Maksim I

3. Categories and Functors

You probably already heard about such a thing as Functor. What is this actually? Before diving into it, let’s talk a little about the basic theory.

3.1 Category

3.1.1 What is a category

In simple words, the definition of the category is quite simple, and you are already familiar with it - it’s the composition! So, let’s look up again what is a composition:

f: A -> B
g: B -> C

g∘f: A -> C

The A B and C are objects of the category (types in terms of TypeScript). While arrows f and g are morphisms (functions).

Objects in category theory have nothing in common with objects in OOP! It is more of an "entity" or just a "black box".

So a category is a collection of objects and morphisms. And category has two basic properties: the ability to compose morphisms, and the existence of identity morphism. What is identity morphism? It’s just an arrow which points to the object itself. identity: A -> A. Nothing more actually.


3.1.2 TypeScript as a category

A category can be interpreted as a simplified model of a typed programming language where:

So, what is this all about and why it might be interesting to us? Because this is the way how the programs are working in FP! All programs are just categories within which other categories are managed. Let’s recall the initial explanation of what is the program: manipulating the data. And here, the data is objects, and manipulative actions is morphisms.

3.1.3 Properties of composition

The composition also has some properties, or rules, which are:

  1. Associativity: (f ∘ g) ∘ h = f ∘ (g ∘ h)
  2. Identity: f ∘ identity = f, where identity: A -> A

Associativity means that the order of composition doesn’t matter. So, f ∘ g ∘ h is the same as h ∘ g ∘ f. And the identity means that the composition with the identity morphism doesn’t change the morphism itself. So, f ∘ identity = f.

Associativity might seem complex, but it’s very important to understand, here’s a diagram which shows how it works: associativity

And a simple example of associativity in TypeScript:

Try in the playground: Listing 3.1.1 - Associativity example

const f = (a: number): string => a.toString();
const g = (b: string): boolean => b.length % 2 === 0;
const h = (c: boolean): 0 | 1 => c ? 1 : 0;

const fgh = compose(h, compose(g, f));
const ghf = compose(compose(h, g), f);

fgh(10) === ghf(10); // true

3.2 Functor

3.2.1 The problem

But the programs are not so simple, and we’re not always dealing with ideal cases like: f: A -> B; g: C -> D where B === C. What if B !== C? How to deal with such a case? What if B is actually F<B>, where F is some type of constructor or another category? Let’s sum up everything above and imagine an example:

type A = number;
type B = string;
type C = boolean;

const f = (a: A): B => a.toString(); // convert number to string
const g = (b: B): C => b.length % 2 === 0; // is the string length an even number?

const gf = compose(g, f);

f(10) === '10';
g('10') === true;

gf(10) === true;

Here’s everything all right. But what if f would return not a B, but F<B>?

Keep in mind, the code below can't be compiled and contains an error. TypeScript does not support Higher-Kind Types (passing type constructors as `Array` to parameters or assigning it to variables). Detailed realization of some functions is omitted in favour of simplicity.
type A = number;
type B = string;
type C = boolean;
type F = Array // Unfortunately we cannot create an alias for generics (type constructors) 

const f = (a: A): F<B> => [a.toString()];
const g = (b: B): C => b.length % 2 === 0; // is the string length an even number?

const gf = compose(g, f); // Error! The returned object of "f" is not assignable to the input object of "g"!  

What can help us here? How can we make it work? Here’s the Functor come in a game.

3.2.2 The solution

What do we need to do to make Listing 3.1.2 work? The g morphism should be transformed from g: B -> C to g: F<B> -> F<C>. But how to do this? How to map one category to another?

This is what the Functor is about. It is a mapping between categories. It has a map operator, which allows it to perform such mapping.

type Functor<Domain> = {
  map: <ObjectA, ObjectB>(
    f: (a: ObjectA) => ObjectB // Takes the morphism f: A -> B
  ) => (fa: Domain<ObjectA>) => Domain<ObjectB> // Returns a new morphism F<f>: F<A> -> F<B>

Try in the playground: Listing 3.2.4 - Usage of Functor

type Functor<M> = {
  //          f: A -> B       map(f): F<A> -> F<B>
  map: <A, B>(f: (a: A) => B) => (fa: M<A>) => M<B>

// Array - a list of values
const functorArray: Functor<Array> = {
  map: <A, B>(f: (a: A) => B) => (fa: Array<A>) => ( // As you perfectly know, Array already has a map operator, so it means that Array in JS is a Functor by default!

const fA = (a: number): Array<string> => [a.toString()];
const g = (b: string): boolean => b.length % 2 === 0;

fA(10) === ['10'];['10']) === [true];['10', '222', '3', '1111']) === [true, false, false, true];

const gfA = compose(, fA);

gfA(10) === [true];

Let me make this more abstract:

# Our situation becomes more complex:
f: A -> F<B>
g: B -> C

# We cannot create a composition, because the output of `f` is not a `B` but `F<B>`
🛑 g ∘ f: A -> F<C>

# But how Functor can help us? The Functor basically makes from morphism A -> B a new morphism F<A> -> F<B>
F<g>: F<B> -> F<C>

# But we can create a composition with the help of Functor
✅ F<g> ∘ f: A -> F<C>

So, in our example above, we’re creating a Functor for the Array type (more strictly say: Functor whose domain is the type Array). Then, we’re creating two functions, fA and g, and then create our composition function with the help of the Functor. And it works! We can create Functor for any type, not only for Array. For example, for Promise:

