# Applicative

`Applicative`

extends `Functor`

with an `ap`

and `pure`

method.

```
import cats.Functor
trait Applicative[F[_]] extends Functor[F] {
def ap[A, B](ff: F[A => B])(fa: F[A]): F[B]
def pure[A](a: A): F[A]
def map[A, B](fa: F[A])(f: A => B): F[B] = ap(pure(f))(fa)
}
```

`pure`

wraps the value into the type constructor - for `Option`

this could be `Some(_)`

, for `Future`

`Future.successful`

, and for `List`

a singleton list.

`ap`

is a bit tricky to explain and motivate, so we’ll look at an alternative but equivalent
formulation via `product`

.

```
trait Applicative[F[_]] extends Functor[F] {
def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
def pure[A](a: A): F[A]
}
// Example implementation for right-biased Either
implicit def applicativeForEither[L]: Applicative[Either[L, ?]] = new Applicative[Either[L, ?]] {
def product[A, B](fa: Either[L, A], fb: Either[L, B]): Either[L, (A, B)] = (fa, fb) match {
case (Right(a), Right(b)) => Right((a, b))
case (Left(l) , _ ) => Left(l)
case (_ , Left(l) ) => Left(l)
}
def pure[A](a: A): Either[L, A] = Right(a)
def map[A, B](fa: Either[L, A])(f: A => B): Either[L, B] = fa match {
case Right(a) => Right(f(a))
case Left(l) => Left(l)
}
}
```

Note that in this formulation `map`

is left abstract, whereas in the previous one with `ap`

`map`

could be implemented in terms of `ap`

and `pure`

. This suggests that `ap`

is equivalent to
`map`

and `product`

, which is indeed the case.

Such an `Applicative`

must obey three laws:

- Associativity: No matter the order in which you product together three values, the result is isomorphic
`fa.product(fb).product(fc) ~ fa.product(fb.product(fc))`

- With
`map`

, this can be made into an equality with`fa.product(fb).product(fc) = fa.product(fb.product(fc)).map { case (a, (b, c)) => ((a, b), c) }`

- Left identity: Zipping a value on the left with unit results in something isomorphic to the original value
`pure(()).product(fa) ~ fa`

- As an equality:
`pure(()).product(fa).map(_._2) = fa`

- Right identity: Zipping a value on the right with unit results in something isomorphic to the original value
`fa.product(pure(())) ~ fa`

- As an equality:
`fa.product(pure(())).map(_._1) = fa`

## Applicatives for effect management

If we view `Functor`

as the ability to work with a single effect, `Applicative`

encodes working with
multiple **independent** effects. Between `product`

and `map`

, we can take two separate effectful values
and compose them. From there we can generalize to working with any N number of independent effects.

```
import cats.Applicative
def product3[F[_]: Applicative, A, B, C](fa: F[A], fb: F[B], fc: F[C]): F[(A, B, C)] = {
val F = Applicative[F]
val fabc = F.product(F.product(fa, fb), fc)
F.map(fabc) { case ((a, b), c) => (a, b, c) }
}
```

## What is ap?

Let’s see what happens if we try to compose two effectful values with just `map`

.

```
import cats.instances.option._
val f: (Int, Char) => Double = (i, c) => (i + c).toDouble
val int: Option[Int] = Some(5)
val char: Option[Char] = Some('a')
```

```
int.map(i => (c: Char) => f(i, c)) // what now?
// res3: Option[Char => Double] = Some($$Lambda$2751/1113069537@3f09635d)
```

We have an `Option[Char => Double]`

and an `Option[Char]`

to which we want to apply the function to,
but `map`

doesn’t give us enough power to do that. Hence, `ap`

.

## Applicatives compose

Like `Functor`

, `Applicative`

s compose. If `F`

and `G`

have `Applicative`

instances, then so
does `F[G[_]]`

.

```
import cats.data.Nested
import cats.instances.future._
import scala.concurrent.Future
import scala.concurrent.ExecutionContext.Implicits.global
val x: Future[Option[Int]] = Future.successful(Some(5))
val y: Future[Option[Char]] = Future.successful(Some('a'))
```

```
val composed = Applicative[Future].compose[Option].map2(x, y)(_ + _)
// composed: scala.concurrent.Future[Option[Int]] = Future(<not completed>)
val nested = Applicative[Nested[Future, Option, ?]].map2(Nested(x), Nested(y))(_ + _)
// nested: cats.data.Nested[scala.concurrent.Future,Option,Int] = Nested(Future(<not completed>))
```

## Traverse

The straightforward way to use `product`

and `map`

(or just `ap`

) is to compose `n`

independent effects,
where `n`

is a fixed number. In fact there are convenience methods named `apN`

, `mapN`

, and `tupleN`

(replacing
`N`

with a number 2 - 22) to make it even easier.

Imagine we have one `Option`

representing a username, one representing a password, and another representing
a URL for logging into a database.

```
import java.sql.Connection
val username: Option[String] = Some("username")
val password: Option[String] = Some("password")
val url: Option[String] = Some("some.login.url.here")
// Stub for demonstration purposes
def attemptConnect(username: String, password: String, url: String): Option[Connection] = None
```

We know statically we have 3 `Option`

s, so we can use `map3`

specifically.

```
Applicative[Option].map3(username, password, url)(attemptConnect)
// res8: Option[Option[java.sql.Connection]] = Some(None)
```

Sometimes we don’t know how many effects will be in play - perhaps we are receiving a list from user input or getting rows from a database. This implies the need for a function:

```
def sequenceOption[A](fa: List[Option[A]]): Option[List[A]] = ???
// Alternatively..
def traverseOption[A, B](as: List[A])(f: A => Option[B]): Option[List[B]] = ???
```

Users of the standard library `Future.sequence`

or `Future.traverse`

will find these names and signatures
familiar.

