# An Introduction to Functors

# An Introduction to Functors

### This comprehensive guide to Functors examines their usefulness in dealing with containers, their constraints, and some unbreakable laws.

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Join For FreeIn a previous article, we talked about Semigroups and Monoids, which are abstractions that let us combine values of the same type together. In this post, we’ll take a look at **Functors**, which allow us to operate on values inside containers without having to know any of the implementation details of the containers themselves.

**Note**: This article is based on an Axosoft Dev Talk titled

*Practical Category Theory: Functors*. Watch that video or keep reading!

**Higher-Kinded Types**

When we program in a language like C# or Java, we often run into “concrete” types like `int`

, `string`

, and `bool`

. The neat thing about concrete types is that we always know all the operations that we can perform on them (`int`

s can be added, `bool`

s can be negated, and so on).

One step up on the abstraction ladder are “generic” types, like `List<T>`

. A fancy term for generic types is “parametric polymorphism:” “parametric” because we’re working with a type parameter, and “polymorphism” because the parameter in question can have multiple (“poly-“) shapes (“-morphism”). So, we know what operations we can perform on the `List`

itself (iterate over all its elements, `Add`

an item, etc.), but we know absolutely nothing about the type represented by `T`

. This gives us a lot more power of abstraction because we can write methods that manipulate these generic structures and have them guarantee to work no matter what type we eventually fill in for the type parameter.

`List<T>`

as
`List t`

: the name of the generic type (
`List`

) comes first, and the name of the type parameter (
`t`

) comes next, separated by a space. Haskell and Purescript know that
`t`

isn’t the name of a concrete type (like if we had written
`Int`

or
`String`

) because it starts with a lowercase letter.
We can go one step further and write a type like `IEnumerable<T>`

, where our container type isn’t a concrete type but is only an interface. Now, we know only a little bit about the container type itself (specifically, that it has elements over which we can iterate), and we still know nothing about `T`

. The number of operations we can perform on a value of type `IEnumerable<T>`

is even smaller than those for `List<T>`

. This limitation is actually a good thing because now we can pass in a `Stack`

or a `Queue`

to a method that takes an `IEnumerable`

, for example, and the method will work as expected.

Usually, this is where we would have to stop because most languages don’t let us go any more abstract. However, Haskell and Purescript don’t have this restriction and support so-called “higher-kinded” types, where we can make both the internal type *and* the container type fully generic. If C# had syntactical support for this, it might look something like `T1<T2>`

. `T1`

could be `IEnumerable`

, `Queue`

, etc., while `T2`

could be `int`

, `string`

, etc. Haskell and Purescript, however, *do* support this concept of higher-kinded types and use this syntax: `f t`

(where `f`

is the container type and `t`

is the type parameter). Because `f`

is lowercase, we know that it’s generic as well as `t`

, and so can be any type that requires exactly one type parameter. For example, the following types would fulfill `f`

:

`List`

, which holds zero or more values of the same type.`Queue`

, which also holds zero or more values of the same type, but doesn’t allow random access.`Maybe`

(A.K.A.`Option`

or`Optional`

), which can contain either one value or be empty.`Promise`

, which eventually produces a value.- A Redux store, which we can think of being parameterized by the type of the state it contains.

Whereas the following *cannot* be used for `f`

:

`Map`

, because that requires*two*type parameters, one for the key and one for the value.`string`

, because that doesn’t have any type parameters (another way of saying that is that the type string is already fully concrete).`Tuple`

, because that also requires two or more type parameters (depending on the number of elements it contains).- A Redux reducer, which requires two type parameters, one for the message and one for the state.

### Fun Fact!

In Haskell or Purescript, the higher-kinded type parameter (`f`

in `f t`

) is usually named `f`

or `m`

, while the type parameter it takes (`t`

in `f t`

) is usually named `a`

(or `b`

or `c`

if more than one is needed).

