(f: A => B): F[B] Functor
6](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-6-320.jpg)
(f: A => B): F[B] Functor
def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply
7](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-7-320.jpg)
(f: A => B): F[B] Functor
def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply
def pure[A](value: A): F[A] Applicative
8](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-8-320.jpg)
(f: A => B): F[B] Functor
def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply
def pure[A](value: A): F[A] Applicative
def flatMap[A,B](fa: F[A])(f: A => F[B]): F[B] Monad
9](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-9-320.jpg)
(f: A => B): F[B] Functor
def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply
def pure[A](value: A): F[A] Applicative
def flatMap[A,B](fa: F[A])(f: A => F[B]): F[B] Monad
10](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-10-320.jpg)
![Covariant Functors Signatures - Pattern
def map A => B => F[B] Functor
def ap F[A => B] => F[B] Apply
def pure () => B => F[B] Applicative*
def flatMap A => F[B] => F[B] Monad
http://blog.tmorris.net/posts/scala-type-class-hierarchy/index.html
11](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-11-320.jpg)
(f: A => B): F[B] Functor
def coflatMap[A, B](fa: F[A])(f: F[A] => B): F[B] CoflatMap
def coflatten[A](fa: F[A]): F[F[A]] CoflatMap
def extract[A](x: F[A]): A Comonad
13](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-13-320.jpg)
![Comonads - Signatures - Pattern
//def flatMap (F[A], A => F[B]): F[B] FlatMap
def coflatMap (F[A], F[A] => B): F[B] CoflatMap
//def flatten F[F[A]]: F[A] Monad
def coflatten F[A]: F[F[A]] CoflatMap
//def pure A : F[A] Applicative
def extract F[A]: A Comonad
14](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-14-320.jpg)
![Contravariant - signature
trait Contravariant[F[_]] {
def contramap[A, B] (fa: F[A])(f: B => A): F[B]
}
17](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-17-320.jpg)
![Contravariant - like a Functor but flip!
trait Functor[F[_]] {
def map[A, B] (fa: F[A]) (f: A => B): F[B]
}
trait Contravariant[F[_]] {
def contramap[A, B] (fa: F[A]) (f: B => A): F[B]
}
18](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-18-320.jpg)
![Contravariant Functor - input + ability to prepend
Functor is “full of” A’s (container A, function producing A)
F[A] we can map A => B and we get F[B]
F[A] we can contramap B => A and we get F[B]
Contravariant “needs” A (index of container, function
consuming A)
19](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-19-320.jpg)
Example in GIthub:
https://github.com/lemastero/scala_typeclassopedia/blob/master/src/test/scala/contravariant/Contravaria
ntSpec.scala
Contravariant - example Predicate (1)
20](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-20-320.jpg)
val predicateContravariantFunctor = new Contravariant[Predicate] {
def contramap[A, B](pred: Predicate[A])(fba: B => A): Predicate[B] =
Predicate[B](fba andThen pred.fun)
}
Contravariant - example Predicate (2)
21](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-21-320.jpg)
pl.fun("Contravariant") // true
val pr1 = Predicate[Int](_ > 5)
pr1.fun(42) // true
Contravariant - example Predicate (3)
22](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-22-320.jpg)
pl.fun("Contravariant") // true
val pr1 = Predicate[Int](_ > 5)
pr1.fun(42) // true
val len: String => Int = _.length
val pr = Contravariant[Predicate].contramap(pr1)(len)
pr.fun("Contravariant") // true
Contravariant - example Predicate (4)
23](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-23-320.jpg)
![Contravariant - example Show
trait Show[F] {
def show(f: F): String
}
implicit val showContravariant = new Contravariant[Show] {
def contramap[A, B](r: Show[A])(f: B => A): Show[B] = new Show[B] {
def show(b: B): String = r.show(f(b))
}
}
Scalaz github.com/scalaz/scalaz/blob/series/7.3.x/core/src/main/scala/scalaz/Show.scala#L51-L53
24](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-24-320.jpg)
def opContravariant [R] = new Contravariant[Op[ R, ?]] {
def contramap[A, B](fa: Op[R, A])(f: B => A): Op[R, B] =
Op(f andThen fa.getOp)
}
25](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-25-320.jpg)
![Contravariant - example Function1
def function1Contravariant[R]: Contravariant[? => R] =
new Contravariant[? => R] {
def contramap[A, B](r: A => R)(f: B => A) = r compose f
}
26](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-26-320.jpg)
![Contravariant - FunctionN parameters all but last
trait Function2[-T1, -T2, +R]{
def apply(v1: T1, v2: T2): R
}
trait Function3[-T1, -T2, -T3, +R]{
def apply(v1: T1, v2: T2, v3: T3): R
}
//...
