Video and slides synchronized, mp3 and slide download available at URL http://bit.ly/2DLgTqd. Philip Wadler explains why category theory is of interest for developers and how categories model three basic data types: products (logical and), sums (logical or), and functions (logical implication). Filmed at qconsf.com. Philip Wadler is Professor of Theoretical Computer Science at the University of Edinburgh and Senior Research Fellow at IOHK. He contributed to the designs of Haskell, Java, and XQuery, and is a co-author of Introduction to Functional Programming (Prentice Hall, 1988), XQuery from the Experts (Addison Wesley, 2004) and Generics and Collections in Java (O'Reilly, 2006).

1. Categories for the
Working Hacker
Philip Wadler
University of Edinburgh
QCon SF, 15 Nov 2017

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Watch the video with slide
synchronization on InfoQ.com!
https://www.infoq.com/presentations/
category-theory-data-types

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14. Products in Java
public class Product<A,B> {
private A fst;
private B snd;
public Product(A fst, B snd) {
this.fst = fst; this.snd = snd;
}
public A getFst() {
return this.fst;
}
public B getSnd() {
return this.snd;
}
}

15. Products in Java
public class Test {
public Product<Integer,String> pair =
new Product(1, “two”);
public Integer one = pair.getFst();
public String two = pair.getSnd();
}

16. Products in Haskell
data Product a b =
Pair { fst :: a, snd :: b }

17. Products in Haskell
pair :: Product Int String
pair = Pair 1 “two”
one :: Int
one = fst pair
two :: String
two = snd pair

24. Sums in Java
public interface Sum<A,B> {
public <C> C caseExpr(Function<A,C> f,
Function<B,C> g);
}
public class Left<A,B> implements Sum<A,B> {
private A x;
public Left(A x) { this.x = x; }
public <C> C caseExpr(Function<A,C> f,
Function<B,C> g) {
return f.apply(x);
}
}
public class Right<A,B> implements Sum<A,B> {
private B y;
public Right(B y) { this.y = y; }
public <C> C caseExpr(Function<A,C> f,
Function<B,C> g) {
return g.apply(y);
}
}

25. Sums in Java
public class ErrInt extends Sum<String,Integer> {
public ErrInt err = new Left(“error”);
public ErrInt one = new Right(1);
public ErrInt add(ErrInt that) {
return this.caseExpr(
e -> new Left(e),
m -> that.caseExpr(
e -> new Left(e),
n -> new Right(m+n)
)
);
public ErrInt test = one.add(err);
}

26. Sums in Haskell
data Sum a b = Left a | Right b

27. Sums in Haskell
type ErrInt = Sum String Int
err = Left “error”
one = Right 1
add :: ErrInt -> ErrInt -> ErrInt
add (Left e) that = Left e
add this (Left e) = Left e
add (Right m) (Right n) = Right (m+n)
test = add one err

32. Exponentials in Java
public class Test {
public Function<Integer,Integer>
add (Integer n) {
return x -> x + n;
}
public Function<Integer,Integer> incr =
add(1);
public Integer three = incr.apply(2);
}

33. Exponentials in Haskell
add :: Int -> (Int -> Int)
add n = x -> n + x
incr :: Int -> Int
incr = add 1
three :: Int
three = incr 2

34. Exponentials in Haskell
add :: Int -> Int -> Int
add n x = n + x
incr :: Int -> Int
incr = add 1
three :: Int
three = incr 2

38. Further Reading
• Saunders MacLane, Categories for the Working
Mathematician
• Benjamin Pierce, Basic Category Theory for Computer
Scientists
• Bartosz Milewski, Programming Cafe (blog)
Bartosz Milewski's Programming Cafe

39. Watch the video with slide
synchronization on InfoQ.com!
https://www.infoq.com/presentations/
category-theory-data-types