## Constraints

Granted, we really can’t do very much with a value of type `f t`

because we intentionally don’t know very much at all about it. However, we can put constraints on our type parameters. In C#:

`class Foo<T> where T : SomeConstraint`

Now, `T`

can’t be just anything but instead has some restrictions on what can be plugged in for it. After we’ve put one or more constraints on `T`

, we now know more about what we can do with it. For example, if `T`

is constrained to be an instance of `IComparable`

, that means `Foo`

will only accept types that support the `CompareTo`

method, like `int`

or `char`

(but *not*, say, `List<string>`

). In Haskell or Purescript, this type can be written `SomeConstraint t => Foo t`

, which means that type `t`

*must* support the operations in `SomeConstraint`

.

In Haskell and Purescript, we can also place constraints on our higher-kinded type parameters, like so: `SomeConstraint f => f t`

. Now, we know that `f`

supports whatever operations are included in `SomeConstraint`

, and we can plug in any type that supports them. This is approximately analogous to the idea behind `IEnumerable<T>`

, where the container type is abstract and so we can choose a specific type, like `List`

or `Queue`

, to make the type concrete. (We might write that example in Haskell and Purescript as `IEnumerable f => f t`

.) We can also use more than one constraint on a single parameter or place constraints on more than one parameter at a time, like so:

`(SomeConstraint f, SomeOtherConstraint f, AnotherConstraint t, YetAnotherConstraint t) => f t`

## Recap

What all this allows us to do is to have a system of specifying the *structures* of values, which will come in handy when we get to Functors below. Higher-kinded types allow us to focus on values that have general structure, and we can narrow the possibilities down with constraints.

Phew, that’s a lot to digest. Here’s a summary of how the different concepts (roughly) translate between C# and Haskell and Purescript.

### Concrete Types

We know everything there is to know about these types.

#### C#

`int`

#### Haskell, Purescript

`Int`

### Generic Types

We know everything there is to know about *part* of these types.

#### C#

`Foo<T>`

#### Haskell, Purescript

`Foo t`

### Constraints

We know *some* things about *various parts* of these types.

#### C#

```
// Constraint on the "traditional" type parameter
Foo<T> where T : IBar
// Constraint on the container type itself
// Approximately:
IFoo<T>
```

#### Haskell, Purescript

```
-- Constraint on the "traditional" type parameter
IBar t => Foo t
-- Constraint on the container type
IFoo f => f t
```

### Higher-Kinded Types

We only know about the very general shape of these types, but we can place constraints on them in order to do useful things with them.

#### C#

```
// Doesn't exist, but might look something like:
T1<T2>
```

#### Haskell, Purescript

`f t`

### Functors

Lo and behold, we’ve made it to Functors!

We’ve been making statements about the “structures” of types without a clear definition, so here goes:

### Structure

Rules about how you can use a type (i.e. the operations that can be performed on it). In an object-oriented (OO) language, this might be represented by interfaces; in a functional language, this might be represented by modules or typeclasses.

`IEnumerable<T>`

, for example, has a different structure from `IComparable<T>`

, because `IEnumerable`

s can be iterated over, sorted, etc., while `IComparable`

s can be compared.

Functor is a specific structure that supports exactly one operation: `map`

. `map`

allows us to operate on value(s) inside a Functor (which we can think of as a container of some sort) without knowing (or caring) how the Functor is implemented. Each Functor has its own rules for how `map`

works, but this general theme of manipulating contained values is the same.

You’ve already used `map`

if you’ve ever used `Array.prototype.map`

in JavaScript or LINQ’s `IEnumerable.Select`

in C#, for example. You didn’t have to know how the container (the `Array`

or specific `IEnumerable`

) was implemented, since its implementation of `map`

took care of that for you. All you needed to do was provide a function that takes an element of the list and returns something else, and the container handled the rest.

Without further ado, let’s take a look at `map`

‘s type signature!