All parameters are in negative position, co we could define Contravariant
instance for it.
(Or even BiContravariant, TriContravariant, ... if they existed :)
27](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-27-320.jpg)
28](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-28-320.jpg)
def readerFunctor[C] = new Functor[Reader[C,?]] {
def map[A, B](x: Reader[C, A])(f: A => B): Reader[C, B] =
Reader(x.run andThen f)
}
29](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-29-320.jpg)
def readerMonad[C] = new Monad[Reader[C, ?]] {
def map[A, B](x: Reader[C, A])(f: A => B): Reader[C, B] =
Reader(x.run andThen f)
def pure[A](a: A): Reader[C, A] =
new Reader(_ => a)
def flatMap[A, B](ma: Reader[C, A])(f: A => Reader[C, B]) = ???
}
30](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-30-320.jpg)
def readerContra[V] = new Contravariant[Reader[?, V]] {
def contramap[A, B](fa: Reader[A, V])(f: B => A):
Reader[B, V] = Reader(f andThen fa.run)
}
31](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-31-320.jpg)
val strSerial = Serializer[String](_.getBytes)
val intSerial = Serializer[Int](_.toString.getBytes)
Github:
https://github.com/lemastero/scala_typeclassopedia/blob/master/src/test/scala/contravaria
nt/DivideSpec.scala
Divide (1) - Serialization
39](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-39-320.jpg)
![val fragmentSerial = Serializer[Fragment] { frag =>
val a1 = strSerial.run(frag.name)
val a2 = intSerial.run(frag.size)
a1 ++ a2
}
val serialized = fragmentSerial.run(Fragment("Area", 52))
new String(serialized ) mustBe "Area52"
Divide (2) - How to combine serializers?
40](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-40-320.jpg)
![trait Divide[F[_]] extends Contravariant[F] {
def divide[A,B,C](f: A => (B,C), fb: F[B], fc: F[C]): F[A]
}
Divide (3) - abstraction
41](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-41-320.jpg)
![val fragmentDivide: Divide[Serializer] = new Divide[Serializer] {
def divide2[A1, A2, Z](s1: => Serializer[ A1], s2: => Serializer[ A2])(f: Z
=> (A1, A2)): Serializer[ Z] = Serializer{ frag =>
val (a1,a2) = f(frag)
s1.run(a1) ++ s2.run(a2)
}
}
Divide (4)
42](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-42-320.jpg)
![val fragAsTuple: Fragment => (String, Int) =
frag => (frag.name, frag.size)
val fragSerial: Serializer[Fragment] =
Divide[Serializer].divide(strSerial, intSerial)(fragAsTuple)
Divide (5)
43](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-43-320.jpg)
![val fragAsTuple: Fragment => (String, Int) =
frag => (frag.name, frag.size)
val fragSerial: Serializer[Fragment] =
Divide[Serializer].divide(strSerial, intSerial)(fragAsTuple)
val serialized = fragSerial.run(Fragment("Area", 52))
new String(serialized ) mustBe "Area52"
Divide - Run
44](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-44-320.jpg)
![Contravariant Functors (1)
def contramap(fa: F[A],f: B => A): F[B] Contravariant
def divide(f: A => (B,C), fb: F[B],fc: F[C]): F[A] Divide
def conquer: F[A] Divisible
There is no Contravariant Monad
There is Contravariant Traversable, Alternative:
https://www.youtube.com/watch?v=cB8DapKQz-I Edward Kmett, Discrimination is
Wrong, 2015
47](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-47-320.jpg)
![Contravariant Functors (2) - arrows reversed
//def map (F[A], A => B): F[B] Functor
def contramap (F[A], A <= B): F[B] Contravariant (Contravariant Functor)
//def map2 ((A,B) => Z, F[A], F[B]): F[Z] Apply (not ap!)
