(f)
F(g
)
F(g ∘ f)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)

75. g ∘ f
f
g
F(f)
F(g
)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)
F(g) ∘ F(f)

76. g ∘ f
f
g
F(f)
F(g
)
Composition Law
F(g ∘ f) = F(g) ∘ F(f)
F(g) ∘ F(f)
F(g ∘ f)

77. Identity law
F(idA)= idF(A)
A

78. Identity law
F(idA)= idF(A)
A
idA

79. Identity law
F(idA)= idF(A)
A
idA
F(idA)

80. Identity law
F(idA)= idF(A)
A

81. Identity law
F(idA)= idF(A)
A F(A)

82. Identity law
F(idA)= idF(A)
A F(A)
idF(A)

83. Identity law
F(idA)= idF(A)
A F(A)
idF(A)
A
idA
F(idA)

84. Terminology
homomorphism

85. Terminology
homomorphism
Same

86. Terminology
homomorphism
Same-shape-ism

87. Terminology
homomorphism
“structure preserving map”

88. Terminology
homomorphism
Functors are
“category homomorphisms”

89. Functors in code
trait Functor[F[_]] {
def map[A,B](fa: F[A],
f: A => B): F[B]
}

90. Functors in code
trait Functor[F[_]] {
def map[A,B](fa: F[A],
f: A => B): F[B]
}
Objects to objects

91. Functors in code
trait Functor[F[_]] {
def map[A,B](fa: F[A],
f: A => B): F[B]
}
Arrows to arrows

92. Functors in code
trait Functor[F[_]] {
def map[A,B]:
(A => B) => (F[A] => F[B])
}
Arrows to arrows

93. Functors laws in code
fa.map(f).map(g)
==
fa.map(g compose f)

94. Functors laws in code
fa.map(a => a) == fa

95. Terminology
endomorphism

96. Terminology
endomorphism
Within

97. Terminology
endomorphism
Within -shape-ism

98. Terminology
endomorphism
“a mapping from
something back to itself”

99. Terminology
endo
“a mapping from
something back to itself”

100. Endofunctors
In Scala, all our functors
are
actually endofunctors.
Type
F
Category Category
Endofunctor Type

101. Endofunctors
Luckily, we can represent
any functor in our type
system as some F[_]
Type F
Category Category
Endofunctor
Type

102. List Functor
sealed trait List[+A]
case class Cons(head: A, tail: List[A])
extends List[A]
case object Nil extends List[Nothing]

103. List Functor
sealed trait List[+A] {
def map[B](f: A => B): List[B] =
this match {
case Cons(h,t) => Cons(f(h), t map f)
case Nil => Nil
}
}
}

104. List Functor
potatoList
.map(mashEm)
.map(boilEm)
.map(stickEmInAStew)

105. List Functor
userList
.map(_.name)
.map(_.length)
.map(_ + 1)
.map(_.toString)

106. Other functors
trait Tree[A]
trait Future[A]
trait Process[A]
trait Command[A]
X => A
(X, A)
trait Option[A]

107. Functors
Fundamental concept in Category Theory
Super useful
Everywhere
Staple of functional programming
Write code that’s ignorant of unnecessary context

108. III. Monoids

109. Monoids
Some set we’ll call M
Compose
• : M × M  M
Identity
id : M

110. Monoid Laws
Associative Law
(f • g) • h = f • (g • h )
Identity Laws
f • id = id • f = f

111. Category Laws
Associative Law
(f ∘ g) ∘ h = f ∘ (g ∘ h )
Identity Laws
f ∘ id = id ∘ f = f

112. Monoids
Compose
• : M × M  M
Identity
id : M

113. Category
Compose
∘ : (B  C)  (A  B)  (A 
C)
Identity
id : A  A

114. Category with 1 object
Compose
∘ : (A  A)  (A  A)  (A 
A)
Identity
id : A  A

115. Category with 1 object
Compose
∘ : M  M M
Identity
id : M

116. Monoids are categories
Each arrow is an element
in the monoid
Only one object

117. Monoids are categories
Objects = placeholder singleton
Arrows = elements of the monoid
Composition = •
Identity = id
Only one object
Each arrow is an element
in the monoid

