Add document about Arrow Choice (#3390)

* add arrow choice docs

* add menu for choice

* address comments;using correct context bond;adding response from amm

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

@@ -115,6 +115,10 @@ options:
     url: typeclasses/arrow.html
     menu_type: typeclasses
 
+  - title: Arrow Choice
+    url: typeclasses/arrowchoice.html
+    menu_type: typeclasses
+
   - title: Law Testing
     url: typeclasses/lawtesting.html
     menu_type: typeclasses

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

@@ -0,0 +1,122 @@
+---
+layout: docs
+title:  "Arrow Choice"
+section: "typeclasses"
+source: "core/src/main/scala/cats/arrow/ArrowChoice.scala"
+scaladoc: "#cats.arrow.ArrowChoice"
+---
+
+
+# `Choice`
+
+Usually we deal with function more often, we're so familiar with `A => B`.
+
+If we have two functions `A => C` and `B => C`, how can we compose them into a single function that can take either A or B and produce a C?
+
+So basically we just look for a function that has type `(A => C) => (B => C) => (Either[A, B] => C)`.
+
+This is exactly typeclass `Choice` provided, if we make `=>` more generic such as `F[_,_]`, you will get a `Choice`
+
+```scala
+trait Choice[F[_, _]] {
+  def choice[A,B,C,D](fac: F[A, C], fbc: F[B, C]): F[Either[A, B], C]
+}
+```
+
+Note the **infix** notation of `choice` is `|||`.
+
+## Middleware
+A very useful case of `Choice` is middleware in HTTP server.
+
+Take Http4s for example:
+
+HttpRoutes[F] in Http4s is defined as [Kleisli](https://typelevel.org/cats/datatypes/kleisli.html)
+
+```scala
+type HttpRoutes[F[_]] = Kleisli[OptionT[F, *], Request[F], Response[F]]
+// defined type HttpRoutes
+def routes[F[_]]: HttpRoutes[F] = ???
+// defined function routes
+```
+
+If we like to have an authentication middleware that composes the route, we can simply define middleware as:
+
+```scala
+type Middleware[F[_]] = Kleisli[OptionT[F, *], Request[F], Either[Response[F], Request[F]]]
+// defined type Middleware
+def auth[F[_]]: Middleware[F] = ???
+// defined function auth
+```
+
+Which means the `Request[F]` goes through the middleware, will become option of `Either[Response[F], Request[F]]`, where `Left` means the request is denied and return immediately, `Right` means the authentication is OK and request will get pass.
+
+Now we need to define what we should do when middleware returns `Left`:
+
+```scala
+def reject[F[_]:Monad]: Kleisli[OptionT[F, *], Response[F], Response[F]] = Kleisli.ask[OptionT[F, *], Response[F]]
+// defined function reject
+```
+
+Now compose middleware with route
+
+```scala
+def authedRoute[F[_]:Monad] = auth[F] andThen (reject[F] ||| routes[F])
+// defined function authedRoute
+```
+
+You will then get a new route that has authentication ability by composing Kleisli.
+
+## HTTP Response
+
+Another example will be HTTP response handler.
+
+```scala
+val resp: IO[Either[Throwable, String]] = httpClient.expect[String](uri"https://google.com/").attempt
+```
+
+`attempt` is syntax from [`MonadError`](https://typelevel.org/cats/api/cats/MonadError.html)
+
+When we need to handle error, without `Choice` the handler would be something like:
+```scala
+resp.flatMap{
+  case Left => ???
+  case Right => ???
+}
+```
+
+With `Choice` there will be more composable solution without embedded logic in pattern matching:
+
+```scala
+def recover[A](error: Throwable): IO[A] = ???
+def processResp[A](resp: String): IO[A] = ???
+
+resp >>= (recover _ ||| processResp _)
+```
+
+# `ArrowChoice`
+
+`ArrowChoice` is an extended version of `Choice`, which has one more method `choose`, with syntax `+++`
+
+```scala
+trait ArrowChoice[F[_, _]] extends Choice[F] {
+  def choose[A, B, C, D](f: F[A, C])(g: F[B, D]): F[Either[A, B], Either[C, D]]
+}
+```
+
+With the middleware example, you can think `ArrowChoice` is middleware of middleware.
+
+For example if we want to append log to middleware of `auth`, that can log both when rejected response and pass request:
+
+```scala
+def logReject[F[_]]: Kleisli[OptionT[F, *], Response[F], Response[F]] = ???
+// defined function logReject
+def logThrough[F[_]]: Kleisli[OptionT[F, *], Request[F], Request[F]] = ???
+// defined function logThrough
+```
+
+See how easy to compose log functionality into our `authedRoute`:
+
+```scala
+def authedRoute[F[_]:Monad] = auth[F] andThen (logReject[F] +++ logThrough[F]) andThen (reject[F] ||| routes[F])
+// defined function authedRoute
+```