#### C#

```
public interface IFunctor {
// There's no second parameter, because this method is on the `IFunctor` we want to map over
IFunctor<TResult> Map<TResult>(Func<T, TResult> fn);
}
```

#### Haskell, Purescript

```
-- Here, we explicitly write the Functor to map over as the second parameter, because Haskell is functional rather than object-oriented
map :: Functor f => (a -> b) -> f a -> f b
```

`map`

takes a function and a Functor and returns the Functor after the function has been applied to its element(s). The type of the element(s) inside the Functor is allowed to change.

So, Functors are really, really simple. Their only special “skill” is that they allow you to apply a function to what’s inside them.

**What Can (and Can’t) Be a Functor?**

So far, we’ve only seen `List`

‘s implementation of Functor, but what else could be a Functor?

`Queue`

can, because it can allow a function to be applied to all its elements. `Maybe`

can also be a Functor: if there is no value, then an empty `Maybe`

is returned; otherwise, a `Maybe`

with its value modified by the provided function is returned. `Promise`

can implement map by allowing you to modify the value that the `Promise`

will eventually resolve to (e.g. `somePromise.then(x => x + 1)`

).

So what *can’t* be a Functor? Well, a Redux store probably can’t. While it certainly allows its state to be modified, being a Functor would mean that the type of the state could change, since any function we use with `map`

with must be allowed to change the type of the elements inside the Functor. We usually don’t want the *type* of our Redux state to change over time, so a Redux store isn’t a valid Functor.

Let’s see some quick examples of Functors in use. Here’s what it looks like to map over a list in JavaScript, C#, and Haskell and Purescript:

#### JavaScript

`[1, 2, 3].map(x => x.toString()) // ["1", "2", "3"]`

#### C#

`new[] { 1, 2, 3 }.Select(x => x.ToString()).ToList() // new[] { "1", "2", "3" }`

#### Haskell, Purescript

`map show [1, 2, 3] -- ["1", "2", "3"]`

Note that in the above example, the `List`

started out as a `List<int>`

but was converted to a `List<string>`

. This is another important property of `map`

that we’ll get to below.

And here’s what it looks like to `map`

over JavaScript’s `Promise`

type instead:

#### JavaScript

```
Promise.resolve(1)
// Think of `then` as `map`
.then(x => x.toString()) // resolves to "2"
```

In both cases, we’re handing `map`

a function and each Functor is doing the rest for us in its own specific way.

As we can see, the concept of a Functor is intentionally very general, and this idea of very general abstractions is what makes category theory so powerful.

**Example**

To summarize: Functors allow us to treat “what to do” and “how to do it” as separate concerns. After a `map`

implementation is written for a type, it can be used the same way with any function (as long as its parameters are the correct types, of course).

Now for a real-world example! Let’s pretend that we have a `Maybe`

type of some sort in JavaScript, along with a corresponding `fromMaybe`

function that takes a modifying function, a default value, and the `Maybe`

to map over. If the `Maybe`

contains a value, `fromMaybe`

will return a modified version of that value; if, however, the `Maybe`

does *not* contain a value, `fromMaybe`

will return the default value that it was passed.

As an example, we might traditionally work with possibly nonexistent values like so:

```
const result = thingThatMightFail();
return result.value
? result.value + otherValue
: 0;
```

The issue with this approach, however, is that we can forget to check that `result`

is `null`

or `undefined`

(‘cannot read property of undefined,’ anyone?). If, however, we use some sort of `Maybe`

object, we might be able to rewrite the above using `fromMaybe`

as follows:

```
const result = thingThatReturnsAMaybe();
return fromMaybe(
v => v + otherValue, // Function to apply if we have a value
0, // Default value
result // Maybe object
);
```

Now, if we forget to use `fromMaybe`

on our `result`

, we’ll catch it when we try to actually use this value. Granted, this isn’t as useful in JavaScript as it is in a statically-typed language like C# where we’d be able to catch such errors at compile-time, but using `Maybe`

like this everywhere we’d use `null`

or `undefined`

still allows us to reason about our code and know what can return a value that might not exist (instead of just hoping that we remembered to check for `null`

or `undefined`

everywhere we use a potentially nonexistent value).