def divide (Z => (A,B), F[A], F[B]): F[Z] Divide (Contravariant Apply)
//def pure: ( () => A ) => F[A] Applicative
def conquer: ( A <= () ) => F[A] Divisible (Contravariant Applicative)
48](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-48-320.jpg)
![trait Profunctor[P[_, _]] {
def dimap[X,Y,Z,W](ab: X => Y, cd: Z => W): P[Y, Z] => P[X, W]
}
Profunctor
51](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-51-320.jpg)
![trait Profunctor[P[_, _]] {
def dimap[X,Y,Z,W](ab: X => Y, cd: Z => W): P[Y, Z] => P[X, W]
}
● contramap on first type ( P[Y, _] is Contravariant )
● map on second type ( P[_, Z] is Functor )
Profunctor = Contravariant + Functor
52](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-52-320.jpg)
![trait Profunctor[P[_, _]] {
def dimap[X,Y,Z,W](ab: X => Y, cd: Z => W): P[Y, Z] => P[X, W]
// derived methods
def lmap[A,B,C](f: A => B): P[B,C] => P[A,C] = dimap[A,B,C,C](f,identity[C])
def rmap[A,B,C](f: B => C): P[A,B] => P[A,C] = dimap[A,A,B,C](identity[A],
f)
}
Profunctor - derived methods
53](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-53-320.jpg)
![val function1: Profunctor[Function1] = new Profunctor[Function1] {
def dimap[X, Y, Z, W](f: X => Y, g: Z => W): (Y => Z) => (X => W) =
h => f andThen (g compose h)
def lmap[A,B,C](f: A => B): (B => C) => (A => C) = f andThen
def rmap[A,B,C](f: B => C): (A => B) => (A => C) = f compose
}
Profunctor - Example Function1
55](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-55-320.jpg)
![val f: String => Int = _.length
case class Person(name: String, age: Int)
val preF: Person => String = _.name
val postF: Int => Boolean = _ > 5
Profunctor[Function1].dimap(f)(preF)(postF)(
Person("Foo", 100)
)
Profunctor - Example Function1 - true ?
57](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-57-320.jpg)
![val f: String => Int = _.length
case class Person(name: String, age: Int)
val preF: Person => String = _.name
val postF: Int => Boolean = _ > 5
Profunctor[Function1].dimap(f)(preF)(postF)(
Person("Foo", 100)
)
// false
Profunctor - Example Function1
58](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-58-320.jpg)
![trait Strong[P[_, _]] extends Profunctor[P] {
def first[X,Y,Z](pab: P[X, Y]): P[(X, Z), (Y, Z)]
}
Profunctor - Strong
61](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-61-320.jpg)
![trait Strong[P[_, _]] extends Profunctor[P] {
def first[X,Y,Z](pab: P[X, Y]): P[(X, Z), (Y, Z)]
// derived methods
def second[X,Y,Z](pab: P[X, Y]): P[(Z, X), (Z, Y)]
}
Profunctor - Strong - Derived methods
62](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-62-320.jpg)
![val functionStrong: Strong[Function1] = new Strong[Function1]
with Function1Profunctor {
def first[X, Y, Z](pab: X => Y): Function1[( X, Z), (Y, Z)] =
(xz: (X, Z)) => (pab(xz._1) , xz._2)
}
Profunctor - Strong - Function1
63](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-63-320.jpg)
![val functionStrong: Strong[Function1] = new Strong[Function1]
with Function1Profunctor {
def first[X, Y, Z](pab: X => Y): Function1[( X, Z), (Y, Z)] =
(xz: (X, Z)) => (pab(xz._1) , xz._2)
}
Profunctor[Function1].first(len)(( "foo", 42)) == (3,42)
Profunctor - Strong - Function1
64](https://image.slidesharecdn.com/bigpictureofcategorytheoryinscalawithdeepdiveintocontravariantandprofunctors-190402202059/85/Big-picture-of-category-theory-in-scala-with-deep-dive-into-contravariant-and-profunctors-64-320.jpg)