118. M
H
N
Monoid Monoid
Monoid homomorphism
(SM, •M, idM) (SN, •N, idN)

119. M
H
N
Monoid Monoid
Monoid homomorphism
(SM, •M, idM) (SN, •N, idN)
MonCategory of monoids

120. M
H
N
Monoid Monoid
Monoid homomorphism
(SM, •M, idM) (SN, •N, idN)
MonCategory of monoids
Objects = monoids
Arrows = monoid homomorphisms
Composition = function composition
Identity = Identity function

121. M H
N
SM SN
“structure-preserving map”
Set Set
function
h
Sets
Where h preserves composition &
identity

122. Example
String length is a monoid
homomorphism from
(String, +, "") to
(Int, +, 0)

123. Preserves identity
Preserves composition
"".length == 0
(str1 + str2).length =
str1.length + str2.length

124. Monoids in code
trait Monoid[M] {
def compose(a: M, b: M): M
def id: M
}

125. Monoids in code
def foldMonoid[M: Monoid](
ms: Seq[M]): M = {
ms.foldLeft(Monoid[M].id)
(Monoid[M].compose)
}

126. Int / 0 / +
import IntAddMonoid._
foldMonoid[Int](Seq(
1,2,3,4,5,6))
 21

127. Int / 1 / *
import IntMultMonoid._
foldMonoid[Int](Seq(
1,2,3,4,5,6))
 720

128. String / "" / +
foldMonoid[String](Seq(
"alea",
"iacta",
"est"))
 ”aleaiactaest"

129. Endos / id / ∘
def mash: Potato => Potato
def addOne: Int => Int
def flipHorizontal: Shape => Shape
def bestFriend: Person => Person

130. A=>A / a=>a / compose
foldMonoid[Int => Int](Seq(
_ + 12,
_ * 2,
_ - 3))
 (n: Int) => ((n + 12) * 2) - 3

131. Are chairs monoids?

132. Chair
Composition =
You can’t turn two chairs into one
Identity =

133. Chair stack

134. Chair stack
Composition = stack them on top
Identity = no chairs

135. Chair Stack is
the
free monoid of
chairs
Protip: just take 0-to-many of
anything, and you get a
monoid for free

136. …almost
Real monoids don’t topple;
they keep scaling

137. Monoids embody the
principle of
weakly-typed
composability

138. IV. Products & sums

139. Algebraic Data Types
List[A]
- Cons(A, List[A])
- Nil
Option[A]
- Some(A)
- None
BusinessResult[A]
- OK(A)
- Error
Wiggles
- YellowWiggle
- BlueWiggle
- RedWiggle
- PurpleWiggle
Address(Street, Suburb,
Postcode, State)

140. Algebraic Data Types
Cons(A × List[A])
+ Nil
Some(A)
+ None
OK(A)
+ Error
YellowWiggle
+ BlueWiggle
+ RedWiggle
+ PurpleWiggle
Street × Suburb × Postcode × State

141. Algebraic Data Types
A × List[A] + 1
A + 1
A + 1
4
Street × Suburb × Postcode × State

142. Algebraic Data Types
A × List[A] + 1
A + 1
A + 1
4
Street × Suburb × Postcode × State
isomorphic

143. Terminology
isomorphism

144. Terminology
isomorphism
Equal

145. Terminology
isomorphism
Equal-shape-ism

146. Terminology
isomorphism
“Sorta kinda the same-ish”
but I want to sound really
smart
- Programmers

147. Terminology
isomorphism
“Sorta kinda the same-ish”
but I want to sound really
smart
- Programmers

148. Terminology
isomorphism
One-to-one mapping
between two objects so
you can go back-and-forth
without losing information

