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5.3.1 Co-Kleisli morphisms
Definition 5.3.5. Let (C, ε, ν) be a comonad on C. A co-Kleisli morphism of C from X to Y is a
morphism CX → Y of C.
Here is a typical situation in science. Suppose that, during an experiment, we have a process
f taking an input x ∈ X and giving an output y ∈ Y. This could be for example a survey in
which we ask people of different age (X) what their political views are (Y). Suppose that we
repeat the experiment, feeding again the same input x to f . It could happen that this time we
get a different outcome than before, y
′ , y. For example, suppose a person of age x expresses
political view y. Another person of the same age x may express a different political view y

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error_log /usr/local/var/log/nginx/error.log;

worker_processes 4;
#worker_shutdown_timeout 1s;

events {
    worker_connections 4096;

}

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trait ActionBuilder2[+R[_]] extends ActionBuilder[R] { self =>
  def async2(block: => Task[Result]): Action[AnyContent] =
    async(block runAsync)

  def async2(block: R[AnyContent] => Task[Result]): Action[AnyContent] =
    async(block(_) runAsync)

  def async2[A](bodyParser: BodyParser[A])(block: R[A] => Task[Result]): Action[A] =
    async(bodyParser)(block(_) runAsync)

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implicit def taggedStringFormat[T]: Format[String @@ T] = new Format[String @@ T] {
    def reads(json: JsValue): JsResult[String @@ T] = json match {
      case JsString(v) => JsSuccess(tag[T](v))
      case unknown => JsError(s"String value expected, got: $unknown")
    }

    def writes(v: String @@ T): JsValue = JsString(v)
  }

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package models

import org.scalacheck.Arbitrary._
import org.scalacheck.Shapeless._
import org.scalacheck._
import shapeless.tag
import shapeless.tag.{apply => _, _}

object ArbitraryModelsSpecification extends Properties("ArbitraryModelsSpecification") {

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import shapeless.tag
import shapeless.tag._

implicit def toTaggedType[U, T](u: U): U @@ T = tag[T](u)
//implicit def toTaggedString[T](s: String): String @@ T = tag[T](s)

trait MyTag

def foo(s: String @@ MyTag): String @@ MyTag = s

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import shapeless.tag.@@
import org.scalacheck.Prop.forAll
import org.scalacheck.Shapeless._

trait Baz

case class Foo(s: String @@ Baz)

case class Bar(f: Foo)

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import play.api.libs.json.{Format, JsResult, JsValue, Json}
import shapeless.tag.@@

trait Money

case class Amount(price: String @@ Money)

private val amountWrites = Json.writes[Amount]
private val amountReads = Json.reads[Amount]

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import org.scalacheck.Arbitrary
import org.scalacheck.Arbitrary._
import org.scalacheck.Gen

def arbitraryCaseClass[A,C](f: A => C)(implicit t: Arbitrary[A]): Arbitrary[C] = Arbitrary(for(v <- arbitrary[A]) yield f(v))

type MyIdType = String

implicit val arbMyIdType: Arbitrary[MyIdType] = Arbitrary(Gen.identifier)

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sealed trait A
case class B() extends A
case class C() extends A

val xs = List(B(), B(), C())

xs.groupBy(_.getClass)