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Big picture of category theory in scala with deep dive into contravariant and profunctors
- 1. Big picture of Category Theory in Scala with deep dive into: - Contravariant - Profunctors Piotr Paradziński ScalaC27.03.2019 1
- 2. ● PR’s to Scalaz 7 ○ github.com/scalaz/scalaz/pull/2020 Day convolution (0,5 year fight translate from Haskell) ○ github.com/scalaz/scalaz/pull/2028 Strong Profunctor laws - they were none ○ github.com/scalaz/scalaz/pull/2029 Density comonad (abstraction - no practical applications yet) ● PR’s to Cats ○ github.com/typelevel/cats/pull/2640 Strong Profunctor laws (based on CT replace ad hoc ones) ● PR’s to Haskell Profunctors ○ github.com/ekmett/profunctors/pull/65 broken link (this is how you start!) ○ github.com/ekmett/profunctors/pull/66 small fix in docs for Strong Profunctor laws ● type_classopedia - wiki about Category Theory abstraction in Scala ○ github.com/lemastero/scala_typeclassopedia ● other OSS work: resources about effects, programming language theory, streaming benchmarks ○ Sclaz ZIO github.com/scalaz/scalaz-zio/pulls?utf8=%E2%9C%93&q=+author%3Alemastero SclaC Hackaton ○ yallop/effects-bibliography github.com/yallop/effects-bibliography/pulls?utf8=✓&q=+author%3Alemastero add Idris Effects, Scala Eff Monad) wiki about effects by Jeremy Yallop (University of Cambridge) ○ steshaw.org/plt/ (github.com/steshaw/plt/pulls?utf8=✓&q=+author%3Alemastero) add 7Sketches, TAC Journal ○ monix/streaming-benchmarks github.com/monix/streaming-benchmarks/pull/1 updated benchmarks ○ cohomolo-gy/haskell-resources github.com/cohomolo-gy/haskell-resources/pull/3 add Haskell papers ○ lauris/awesome-scala github.com/lauris/awesome-scala/pull/425 add ZIO, update Monix, RxScala ○ passy/awesome-recurion-schemas github.com/passy/awesome-recursion-schemes/pull/22 add droste
- 4. Category Theory abstractions in Scala - Functor, Apply, Applicative 4
- 5. Category Theory abstractions there is more - Functor, Apply, Applicative - Monads: State, Writer, Reader, IO, Option, Eiter, Validated, List/Vector 5
- 6. Functor - Signature def map[A,B](fa: F[A])(f: A => B): F[B] Functor 6
- 7. Apply - Signature def map[A,B](fa: F[A])(f: A => B): F[B] Functor def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply 7
- 8. Applicative - Signature def map[A,B](fa: F[A])(f: A => B): F[B] Functor def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply def pure[A](value: A): F[A] Applicative 8
- 9. Monad - Signature def map[A,B](fa: F[A])(f: A => B): F[B] Functor def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply def pure[A](value: A): F[A] Applicative def flatMap[A,B](fa: F[A])(f: A => F[B]): F[B] Monad 9
- 10. Signatures - similar? def map[A,B](fa: F[A])(f: A => B): F[B] Functor def ap[A,B](ff: F[A => B])(fa: F[A]): F[B] Apply def pure[A](value: A): F[A] Applicative def flatMap[A,B](fa: F[A])(f: A => F[B]): F[B] Monad 10
- 11. Covariant Functors Signatures - Pattern def map A => B => F[B] Functor def ap F[A => B] => F[B] Apply def pure () => B => F[B] Applicative* def flatMap A => F[B] => F[B] Monad http://blog.tmorris.net/posts/scala-type-class-hierarchy/index.html 11
- 12. Category Theory abstractions in Scala - Functor, Apply, Applicative - Monads: State, Writer, Reader, IO, Option, Eiter, Validated, List/Vector, Free - Comonads: NonEmptyList, Stream, Store, Context, Cofree 12