149. Isomorphism
object
object
arrows

150. Isomorphism
Same as
identity

151. Isomorphism
Same as
identity

152. These 4
Shapes
Wiggles
Set
functions
Set

153. These 4
Shapes
Wiggles

154. These 4
Shapes
Wiggles

155. There can be lots of
isos between two
objects!
If there’s at least one, we
can say they are
isomorphic
or A ≅ B

156. Products
A × BA B
first seco
nd
Given the product of A-and-B,
we can obtain both A and B

157. Sums
A + BA B
left right
Given an A, or a B, we have
the sum A-or-B

158. Opposite categories
C Cop
A
B
C
g ∘ f
f
g
A
B
C
fop ∘ gop
fop
gop
Isomorphic!

159. A
B
C
g ∘ f
f
g
A
B
C
f ∘ g
f
g
Just flip the arrows, and
reverse composition!

160. A
A×B
B
A product in C is a sum in Cop
A sum in C is a product in Cop
A+B
B
A
C Cop

161. Sums ≅ Products!

162. Terminology
dual
An object and its equivalent in the
opposite category are
to each other.

163. Terminology
Co-(thing)
Often we call something’s dual a

164. Terminology
Coproducts
Sums are also called

165. V. Composable
systems

166. Growing a system
Banana

167. Growing a system

168. Growing a system

169. Growing a system
Bunch

170. Growing a system
Bunch
Bunch

171. Growing a system
Bunch
Bunch
Bunch

172. Growing a system
Bunch
Bunch
Bunch
BunchManager

173. Growing a system
Bunch
Bunch
Bunch
BunchManager
AnyManagers

177. compose

179. compose

180. etc…

181. Using composable
abstractions means your
code can grow without
getting more complex
Categories and Monoids
capture the essence of
composition in software!

182. Look for Monoids and
Categories in your domain
where you can
You can even bludgeon non-
composable things into free
monoids and free categories

183. VI. Abstraction

184. Spanner

185. Spanner
AbstractSpanner

186. Spanner
AbstractSpanner
AbstractToolThing

187. Spanner
AbstractSpanner
AbstractToolThing
GenerallyUsefulThing

188. Spanner
AbstractSpanner
AbstractToolThing
GenerallyUsefulThing
AbstractGenerallyUsefulThingFactory

189. Spanner
AbstractSpanner
AbstractToolThing
GenerallyUsefulThing
AbstractGenerallyUsefulThingFactory
WhateverFactoryBuilder

191. That’s not what
abstraction
means.

192. Code shouldn’t know
things that aren’t
needed.

193. def getNames(users: List[User]):
List[Name] = {
users.map(_.name)
}

194. def getNames(users: List[User]):
List[Name] = {
println(users.length)
users.map(_.name)
}
Over time…

195. def getNames(users: List[User]):
List[Name] = {
println(users.length)
if (users.length == 1) {
s”${users.head.name} the one and only"
} else {
users.map(_.name)
}
}

196. “Oh, now we need
the roster of names!
A simple list won’t
do.”

197. def getRosterNames(users: Roster[User]):
Roster[Name] = {
users.map(_.name)
}

198. def getRosterNames(users: Roster[User]):
Roster[Name] = {
LogFactory.getLogger.info(s”When you
party with ${users.rosterTitle}, you must
party hard!")
users.map(_.name)
}
Over time…

199. def getRosterNames(users: Roster[User]):
Roster[Name] = {
LogFactory.getLogger.info(s"When you
party with ${users.rosterTitle}, you must
party hard!")
if (users.isFull) EmptyRoster("(everyone)
")
else users.map(_.name)
}

200. When code knows too much,
soon new things will appear
that actually require the other
stuff.

201. Coupling has
increased. The mixed
concerns will tangle
and snarl.

202. Code is rewritten each time for
trivially different requirements

203. def getNames[F: Functor](users: F[User]):
F[Name] = {
Functor[F].map(users)(_.name)
}
getNames(List(alice, bob, carol))
getNames(Roster(alice, bob, carol))

204. Not only is the abstract code
not weighed down with
useless junk, it can’t be!
Reusable out of the box!

205. Abstraction is
about hiding
unnecessary
information. This
a good thing.
We actually know more about what the code
does, because we have stronger
guarantees!

206. We’ve seen deep
underlying patterns
beneath superficially
different things
A×B A+B

207. Just about everything
ended up being in a
category, or being one.

208. There is no better
way to understand
the patterns
underlying software
than studying
Category Theory.

209. Further reading
Awodey, “Category Theory”
Lawvere & Schanuel, “Conceptual Mathematics: an
introduction to categories”
Jeremy Kun, “Math ∩ Programming” at
http://jeremykun.com/
Gabriel Gonzalez “Haskell for all”
http://www.haskellforall.com/2012/08/the-category-design-
pattern.html
http://www.haskellforall.com/2014/04/scalable-program-
architectures.html