1.
Category Theory
in 10 Minutes
Jordan Parmer
@ProfParmer

2.
What’s Category Theory?

3.
What’s Category Theory?
Mathematical study of abstract
algebras of functions
Systems of functions among some
objects

4.
What’s a Category?

5.
A B
f

6.
A B
fObject Object
Arrow/Morphism

7.
A B
fObject
Arrow/Morphism
Object
Domain (f) Codomain (f)

8.
A B
C
f
g

9.
A B
C
f
g
g ∘ f
g ∘ f = g ( f ( a ) )

10.
A B
C
f
g
g ∘ f
Object
domain(f)
Arrow
Composite
Object
codomain(f)
domain(g)
Arrow
Object
codomain(g)( g ◦ f )( a ) = g ( f ( a ) ) a ∈ A

11.
A B
C
f
g
g ∘ f
● Unit (identity) law: 1A : A → A, f ∘ 1A = f = 1B ∘ f
● Associativity law: h ∘ (g ∘ f) = (h ∘ g) ∘ f

12.
A B
C h
g
g ∘ f
D
f
h ∘ g

13.
Category theory is just
mathematicians playing Portal.

14.
Oh the names we’ll use...
Category theory is comprised of many small, simple laws.
Adding a certain combination of laws together forms various
algebraic structures each with their own name.

15.
https://medium.com/@sinisalouc/demistifying-the-monad-in-scala-part-2-a-category-theory-approach-2f0a6d370eff

16.
Functional Concepts
Functor
Transforms from one
category to another.
A => B
B => C
Endofunctor
Transforms objects between
same category. (e.g. Functor
in the category of types)
Int => Int
String => Int
String => TaskResult

17.
Functional Concepts
Functor
Transforms from one
category to another.
A => B
B => C
Endofunctor
Transforms objects between
same category. (e.g. Functor
in the category of types)
Int => Int
String => Int
String => TaskResult
Functors in most programming languages are
all endofunctors because most programming
languages are in the category of Types.

18.
Functional Concepts
Functor
Transforms from one
category to another.
A => B
B => C
Endofunctor
Transforms objects between
same category. (e.g. Functor
in the category of types)
Int => Int
String => Int
String => TaskResult
Often called ‘map’ or ‘fmap’ function
Ex: “pickle”.map(c => c + “z”)
Yields: (“pz”, “iz”, “cz”, “kz”, “lz”, “ez”)

19.
Functional Concepts
Semigroup
◦ Append
Append
1 + 2
“Hello” + “World”
[1, 2, 3] + [4, 5, 6]
Monoid
◦ Identity
◦ Append
Identity
0 + 1 == 1 + 0 == 1
“” + “A” == “A” + “” == “A”
[ ] + [1] == [1] + [ ] == [1]
*Append -> binary associative operator

20.
Functional Concepts
Monad
◦ Identity as Apply
◦ Append as
Flatten
Monoid in the
category of
endofunctors.
For all practical purposes,
it is a monoid that can be
flattened if the inner type
is itself a monoid.
Maybe(Maybe(T)) to Maybe(T)
List(List(T)) to List(T)
List(Maybe(T)) to List(T)

21.
Chaining
Monads
Pictures by Aditya Bhargava
https://tinyurl.com/cmvrfpb
“Bind” operation
plucks value from
monadic context
and applies it to
next function
returning it back in
context.

22.
Thanks!
ANY QUESTIONS?
You can find me at
@ProfParmer