- 13. Comonads - signatures def map[A,B](fa: F[A])(f: A => B): F[B] Functor def coflatMap[A, B](fa: F[A])(f: F[A] => B): F[B] CoflatMap def coflatten[A](fa: F[A]): F[F[A]] CoflatMap def extract[A](x: F[A]): A Comonad 13
- 14. Comonads - Signatures - Pattern //def flatMap (F[A], A => F[B]): F[B] FlatMap def coflatMap (F[A], F[A] => B): F[B] CoflatMap //def flatten F[F[A]]: F[A] Monad def coflatten F[A]: F[F[A]] CoflatMap //def pure A : F[A] Applicative def extract F[A]: A Comonad 14
- 15. Category Theory abstractions in Scala - Functor, Apply, Applicative - Monads: State, Writer, Reader, IO, Option, Eiter, Validated, List/Vector - Comonads: NonEmptyList, Stream, Store, Context - Contravariant, Divide, Divisible 15
- 17. Contravariant - signature trait Contravariant[F[_]] { def contramap[A, B] (fa: F[A])(f: B => A): F[B] } 17
- 18. Contravariant - like a Functor but flip! trait Functor[F[_]] { def map[A, B] (fa: F[A]) (f: A => B): F[B] } trait Contravariant[F[_]] { def contramap[A, B] (fa: F[A]) (f: B => A): F[B] } 18
- 19. Contravariant Functor - input + ability to prepend Functor is “full of” A’s (container A, function producing A) F[A] we can map A => B and we get F[B] F[A] we can contramap B => A and we get F[B] Contravariant “needs” A (index of container, function consuming A) 19
- 20. case class Predicate[A](fun: A => Boolean) Example in GIthub: https://github.com/lemastero/scala_typeclassopedia/blob/master/src/test/scala/contravariant/Contravaria ntSpec.scala Contravariant - example Predicate (1) 20
- 21. case class Predicate[A](fun: A => Boolean) val predicateContravariantFunctor = new Contravariant[Predicate] { def contramap[A, B](pred: Predicate[A])(fba: B => A): Predicate[B] = Predicate[B](fba andThen pred.fun) } Contravariant - example Predicate (2) 21
- 22. val pl = Predicate[String](_.length > 5) pl.fun("Contravariant") // true val pr1 = Predicate[Int](_ > 5) pr1.fun(42) // true Contravariant - example Predicate (3) 22
- 23. val pl = Predicate[String](_.length > 5) pl.fun("Contravariant") // true val pr1 = Predicate[Int](_ > 5) pr1.fun(42) // true val len: String => Int = _.length val pr = Contravariant[Predicate].contramap(pr1)(len) pr.fun("Contravariant") // true Contravariant - example Predicate (4) 23
- 24. Contravariant - example Show trait Show[F] { def show(f: F): String } implicit val showContravariant = new Contravariant[Show] { def contramap[A, B](r: Show[A])(f: B => A): Show[B] = new Show[B] { def show(b: B): String = r.show(f(b)) } } Scalaz github.com/scalaz/scalaz/blob/series/7.3.x/core/src/main/scala/scalaz/Show.scala#L51-L53 24
- 25. Contravariant - example Op (Haskell like) case class Op[R,A](getOp: A => R) def opContravariant [R] = new Contravariant[Op[ R, ?]] { def contramap[A, B](fa: Op[R, A])(f: B => A): Op[R, B] = Op(f andThen fa.getOp) } 25
- 26. Contravariant - example Function1 def function1Contravariant[R]: Contravariant[? => R] = new Contravariant[? => R] { def contramap[A, B](r: A => R)(f: B => A) = r compose f } 26
- 27. Contravariant - FunctionN parameters all but last trait Function2[-T1, -T2, +R]{ def apply(v1: T1, v2: T2): R } trait Function3[-T1, -T2, -T3, +R]{ def apply(v1: T1, v2: T2, v3: T3): R } //... All parameters are in negative position, co we could define Contravariant instance for it. (Or even BiContravariant, TriContravariant, ... if they existed :) 27
