Add a documentation page for Nested (#2369)

docs/src/main/tut/datatypes/nested.md

@@ -0,0 +1,119 @@
+---
+layout: docs
+title:  "Nested"
+section: "data"
+source: "core/src/main/scala/cats/data/Nested.scala"
+scaladoc: "#cats.data.Nested"
+---
+
+# Motivation
+
+In day-to-day programming we quite often we end up with data inside nested
+effects, e.g. an integer inside an `Either`, which in turn is nested inside
+an `Option`:
+
+```tut:silent
+import cats.data.Validated
+import cats.data.Validated.Valid
+val x: Option[Validated[String, Int]] = Some(Valid(123))
+```
+
+This can be quite annoying to work with, as you have to traverse the nested
+structure every time you want to perform a `map` or something similar:
+
+```tut:book
+x.map(_.map(_.toString))
+```
+
+`Nested` can help with this by composing the two `map` operations into one:
+
+```tut:silent
+import cats.data.Nested
+import cats.instances.option._
+import cats.syntax.functor._
+val nested: Nested[Option, Validated[String, ?], Int] = Nested(Some(Valid(123)))
+```
+
+```tut:book
+nested.map(_.toString).value
+```
+
+In a sense, `Nested` is similar to monad transformers like `OptionT` and
+`EitherT`, as it represents the nesting of effects inside each other. But
+`Nested` is more general - it does not place any restriction on the type of the
+two nested effects:
+
+```scala
+final case class Nested[F[_], G[_], A](value: F[G[A]])
+```
+
+Instead, it provides a set of inference rules based on the properties of `F[_]`
+and `G[_]`. For example:
+
+* If `F[_]` and `G[_]` are both `Functor`s, then `Nested[F, G, ?]` is also a
+    `Functor` (we saw this in action in the example above)
+* If `F[_]` and `G[_]` are both `Applicative`s, then `Nested[F, G, ?]` is also an
+    `Applicative`
+* If `F[_]` is an `ApplicativeError` and `G[_]` is an `Applicative`, then
+    `Nested[F, G, ?]` is an `ApplicativeError`
+* If `F[_]` and `G[_]` are both `Traverse`s, then `Nested[F, G, ?]` is also a
+    `Traverse`
+
+You can see the full list of these rules in the `Nested` companion object.
+
+## A more interesting example
+
+(courtesy of [Channing Walton and Luka Jacobowitz via
+Twitter](https://twitter.com/LukaJacobowitz/status/1017355319473786880),
+slightly adapted)
+
+Say we have an API for creating users:
+
+```tut:silent
+import scala.concurrent.Future
+
+case class UserInfo(name: String, age: Int)
+case class User(id: String, name: String, age: Int)
+
+def createUser(userInfo: UserInfo): Future[Either[List[String], User]] =
+  Future.successful(Right(User("user 123", userInfo.name, userInfo.age)))
+```
+
+Using `Nested` we can write a function that, given a list of `UserInfo`s,
+creates a list of `User`s:
+
+```tut:silent
+import scala.concurrent.Await
+import scala.concurrent.ExecutionContext.Implicits.global
+import scala.concurrent.duration._
+import cats.Applicative
+import cats.data.Nested
+import cats.instances.either._
+import cats.instances.future._
+import cats.instances.list._
+import cats.syntax.traverse._
+
+def createUsers(userInfos: List[UserInfo]): Future[Either[List[String], List[User]]] =
+  userInfos.traverse(userInfo => Nested(createUser(userInfo))).value
+
+val userInfos = List(
+  UserInfo("Alice", 42),
+  UserInfo("Bob", 99)
+)
+```
+
+```tut:book
+Await.result(createUsers(userInfos), 1.second)
+```
+
+Note that if we hadn't used `Nested`, the behaviour of our function would have
+been different, resulting in a different return type:
+
+```tut:silent
+def createUsersNotNested(userInfos: List[UserInfo]): Future[List[Either[List[String], User]]] =
+  userInfos.traverse(createUser)
+```
+
+```tut:book
+Await.result(createUsersNotNested(userInfos), 1.second)
+```