Let’s implement `traverseOption`

(you can implement `sequenceOption`

in terms of `traverseOption`

).

```
def traverseOption[A, B](as: List[A])(f: A => Option[B]): Option[List[B]] =
as.foldRight(Some(List.empty[B]): Option[List[B]]) { (a: A, acc: Option[List[B]]) =>
val optB: Option[B] = f(a)
// optB and acc are independent effects so we can use Applicative to compose
Applicative[Option].map2(optB, acc)(_ :: _)
}
```

```
traverseOption(List(1, 2, 3))(i => Some(i): Option[Int])
// res11: Option[List[Int]] = Some(List(1, 2, 3))
```

This works…but if we look carefully at the implementation there’s nothing `Option`

-specific going on. As
another example let’s implement the same function but for `Either`

.

```
import cats.instances.either._
def traverseEither[E, A, B](as: List[A])(f: A => Either[E, B]): Either[E, List[B]] =
as.foldRight(Right(List.empty[B]): Either[E, List[B]]) { (a: A, acc: Either[E, List[B]]) =>
val eitherB: Either[E, B] = f(a)
Applicative[Either[E, ?]].map2(eitherB, acc)(_ :: _)
}
```

```
traverseEither(List(1, 2, 3))(i => if (i % 2 != 0) Left(s"${i} is not even") else Right(i / 2))
// res13: Either[String,List[Int]] = Left(1 is not even)
```

The implementation of `traverseOption`

and `traverseEither`

are more or less identical, modulo the initial
“accumulator” to `foldRight`

. But even that could be made the same by delegating to `Applicative#pure`

!
Generalizing `Option`

and `Either`

to any `F[_]: Applicative`

gives us the fully polymorphic version.
Existing data types with `Applicative`

instances (`Future`

, `Option`

, `Either[E, ?]`

, `Try`

) can call it by fixing `F`

appropriately, and new data types need only be concerned with implementing `Applicative`

to do so as well.

```
def traverse[F[_]: Applicative, A, B](as: List[A])(f: A => F[B]): F[List[B]] =
as.foldRight(Applicative[F].pure(List.empty[B])) { (a: A, acc: F[List[B]]) =>
val fb: F[B] = f(a)
Applicative[F].map2(fb, acc)(_ :: _)
}
```

This function is provided by Cats via the `Traverse[List]`

instance and syntax, which is covered in another
tutorial.

```
import cats.instances.list._
import cats.syntax.traverse._
```

```
List(1, 2, 3).traverse(i => Some(i): Option[Int])
// res14: Option[List[Int]] = Some(List(1, 2, 3))
```

With this addition of `traverse`

, we can now compose any number of independent effects, statically known or otherwise.

## Apply - a weakened Applicative

A closely related type class is `Apply`

which is identical to `Applicative`

, modulo the `pure`

method. Indeed in Cats `Applicative`

is a subclass of `Apply`

with the addition of this method.

```
trait Apply[F[_]] extends Functor[F] {
def ap[A, B](ff: F[A => B])(fa: F[A]): F[B]
}
trait Applicative[F[_]] extends Apply[F] {
def pure[A](a: A): F[A]
def map[A, B](fa: F[A])(f: A => B): F[B] = ap(pure(f))(fa)
}
```

The laws for `Apply`

are just the laws of `Applicative`

that don’t mention `pure`

. In the laws given
above, the only law would be associativity.

One of the motivations for `Apply`

’s existence is that some types have `Apply`

instances but not
`Applicative`

- one example is `Map[K, ?]`

. Consider the behavior of `pure`

for `Map[K, A]`

. Given
a value of type `A`

, we need to associate some arbitrary `K`

to it but we have no way of doing that.

However, given existing `Map[K, A]`

and `Map[K, B]`

(or `Map[K, A => B]`

), it is straightforward to
pair up (or apply functions to) values with the same key. Hence `Map[K, ?]`

has an `Apply`

instance.

## Syntax

Syntax for `Applicative`

(or `Apply`

) is available under the `cats.implicits._`

import. The most
interesting syntax is focused on composing independent effects - there are two options for this.

The first is the builder syntax which incrementally builds up a collection of effects until a function is applied to compose them.

```
import cats.implicits._
val o1: Option[Int] = Some(42)
val o2: Option[String] = Some("hello")
```

```
(o1 |@| o2).map((i: Int, s: String) => i.toString ++ s)
// res16: Option[String] = Some(42hello)
(o1 |@| o2).tupled
// res17: Option[(Int, String)] = Some((42,hello))
```

The second expects the effects in a tuple and works by enriching syntax on top of the existing
`TupleN`

types.

```
(o1, o2).map2((i: Int, s: String) => i.toString ++ s)
// res18: Option[String] = Some(42hello)
```

## Further Reading

- Applicative Programming with Effects - McBride, Patterson. JFP 2008.