Try in the playground: Listing 3.2.5 - Functors for Task and Maybe

// @type {Functor<Promise>}
const functorTask = {
  map: <A, B>(f: (a: A) => B) => (fa: Promise<A>) => (
    // In fact, Promise is initially a Functor too, because it has an operator for mapping as well as Array has
    // It is only called "then" instead of "map"

const task = <A>(a: A): Promise<A> => Promise.resolve(a);

const fP = (a: number): Promise<string> => task(a.toString());
const g = (b: string): boolean => b.length % 2 === 0;

const gfP = compose(, fP);

fP(10) === task('10');'10')) === task(true);

gfP(10) === task(true);

 * We can also create our own type, and create a Functor for it!
 * Maybe - the value that might (and might not) be empty or none. Like nullable (Value | null)

type Some<A> = { type: 'some', value: A };
type None = { type: 'none' };
type Maybe<A> = Some<A> | None;

const isSome = <A>(m: Maybe<A>): m is Some<A> => m.type === 'some'

const some = <A>(a: A): Some<A> => ({
  value: a,
  type: 'some',
const none: None = { type: 'none' };

// @type {Functor<Maybe>}
const functorMaybe = {
  map: <A, B>(f: (a: A) => B) => (fa: Maybe<A>) => (
    isSome(fa) ? some(f(fa.value)) : fa

const fM = (a: number): Maybe<string> => some(a.toString());

const gfM = compose(, fM);

fM(10) === some('10');'10')) === some(true); === none;

gfM(10) === some(true);

It might look a little different from the way how we usually think about map. Why we’re using our good old map for Promises or this strange “Maybe”!? Isn’t it about iterating? Isn’t it about making an Array of usernames from my Array of User objects? Yes and no.

To make it easier, there are actually two viewpoints on the map:

  1. Morphism centric: something that makes from morphism of f: A -> B a new morphism of F<f>: F<A> -> F<B>
  2. Functor centric: something that makes from functor F<A> a new functor F<B> using a morphism f: A -> B

Both are correct, but the second one is more practical for us because in programs data transformation is more well spread than function transformation.

3.3 Thinking in Functional programming way

The hardest part of learning new paradigms is to switch your mindset and way of thinking when you code. We are learning how to code and think again.

It is easy to say that functional programming is about “what to do” rather than “how to do”. But this difference is significant!

When we are starting to work with functional programming, one says to us: “Here’s the map function. It is a functional way to iterate over an array, it accepts the callback as a parameter, this callback is applied to each value in the array, and returns a new array without modification of the previous one”. And this explanation is fully correct! But here’s the thing - this description is imperative. It sticks to the detail of how it behaves, not what it is for.

So, the map is for mapping. For mapping the F<A> to F<B>, it doesn’t matter if it is Array<User> to Array<string>, or Promise<User> to Promise<string>. The same operation, the same signature, the same purpose.

Let’s look at another example. What is a + operator in terms of JavaScript? It is an operator of addition. When you see 2 + 3, are you describing it as merging bits from right to left with the &? I guess not, we are saying that this is an addition.

And then, we see 'hello' + ' world'. Is it the same operation? Yes! But is it behave similarly to 2 + 3? Absolutely no, it has nothing in common with mathematical addition. Do you describe this operation as “creating a new string by filling it at the beginning with the chars of the first string, and then the second”? Also, no, just addition or concatenation of strings. And it seems quite natural to us. Even with the fact that this violates the base rule of addition - associativity.

5 + 3 === 3 + 5;

'hello' + ' world' !== ' world' + 'hello';

We just know the abstract meaning of the +. We can quite easily understand what the result this code would produce even without looking at the details:

type Point = { x: number; y: number };

type add = (point1: Point, point2: Point) => Point;
point1 + point2; // add(point1, point2)

Or, when we are talking about equality, it is also quite easy to understand this code:

type equals = (point1: Point, point2: Point) => boolean;

point1 === point2 // equals(point1, point2)

Quite easy, isn’t it? Now you know what Semigroup (addition) and Setoid (equals) are! Congratulations! (details about it are in the next chapter).

That’s it, the most important concept of functional programming - abstraction. You do not care what the actual data is. When you see that something is Functor, Monad or Semigroup, you already know what it does, without even looking at the code. Why?

Because as well as you don't need to think about how the computer handles the "5 + 3" or any other "+", you shouldn't think about how "" or any other "" is working. You just know what it does - mapping

3.3 Immutability

When we are talking about FP, we are always faced with the fact, that everything is immutable. But why?

Actually, it’s obvious. We mentioned above that everything is an expression, and by chaining functions, our values are transformed. Just like the 10 * 2 + 5, there is data (objects): 10, 2, 5; and operations (morphisms): *, + (in such languages as Haskell mathematical operations are indeed functions, which can be carried and follow all other rules).

Do we need explicitly mention that for example 10 is immutable? Of course not! It’s obvious! That’s right, we can’t even imagine how to mutate 10, it is just a constant, it can’t be anything but just the number 10.

In functional programming, the same logic applies to every data unit, everything is a constant. Be this a number, asynchronous data, an array, or data that might have an error or be empty - everything is the same, treated in the same way, and subject to the same rules.