- 28. Contravariant - example Reader (1) case class Reader[C, V](run: C => V) 28
- 29. Contravariant - example Reader - Functor case class Reader[C, V](run: C => V) def readerFunctor[C] = new Functor[Reader[C,?]] { def map[A, B](x: Reader[C, A])(f: A => B): Reader[C, B] = Reader(x.run andThen f) } 29
- 30. Contravariant - example Reader - Monad case class Reader[C, V](run: C => V) def readerMonad[C] = new Monad[Reader[C, ?]] { def map[A, B](x: Reader[C, A])(f: A => B): Reader[C, B] = Reader(x.run andThen f) def pure[A](a: A): Reader[C, A] = new Reader(_ => a) def flatMap[A, B](ma: Reader[C, A])(f: A => Reader[C, B]) = ??? } 30
- 31. Contravariant - example Reader - Contravariant case class Reader[C, V](run: C => V) def readerContra[V] = new Contravariant[Reader[?, V]] { def contramap[A, B](fa: Reader[A, V])(f: B => A): Reader[B, V] = Reader(f andThen fa.run) } 31
- 32. Contravariant - example Encoder Encoder in scodec: github.com/scodec/scodec/blob/series/1.11.x/shared/src/main/scala/scodec/Encoder.scala#L4 0-L47 has Contravariant instance github.com/scodec/scodec-cats/blob/master/shared/src/main/scala/scodec/interop/cats/CatsI nstances.scala#L121-L123 32
- 33. Contravariant - Resources Haskell George Wilson: Contravariant Functors: The Other Side of the Coin : https://www.youtube.com/watch?v=IJ_bVVsQhvc Michael noyman - Covariance and Contravariance fpcomplete.com/blog/2016/11/covariance-contravariance Tom Ellis - 24 Days of Hackage: contravariant https://ocharles.org.uk/blog/guest-posts/2013-12-21-24-days-of-hackage-contravariant.html (nice example with actors producing Behaviour a) https://packdeps.haskellers.com/reverse/contravariant (100+ Haskell libs using Contravariant package) 33
- 34. Contravariant - discrimination sort youtube.com/watch?v=cB8DapKQz-I Sorting in linear time Edward Kmett in Haskell In Scala? 34
- 35. Functor laws fmap id = id fmap f . fmap g = fmap (f . g) Contravariant laws contramap id = id contramap f . contramap g = contramap (g . f) Contravariant - laws in Haskell 35
- 37. Contravariant - laws in Scala 37
- 38. Divide (Contravariant Semigroupal - in Cats) 38
- 39. case class Serializer[A](run: A => Array[Byte]) val strSerial = Serializer[String](_.getBytes) val intSerial = Serializer[Int](_.toString.getBytes) Github: https://github.com/lemastero/scala_typeclassopedia/blob/master/src/test/scala/contravaria nt/DivideSpec.scala Divide (1) - Serialization 39
- 40. val fragmentSerial = Serializer[Fragment] { frag => val a1 = strSerial.run(frag.name) val a2 = intSerial.run(frag.size) a1 ++ a2 } val serialized = fragmentSerial.run(Fragment("Area", 52)) new String(serialized ) mustBe "Area52" Divide (2) - How to combine serializers? 40
- 41. trait Divide[F[_]] extends Contravariant[F] { def divide[A,B,C](f: A => (B,C), fb: F[B], fc: F[C]): F[A] } Divide (3) - abstraction 41
- 42. val fragmentDivide: Divide[Serializer] = new Divide[Serializer] { def divide2[A1, A2, Z](s1: => Serializer[ A1], s2: => Serializer[ A2])(f: Z => (A1, A2)): Serializer[ Z] = Serializer{ frag => val (a1,a2) = f(frag) s1.run(a1) ++ s2.run(a2) } } Divide (4) 42
- 43. val fragAsTuple: Fragment => (String, Int) = frag => (frag.name, frag.size) val fragSerial: Serializer[Fragment] = Divide[Serializer].divide(strSerial, intSerial)(fragAsTuple) Divide (5) 43
