Notes on Category Theory for Programmers

1 Category: The Essence of Composition

  • Categories consist of objects and arrows that go between them
  • The essence of a category is composition, i.e. that A -> B and B -> C implies A -> C

1.1 Arrows as Functions

  • Arrows are morphisms which are equivalent to functions
  • Objects are equivalent to types.
  • If we have f :: A -> B and g :: B -> C then (g . f) :: A -> C

1.2 Properties of Composition

Composition in any category must satisfy two properties:

  • Composition is associative, which means the order of brackets doesn’t matter. This means for functions f g and h, h . (g . f) = (h . g) . f = h . g. f
  • Every object has an identity mapping back to itself. If we have f :: A -> B then f . identityA = f and identityB . f = f. Note the identity is per object.

1.3 Composition is the Essence of Programming

Why does this matter for programming? Because decomposing problems, creating solutions to simpler problems and then composing them is the essence of programming. Categories provide a mathematical basis for thinking about this.

1.4 Challenges

Q: Implement, as best as you can, the identity function in your favorite language (or the second favorite, if your favorite language happens to be Haskell).


const identity = x => x

Q: Implement the composition function in your favorite language.It takes two functions as arguments and returns a function that is their composition.


const comp = (f, g) => x => g(f(x))

Q: Write a program that tries to test that your composition function respects identity.


const double = x => x + x
console.assert(comp(double, identity)(10) === double(10))

Q: Is the world-wide web a category in any sense? Are links morphisms?

A: Yes. If we view links between pages as a function from “visiting page A” to “visiting page B” then links are composable (a crawler could follow a chain of links equivalently to a single link), and pages are a category under links.

Q: Is Facebook a category, with people as objects and friendships as morphisms?

A: No, as friendship isn’t composable (I may be friends with you, but some of your friends are completely unknown to me and thus not my friends), and there’s also no identity function as people can’t friend themselves. We’d need an identity and an axiom such as “the friend of my friend is my friend” for this to be a category.

Q: When is a directed graph a category?

A: A directed graph is a category provided we allow nodes to be adjacent to themselves, otherwise we do not have an identity function.

2 Types and Functions

2.1 Who needs Types?

Thought experiment: millions of monkeys typing random keys, producing programs, compiling and running them.

With machine language, any combination of bytes produced by monkeys would be accepted and run. But with higher level languages, we do appreciate the fact that a compiler is able to detect lexical and grammatical errors… Type checking provides yet another barrier against nonsensical programs.

The usual goal in the typing monkeys thought experiment is the pro- duction of the complete works of Shakespeare. Having a spell checker and a grammar checker in the loop would drastically increase the odds. The analog of a type checker would go even further by making sure that, once Romeo is declared a human being, he doesn’t sprout leaves or trap photons in his powerful gravitational field.

2.2 Types are about composability

  • Arrows can only be composed when targets and sources match, i.e. for programs to be composable the input / output types must match.
  • Strong static typing can eliminate semantically valid programs, but this is rare and escape hatches exist (e.g. unsafeCoerce in Haskell).
  • Type inference removes some friction from strong static typing, though designing types up front can be a valuable design technique.
  • Strong static typing does not replace testing (e.g. it does not guarantee program logic is correct), and testing does not replace strong typing. Tests are probabilistic not deterministic and can’t typically cover all cases.

Testing is a poor substitute for proof

2.3 What are Types?

2.4 Why do we need a mathematical model?

2.5 Pure and dirty functions

2.6 Examples of Types

2.7 Challenges

Q: Define a higher-order function (or a function object) memoize in your favorite language.


var memoize = f => {
	const cache = {}
	return (...args) => {
		if (cache[args]) return cache[args]
		cache[args] = f(...args)
		return cache[args]