Add documentation for FlatMap and Bifunctor (#2459)

* Add documentation for FlatMap and Bifunctor

* Fix page title

docs/src/main/tut/typeclasses/bifunctor.md

@@ -0,0 +1,52 @@
+---
+layout: docs
+title:  "Bifunctor"
+section: "typeclasses"
+source: "core/src/main/scala/cats/Bifunctor.scala"
+scaladoc: "#cats.Bifunctor"
+---
+# Bifunctor
+
+`Bifunctor` takes two type parameters instead of one, and is a functor in both
+of these parameters. It defines a function `bimap`, which allows for mapping over both
+arguments at the same time. Its signature is as follows:
+
+```scala
+def bimap[A, B, C, D](fab: F[A, B])(f: A => C, g: B => D): F[C, D]
+```
+
+## Either as a Bifunctor
+
+Probably the most widely used Bifunctor instance is the Either data type.
+
+Say you have a value that is either an error or a `ZonedDateTime` instance.
+You also want to react to both possibilities - if there was a failure, you want to
+convert it to your own `DomainError`, and if the result was a success, you want to
+convert it to an UNIX timestamp.
+
+```tut:silent
+import cats._
+import cats.implicits._
+import java.time._
+
+case class DomainError(message: String)
+
+def dateTimeFromUser: Either[Throwable, ZonedDateTime] = 
+  Right(ZonedDateTime.now())  // Example definition
+```
+
+```tut:book
+dateTimeFromUser.bimap(
+  error => DomainError(error.getMessage),
+  dateTime => dateTime.toEpochSecond
+)
+```
+
+`Bifunctor` also defines a convenience function called `leftMap`, which is defined as follows:
+
+```scala
+def leftMap[A, B, C](fab: F[A, B])(f: A => C): F[C, B] = bimap(fab)(f, identity)
+```
+
+There is no `rightMap` however - use `map` instead. The reasoning behind this is that in Cats, the instances of
+`Bifunctor` are also mostly instances of `Functor`, as it is the case with `Either`.

docs/src/main/tut/typeclasses/monad.md

@@ -159,3 +159,26 @@ implicit def optionTMonad[F[_]](implicit F : Monad[F]) = {
 This sort of construction is called a monad transformer.
 
 Cats has an [`OptionT`](optiont.html) monad transformer, which adds a lot of useful functions to the simple implementation above.
+
+## FlatMap - a weakened Monad
+A closely related type class is `FlatMap` which is identical to `Monad`, minus the `pure`
+method. Indeed in Cats `Monad` is a subclass of `FlatMap` (from which it gets `flatMap`)
+and `Applicative` (from which it gets `pure`).
+
+```scala
+trait FlatMap[F[_]] extends Apply[F] {
+  def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
+}
+
+trait Monad[F[_]] extends FlatMap[F] with Applicative[F]
+```
+
+The laws for `FlatMap` are just the laws of `Monad` that don't mention `pure`.
+
+One of the motivations for `FlatMap`'s existence is that some types have `FlatMap` instances but not
+`Monad` - 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 `FlatMap` instance.
+