- 44. val fragAsTuple: Fragment => (String, Int) = frag => (frag.name, frag.size) val fragSerial: Serializer[Fragment] = Divide[Serializer].divide(strSerial, intSerial)(fragAsTuple) val serialized = fragSerial.run(Fragment("Area", 52)) new String(serialized ) mustBe "Area52" Divide - Run 44
- 45. https://github.com/lemastero/scala_typeclassopedia#divide-co ntravariant-apply Divide - Laws & derived methods 45
- 46. Define full laws for Divide as in Haskell hackage.haskell.org/package/contravariant/docs/Data-Functor-Contravariant-Divisible.html#g:4 Not simplified as in Scalaz and Cats! Contravariant - Divide - Exercise 46
- 47. Contravariant Functors (1) def contramap(fa: F[A],f: B => A): F[B] Contravariant def divide(f: A => (B,C), fb: F[B],fc: F[C]): F[A] Divide def conquer: F[A] Divisible There is no Contravariant Monad There is Contravariant Traversable, Alternative: https://www.youtube.com/watch?v=cB8DapKQz-I Edward Kmett, Discrimination is Wrong, 2015 47
- 48. Contravariant Functors (2) - arrows reversed //def map (F[A], A => B): F[B] Functor def contramap (F[A], A <= B): F[B] Contravariant (Contravariant Functor) //def map2 ((A,B) => Z, F[A], F[B]): F[Z] Apply (not ap!) def divide (Z => (A,B), F[A], F[B]): F[Z] Divide (Contravariant Apply) //def pure: ( () => A ) => F[A] Applicative def conquer: ( A <= () ) => F[A] Divisible (Contravariant Applicative) 48
- 50. Category Theory abstractions in Scala - Functor, Apply, Applicative - Monads: State, Writer, Reader, IO, Option, Eiter, Validated, List/Vector - Comonads: NonEmptyList, Stream, Store, Context - Contravariant, Divide, Divisible - Profunctor, Profunctor Strong, Profunctor Choice, Arrows, Kleisli 50
- 51. trait Profunctor[P[_, _]] { def dimap[X,Y,Z,W](ab: X => Y, cd: Z => W): P[Y, Z] => P[X, W] } Profunctor 51
- 52. trait Profunctor[P[_, _]] { def dimap[X,Y,Z,W](ab: X => Y, cd: Z => W): P[Y, Z] => P[X, W] } ● contramap on first type ( P[Y, _] is Contravariant ) ● map on second type ( P[_, Z] is Functor ) Profunctor = Contravariant + Functor 52
- 53. trait Profunctor[P[_, _]] { def dimap[X,Y,Z,W](ab: X => Y, cd: Z => W): P[Y, Z] => P[X, W] // derived methods def lmap[A,B,C](f: A => B): P[B,C] => P[A,C] = dimap[A,B,C,C](f,identity[C]) def rmap[A,B,C](f: B => C): P[A,B] => P[A,C] = dimap[A,A,B,C](identity[A], f) } Profunctor - derived methods 53
- 54. Haskell ● dimap id id == id ● dimap (f . g) (h . i) == dimap g h . dimap f i Scala a bit more verbose Profunctor - Laws 54
- 55. val function1: Profunctor[Function1] = new Profunctor[Function1] { def dimap[X, Y, Z, W](f: X => Y, g: Z => W): (Y => Z) => (X => W) = h => f andThen (g compose h) def lmap[A,B,C](f: A => B): (B => C) => (A => C) = f andThen def rmap[A,B,C](f: B => C): (A => B) => (A => C) = f compose } Profunctor - Example Function1 55
- 56. val f: String => Int = _.length case class Person(name: String, age: Int) val preF: Person => String = _.name val postF: Int => Boolean = _ > 5 Profunctor - Example Function1 - setup 56
- 57. val f: String => Int = _.length case class Person(name: String, age: Int) val preF: Person => String = _.name val postF: Int => Boolean = _ > 5 Profunctor[Function1].dimap(f)(preF)(postF)( Person("Foo", 100) ) Profunctor - Example Function1 - true ? 57
