cats.mtl

Ask[F, E] lets you access an E value in the F[_] context.

Intuitively, this means that an E value is required as an input to get "out" of the F[_] context.

Ask[F, E] has one external law:

def askAddsNoEffects[A](fa: F[A]) = {
 (ask *> fa) <-> fa
}

Ask[F, E] has one internal law:

def readerIsAskAndMap[A](f: E => A) = {
 ask.map(f) <-> reader(f)
}

Listen[F, L] is a function F[A] => F[(A, L)] which exposes some state that is contained in all F[A] values, and can be modified using tell.

Listen has two external laws:

def listenRespectsTell(l: L) = {
 listen(tell(l)) <-> tell(l).as(((), l))
}

def listenAddsNoEffects(fa: F[A]) = {
 listen(fa).map(_._1) <-> fa
}

Listen has one internal law:

def listensIsListenThenMap(fa: F[A], f: L => B) = {
 listens(fa)(f) <-> listen(fa).map { case (a, l) => (a, f(l)) }
}

Local[F, E] lets you alter the E value that is observed by an F[A] value using ask; the modification can only be observed from within that F[A] value.

Local[F, E] has three external laws:

def askReflectsLocal(f: E => E) = {
 local(f)(ask) <-> ask map f
}

def localPureIsPure[A](a: A, f: E => E) = {
 local(f)(pure(a)) <-> pure(a)
}

def localDistributesOverAp[A, B](fa: F[A], ff: F[A => B], f: E => E) = {
 local(f)(ff ap fa) <-> local(f)(ff) ap local(f)(fa)
}

Local has one internal law:

def scopeIsLocalConst(fa: F[A], e: E) = {
 scope(e)(fa) <-> local(_ => e)(fa)
}

Encapsulates the notion of a monad, G, which contains all of the effects of some other monad, F. This means that any effect of type F[A] can be lifted to G[A], such that both F and G form monads and the lifting distributes over flatMap and pure.

Original idea by Kris Nuttycombe.

Raise[F, E] expresses the ability to raise errors of type E in a functorial F[_] context. This means that a value of type F[A] may contain no A values but instead an E error value, and further map calls will not have any values to execute the passed function on.

Raise has no external laws.

Raise has two internal laws:

def catchNonFatalDefault[A](a: => A)(f: Throwable => E)(implicit A: Applicative[F]) = {
 catchNonFatal(a)(f) <-> try {
   A.pure(a)
 } catch {
   case NonFatal(ex) => raise(f(ex))
 }
}

def ensureDefault[A](fa: F[A])(error: => E)(predicate: A => Boolean)(implicit A: Monad[F]) = {
 ensure(fa)(error)(predicate) <-> for {
   a <- fa
   _ <- if (predicate(a)) pure(()) else raise(error)
 } yield a
}

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

def failThenFlatMapFails[A, B](ex: E, f: A => F[B]) = {
 fail(ex).flatMap(f) <-> fail(ex)
}
guaranteed by:
 fail[X](ex) <-> fail[F[Y]](ex) // parametricity
 fail[X](ex).map(f) <-> fail[F[Y]](ex)  // map must have no effect, because there's no X value
 fail[X](ex).map(f).join <-> fail[F[Y]].join // add join to both sides
 fail(ex).flatMap(f) <-> fail(ex) // join is equal, because there's no inner value to flatten effects from
 // QED.

Stateful[F, S] is the capability to access and modify a state value from inside the F[_] context, using set(s: S): F[Unit] and get: F[S].

Stateful has four external laws:

def getThenSetDoesNothing = {
 get >>= set <-> pure(())
}
def setThenGetReturnsSetted(s: S) = {
 set(s) *> get <-> set(s) *> pure(s)
}
def setThenSetSetsLast(s1: S, s2: S) = {
 set(s1) *> set(s2) <-> set(s2)
}
def getThenGetGetsOnce = {
 get *> get <-> get
}

Stateful has two internal law:

def modifyIsGetThenSet(f: S => S) = {
 modify(f) <-> (inspect(f) flatMap set)
}

def inspectLaw[A](f: S => A) = {
 inspect(f) <-> (get map f)
}

Tell[F, L] is the ability to "log" values L inside a context F[_], as an effect.

Tell has no external laws.

Tell has one internal law:

def writerIsTellAndMap(a: A, l: L) = {
 (tell(l) as a) <-> writer(a, l)
}

def tupleIsWriterFlipped(a: A, l: L) = {
 writer(a, l) <-> tuple((l, a))
}