If we want to run multiple operations in sequence on a value that might not exist, we might think to do it like this:

```
const json = someAjaxRequest();
let result = defaultUser;
if (json) {
const id = extractId(json);
const user = getUserById(id);
result = upgradeUser(user);
}
```

However, with `Maybe`

, we can write it differently:

**Note:**`R.pipe`

is a function from the Ramda.js library that “composes” functions: it takes several functions and chains them together to create a new function (for more details, see the docs).

Also, for clarity, psuedo Flow function type annotations are included.

```
const maybeJson: Maybe<JSON> = someAjaxRequest(); // someAjaxRequest: () => Maybe<JSON>
const result: User = R.pipe(
(maybeJson: Maybe<JSON>) => map((extractId: (JSON) => Id), maybeJson),
(maybeId: Maybe<Id>) => map((getUserById: (Id) => User), id),
(maybeUser: Maybe<User>) => fromMaybe(
(upgradeUser: (User) => User),
(defaultUser: User),
maybeUser
)
)(maybeJson);
```

And here’s the same example, but this time we’ll remove unnecessary Flow type annotations and take advantage of a technique called currying in order to see what the same code might look like in a real-world situation:

```
const maybeJson: Maybe<JSON> = someAjaxRequest();
const result: User = R.pipe(
map(extractId),
map(getUserById),
fromMaybe(upgradeUser, defaultUser)
)(maybeJson);
```

So, let’s see what we gain from using Functors. As opposed to in the first, ‘traditional’ example, we didn’t have to manually handle the case where no value was returned. The `Maybe`

Functor itself was able to decide how to do so, and all we needed to worry about was giving `map`

the functions that we’d like to perform if a value does happen to exist. We can use any functions we want for each step in the pipeline, as long as they have the correct types (namely, the return type of each one must be the type of the next one’s parameter).

Functors are incredibly common in practice, and a good rule-of-thumb is to ask yourself whenever you begin to write a function that operates on a data structure if the function could be made general enough to operate on a Functor instead. For example, Haskell/Purescript functions like `replace`

and `zip`

*could* be defined to work with lists specifically, but they’re instead implemented to only require that their parameters are Functors; as such, they work on many different structures for free! They don’t need to know anything about how each of those Functors works on the inside because `map`

handles the details for them.

## Laws

Finally, Functors have two associated “laws” to ensure that the Functor will behave as one would expect. Don’t worry, you won’t go to jail if you get these laws wrong (although it is certainly an unpleasant surprise when a Functor is “unlawful”).

### Law #1: Identity

Mapping the identity function over a Functor has no effect.

Fancy pseudo-Haskell way to say it:

`map id = id`

This one’s pretty self-explanatory: if your function doesn’t actually do anything to the element(s) inside the Functor, then it would certainly be surprising if you got back a different result than what you passed in.

#### JavaScript Example

`map(x => x, [1, 2, 3]) // [1, 2, 3]`

### Law 2: Composition

Multiple `map`

s in a row can be composed into one without any changes in the overall behavior.

Fancy pseudo-Haskell way to say it (`(.)`

is the function composition operator):

`map f . map g = map (f . g)`

#### JavaScript Example

We could write our earlier JSON-manipulation example like so if we combine the two-in-a-row `map`

s into a single `map`

, with the new function to use being the composition of the two steps in our pipeline.

`R.pipe(map(extractId), map(getUserById))(json) // map(R.pipe(extractId, getUserById), json)`

## Go Forth and Map!

And that’s it for Functors! They’re very simple structures that give us a lot of power. Just like Semigroups and Monoids, they’re used very commonly because they’re so simple to reason about. And, because they’re so general they allow us to write functions once and use them on all sorts of different data structures, guaranteed!

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