Comonad documentation (#2888)

* Add comonad first draft

* Update menu

* Update with Kailuo's suggestion

* Couple of corrections

docs/src/main/resources/microsite/data/menu.yml

@@ -44,6 +44,11 @@ options:
     menu_type: typeclasses
     menu_section: monads
 
+  - title: Comonads
+    url: typeclasses/comonad.html
+    menu_type: typeclasses
+    menu_section: comonads
+
   - title: Variance and Functors
     url: typeclasses/functor.html
     menu_type: typeclasses

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

@@ -0,0 +1,85 @@
+---
+layout: docs
+title:  "Comonad"
+section: "typeclasses"
+source: "core/src/main/scala/cats/Comonad.scala"
+scaladoc: "#cats.Comonad"
+---
+# Comonad
+
+`Comonad` is a `Functor` and provides duals of the [`Monad`](monad.html) `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 even possible 
+depends on the particular type. For a more formal definition of duality, please 
+refer to [https://ncatlab.org/nlab/show/duality](https://ncatlab.org/nlab/show/duality).
+
+### extract
+
+Monad's 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 Comand 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.
+
+```tut:silent
+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.
+
+```tut:book
+NonEmptyList.of(1,2,3).extract
+```
+
+### 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.
+
+```tut:book
+NonEmptyList.of(1,2,3,4,5).coflatMap(identity)
+```
+
+# 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.
+
+```tut:silent
+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`.
+
+```tut:book
+List(1,2,3,4,5).coflatMap(identity)
+```