Monads

[eparejatobes]

• [x] structure (monoid in an endofunctor category, if possible)
• [x] kleisli categories

[eparejatobes]

There's some casting here due to higher-kinded type members being not checked correctly in refinements. For example, for the Identity functor below:

final
class Identity[Cat <: Category](val cat: Category.is[Cat]) extends Functor {

  type Source = Cat
  val source = cat

  type Target = Cat
  val target = cat

  type F[Z <: Source#Objects] = Z

  def at[X <: Source#Objects, Y <: Source#Objects]: Source#C[X,Y] -> Target#C[F[X], F[Y]] =
    Scala.identity
}

We should have that Identity[Cat] <: Functor.is[Identity[Cat]], as everything is bound at this point; we don't:

// full snippet
type is[functor <: Functor] =
  functor {
    type Source = functor#Source
    type Target = functor#Target
    type F[Z <: Source#Objects] = functor#F[Z]
  }

type weakIs[F0 <: Functor] =
  F0 {
    type Source = F0#Source
    type Target = F0#Target
  }

import scala.Predef.<:<

// compiles
def z[Cat <: Category] =
  scala.Predef.implicitly[Identity[Cat] <:< weakIs[Identity[Cat]]]

// nonono
// def z0[Cat <: Category] =
//   scala.Predef.implicitly[Identity[Cat] <:< is[Identity[Cat]]]

[eparejatobes]

Looking good, will review later and merge.

[eparejatobes]

@agarciamontoro I'm requesting a review from you just to be sure that you understand this.

[eparejatobes]

Tomorrow I'll try to reduce the number of X.is[_] here.

[eparejatobes]

Done. LGTM.