lifting

Type Members

1. trait ApplicativeLayer[M[_], Inner[_]] extends FunctorLayer[M, Inner] with Serializable

   

   ApplicativeLayer[M, Inner] has the following functionality: - the capability to lift values from the Applicative Inner to the Applicative M, preserving the Applicative structure - lifts Applicative isomorphisms in Inner ((Inner ~> Inner, Inner ~> Inner)) into Applicative homomorphisms in M (M ~> M).

   ApplicativeLayer[M, Inner] has the following functionality: - the capability to lift values from the Applicative Inner to the Applicative M, preserving the Applicative structure - lifts Applicative isomorphisms in Inner ((Inner ~> Inner, Inner ~> Inner)) into Applicative homomorphisms in M (M ~> M). This allows you to "map" a natural transformation over the Inner inside M, but only if you can provide an inverse of that natural transformation.

   ApplicativeLayer[M, Inner] has two external laws:

   def layerRespectsPure[A](a: A) = {
     layer(a.pure[Inner]) <-> a.pure[M]
   }
   def layerRespectsAp[A, B](m: Inner[A])(f: Inner[A => B]) = {
     layer(m).ap(layer(f)) <-> layer(m.ap(f))
   }

2. trait ApplicativeLayerFunctor[M[_], Inner[_]] extends ApplicativeLayer[M, Inner] with FunctorLayerFunctor[M, Inner] with Serializable

   

   ApplicativeLayerFunctor is the capability to lift Applicative homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

   ApplicativeLayerFunctor is the capability to lift Applicative homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

   ApplicativeLayerFunctor[M, Inner] has no additional laws.

3. trait FunctorLayer[M[_], Inner[_]] extends Serializable

   

   FunctorLayer[M, Inner] has the following functionality: - lifts values from the Functor Inner to the Functor M.

   FunctorLayer[M, Inner] has the following functionality: - lifts values from the Functor Inner to the Functor M. - lifts Functor isomorphisms in Inner ((Inner ~> Inner, Inner ~> Inner)) into Functor homomorphisms in M (M ~> M). This allows you to "map" a natural transformation over the Inner inside M, but only if you can provide an inverse of that natural transformation.

   FunctorLayer[M, Inner] has two external laws:

   def mapForwardRespectsLayer[A](in: Inner[A])(forward: Inner ~> Inner, backward: Inner ~> Inner) = {
     layer(forward(in)) <-> layerImapK(layer(in))(forward, backward)
   }
   
   def layerImapRespectsId[A](in: M[A]) = {
     in <-> layerImapK(in)(FunctionK.id, FunctionK.id)
   }

   FunctorLayer has one free law, i.e. a law guaranteed by parametricity:

   def layerRespectsMap[A, B](in: Inner[A])(f: A => B) = {
     layer(in.map(f)) <-> layer(in).map(f)
   } // this is free because all `FunctionK`'s (including `layer`, despite not being an instance of the class)
     // are natural transformations (i.e., they follow this law exactly)

4. trait FunctorLayerFunctor[M[_], Inner[_]] extends FunctorLayer[M, Inner] with Serializable

   

   FunctorLayerFunctor is the capability to lift Functor homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

   FunctorLayerFunctor is the capability to lift Functor homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

   FunctorLayerFunctor[M, Inner] has two external laws:

   def layerMapRespectsLayerImapK[A](ma: M[A])(forward: Inner ~> Inner,
                                            backward: Inner ~> Inner) = {
     layerImapK(ma)(forward, backward) <-> layerMapK(ma)(forward)
   }
   
   def layerMapRespectsLayer[A](in: Inner[A])(forward: Inner ~> Inner) = {
     layer(forward(in)) <-> layerMapK(layer(in))(forward)
   }

   FunctorLayerFunctor[M, Inner] has one free law, that is, one law guaranteed by other laws and parametricity:

   def layerMapRespectsId[A](in: M[A]) = {
     in <-> layerImapK(in)(FunctionK.id, FunctionK.id)
   }
   Justification:
   layerImapK(in)(FunctionK.id, FunctionK.id) <-> in [by layerMapRespectsLayerImapK[A]]
   layerMapK(in)(FunctionK.id) <-> layerImapK(FunctionK.id, FunctionK.id) [by layerImapRespectsId[A]]

5. trait MonadLayer[M[_], Inner[_]] extends ApplicativeLayer[M, Inner]

   

   MonadLayer[M, Inner] has the following functionality: - lifts values from the Monad Inner to the Monad M.

   MonadLayer[M, Inner] has the following functionality: - lifts values from the Monad Inner to the Monad M. - lifts Monad isomorphisms in Inner ((Inner ~> Inner, Inner ~> Inner)) into Monad homomorphisms in M (M ~> M). This allows you to "map" a natural transformation over the Inner inside M, but only if you can provide an inverse of that natural transformation.

   MonadLayer has one external law:

   def layerRespectsFlatMap[A, B](m: Inner[A])(f: A => Inner[B]) = {
     layer(m).flatMap(f andThen layer) <-> layer(m.flatMap(f))
   }

6. trait MonadLayerControl[M[_], Inner[_]] extends MonadLayerFunctor[M, Inner]

   

   MonadLayerControl is possible to use to access lower monad layers in invariant position, as in so-called "control operations".

   MonadLayerControl is possible to use to access lower monad layers in invariant position, as in so-called "control operations".

   Three external laws:

   def layerMapRespectsLayerControl[A](m: M[A], f: Inner ~> Inner) = {
     layerMapK(m)(f) <-> layerControl(run => f(run(m)).flatMap(restore)
   }
   
   def distributionLaw[A](nt: State ~> State, st: State[A]) = {
     restore(nt(st)) <-> layerControl[State[A]](_ (restore(st)).map(nt(_))).flatMap(restore)
   }
   
   def layerControlIdentity[A](ma: M[A]) = {
     ma <->
       layerControl[Inner[State[A]]] { cps =>
         cps(ma).pure[Inner]
       }.flatMap(layer).flatMap(restore)
   }

7. trait MonadLayerFunctor[M[_], Inner[_]] extends MonadLayer[M, Inner] with ApplicativeLayerFunctor[M, Inner]

   

   MonadLayerFunctor is the capability to lift Monad homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

   MonadLayerFunctor is the capability to lift Monad homomorphisms in Inner (Inner ~> Inner) into homomorphisms in M (Inner ~> Inner).

   MonadLayerFunctor[M, Inner] has no additional laws.