# Comonad

`Comonad`

is a `Functor`

and provides duals of the `Monad`

`pure`

and `flatMap`

functions. A dual to a function has the same types but the
direction of the arrows are reversed. Whether or not that is useful, or even possible,
depends on the particular type. For a more formal definition of duality, please
refer to https://ncatlab.org/nlab/show/duality.

### extract

Monads have `pure`

from `Applicative`

which gives you the ability to wrap
a value `A`

using the type constructor giving an `F[A]`

. Comonad has
`extract`

which instead takes an `F[A]`

and extracts the `A`

. Therefore, to be
able to implement extract we must have a type of which we are certain
we can get an `A`

from an `F[A]`

. For example we cannot always get an `A`

from a `List[A]`

because if the list is empty there is nothing to get.

For the same reason, `Option`

doesn’t have a `Comonad`

instance, because we
cannot always get an `A`

from an Option, it may be empty too.

Some examples that we can implement `Comonad`

for include `OneAnd`

, `Tuple2`

and the “non empty” collections.

First some imports.

```
import cats._
import cats.data._
import cats.implicits._
import cats.syntax.comonad._
import cats.instances.list._
```

`NonEmptyList`

has a `Comonad`

instance and its implementation of `extract`

simply returns the head element of the list, which we know we will always
have.

```
NonEmptyList.of(1,2,3).extract
// res0: Int = 1
```

### coflatMap

`coflatMap`

is the dual of Monad’s `flatMap`

. While `flatMap`

allows us to chain
together operations in a monadic context, `coflatMap`

takes a value in some context
`F[A]`

and a function `F[A] => B`

and returns a new value in a context `F[B]`

.

The default implementation of `coflatMap`

for `NonEmptyList`

will pass the supplied
function with the whole list, then the tail of that, then the tail of that and so
on. This is illustrated below.

```
NonEmptyList.of(1,2,3,4,5).coflatMap(identity)
// res1: cats.data.NonEmptyList[cats.data.NonEmptyList[Int]] = NonEmptyList(NonEmptyList(1, 2, 3, 4, 5), NonEmptyList(2, 3, 4, 5), NonEmptyList(3, 4, 5), NonEmptyList(4, 5), NonEmptyList(5))
```

# CoflatMap

While `FlatMap`

is a weaker version of `Monad`

that doesn’t have the `pure`

function,
`CoflatMap`

is a `Comonad`

without the `extract`

function. There are many instances
of type classes in Cats that implement `CoflatMap`

but not `Comonad`

.

For example we cannot write `extract`

for `Option[A]`

because there’s no way to
pull an `A`

out of nowhere if the Option is empty.

```
def extract[A](fa : Option[A]): A = fa match {
case Some(a) => a
case None => ??? // What now?
}
```

Another example is `List`

. Remember we cannot write `extract`

for list because lists
can be empty, but we can implement the `coflatMap`

and it works identically to the
one shown above for `NonEmptyList`

.

```
List(1,2,3,4,5).coflatMap(identity)
// res2: List[List[Int]] = List(List(1, 2, 3, 4, 5), List(2, 3, 4, 5), List(3, 4, 5), List(4, 5), List(5))
```