This document discusses category theory concepts including functors, exponential functors, currying, and monads. It provides examples of functors such as the identity functor, product functor, and +1 functor. It explains how currying relates exponential functors to function spaces. It also gives examples of functors between categories of integers and rationals. Finally, it defines monads as endofunctors with natural unit and flatten operations.

1. Formal Methods in
Software
Lecture 8. Category Theory
Vlad Patryshev
SCU
2014

2. In This Lecture
● functors and examples (diagrams; product; exponentiations);
● currying/yoneda lemma;
● example with integers/rationals;
● monad

3. Example: Parametric Class
trait List[A] extends Iterable[A] {
def map[A](f:A=>B):List[B]
}

4. Functor
Given categories A and B, a functor F: A → B
consists of mappings F: Obj(A) → Obj(B)
and F: Fun(A) → Fun(B) ,
so that
F(idX
)=idF(X)
and F(f∘g)=F(f)∘F(g)

5. More Examples
● A= 3 , B=Sets; what is a functor 3 → Sets?
Have to define: sets F(0), F(1), F(2); functions F(f), F(g), F(gf)
● Any f:X→Y in category C is a functor from 2 to C.

6. Cartesian Product is a Functor
Fix an object A.
X ↦ X×A is a functor.
How does it work on functions?
f:X→Y ↦ f×idA
: X×A → Y×A

7. +1 is a Functor
In Sets, take X ↦ X+1. f: X→Y ↦ (f+1): X+1 → Y+1
This functor is called Option[T].

8. Category of Categories
Take functors F:A→B, G:B→C; their composition H=G∘F is also a functor.
H(idX
) = G(F(idX
)) = G(idF(X)
) = idG(F(X))
H(p∘q) = G(F(p∘q)) = G(F(p)∘F(q)) = G(F(p)) ∘G(F(q)) = H(p) ∘H(q)
Identity functor IdA
:A→A Id(f) = f
Cat - a category of all categories.
No barber problems, nobody shaves nobody.

9. Exponential Functor
A, B - objects in category C. BA
- an object of functions from A to B, if such
exists.
E.g. in Set, {f:A→B}.
In Scala In Java (Guava) (https://code.google.com/p/guava-
libraries/wiki/FunctionalExplained)
trait Function[X,Y] {
def apply(x:X): Y
}
interface Function<X,Y> {
Y apply(X x);
}
val len:Function[String, Int] = _.length Function<String, Integer> lengthFunction = new
Function<String, Integer>() {
public Integer apply(String string) {
return string.length();
}
};

10. Define Exponential via Currying
In Set, f:A→CB
≡ f’:A×B→C
f(x)(b) = f’(x,b)
Very similar to P⊢(Q→R) ≡ P∧Q ⊢ R
Actually… it’s the same thing.
Take posets, for example, Boolean lattice,
where a×b ≡ min(a,b) /*see lecture 7*/ ≡ a∧b
a∧b ≤ c ≡ a ≤ (b→c)

11. Currying: Yoneda Lemma
Fun(A×B, C) ≡ Fun(A, CB
)
(Caveat: what is Fun? A set? It’s not always available as a set)

12. Example with Numbers
Category Z = (Z,≤); category Q = (Q,≤)
Inclusion iz
: Z ↣ Q - preserves order, so is a functor.
How about Q → Z? Upb: q ↦ Upb(q)
Upb(q) ≤ n /* this is in Z */ ≡ q ≤ n /* this is in Q */
Take composition q ↦ iZ
(Upb(q)), call it Mint.
We see two properties:
● q ≤ Mint(q)
● Mint(Mint(q)) ≤ Mint(q) /* actually equal */

13. Example with Lists
Category Scala
Take functor List.
We see two properties:
● singleton: X → List[X]
● flatten: List[List[X]] → List[X]
Do you see similarity?

14. Monad
* f:F(X)->G(X) is natural if for all p:X→ Y
Given a category C, an endofunctor M: C → C
is called a Monad if it has two features:
● unitX
: X → M(X)
● flattenX
: M(M(X)) → M(X)
These two functions should be natural*

16. References
http://www.amazon.com/Category-Computer-Scientists-Foundations-Computing/dp/0262660717
http://fundeps.com/tables/FromSemigroupToMonads.pdf
Wikipedia