- 58. val f: String => Int = _.length case class Person(name: String, age: Int) val preF: Person => String = _.name val postF: Int => Boolean = _ > 5 Profunctor[Function1].dimap(f)(preF)(postF)( Person("Foo", 100) ) // false Profunctor - Example Function1 58
- 59. ● Haskell opaleye: https://github.com/tomjaguarpaw/haskell-opaleye/search?q=dimap&unscoped_q=dimap ● Monadic profunctors for bidirectional programming - blog post https://blog.poisson.chat/posts/2017-01-01-monadic-profunctors.html ● Kleisli Arrow Profunctor - Usage 59
- 61. trait Strong[P[_, _]] extends Profunctor[P] { def first[X,Y,Z](pab: P[X, Y]): P[(X, Z), (Y, Z)] } Profunctor - Strong 61
- 62. trait Strong[P[_, _]] extends Profunctor[P] { def first[X,Y,Z](pab: P[X, Y]): P[(X, Z), (Y, Z)] // derived methods def second[X,Y,Z](pab: P[X, Y]): P[(Z, X), (Z, Y)] } Profunctor - Strong - Derived methods 62
- 63. val functionStrong: Strong[Function1] = new Strong[Function1] with Function1Profunctor { def first[X, Y, Z](pab: X => Y): Function1[( X, Z), (Y, Z)] = (xz: (X, Z)) => (pab(xz._1) , xz._2) } Profunctor - Strong - Function1 63
- 64. val functionStrong: Strong[Function1] = new Strong[Function1] with Function1Profunctor { def first[X, Y, Z](pab: X => Y): Function1[( X, Z), (Y, Z)] = (xz: (X, Z)) => (pab(xz._1) , xz._2) } Profunctor[Function1].first(len)(( "foo", 42)) == (3,42) Profunctor - Strong - Function1 64
- 65. PR #2640 Strong Profunctor Laws: https://github.com/typelevel/cats/pull/2640 Profunctor - Strong Laws 65
- 66. Description: https://github.com/lemastero/scala_typeclassopedia#profunctor Translation to Scala early (very early pre-alpha): https://github.com/lemastero/scala_typeclassopedia/tree/master/src/main/scala/profunctor Profunctor hierarchy - big, not in Scala yet 66
- 67. Strong Profunctor == Arrow Model IO using Profunctors/Arrows? 67
- 69. Category Theory abstractions in Scala - Functor, Apply, Applicative - Monads: State, Writer, Reader, IO, Option, Eiter, Validated, List/Vector - Comonads: NonEmptyList, Stream, Store, Context - Contravariant, Divide, Divisible - Profunctor, Profunctor Strong, Profunctor Choice, Arrows, Kleisli - Traversable, Foldable (+ Monoid, Semigroup) - Kan extensions, Yoneda, Coyoneda, Density, Codensity - Free Monads, Free Applicative, Cofree, Free Alternative, Free Arrows - ... Profunctor optics, Cartesian Closed Categories, Day Convolution 69
- 70. Big Picture of Category Theory in Scala 70
- 71. Nice papers - Different encodings of CT like Sjoerd Visscher data-category: hackage.haskell.org/package/data-category-0.7/docs/Data-Category.html - Bifunctors: Joker, Clown, Biff, … - nice papers - Arrows … - nice papers - Distributive (Comonad Applicative) - nice papers - Adjunctions between Monads, Applicative, Arrows - nice papers - Monad, Applicative, Arrow as Monoid in Monoidal Category with given tensor (Functor Composition, Sum, Product, Day convolution) - nice papers - Recursion schemas (Fold, Unfolds, Fix point) - people talk about 3, exists 20+ 71
- 72. Alternative encoding - Different encodings of CT like Sjoerd Visscher data-category: hackage.haskell.org/package/data-category-0.7/docs/Data-Category.html - Proofs of Category Theory in Coq - Proofs of Category Theory using Univalent Foundations (related to HoTT) in Coq (UniMath/UniMath) https://github.com/UniMath/UniMath/tree/master/UniMath/CategoryTheory 72
- 73. How to learn Category Theory 1) Translate to Scala, Kata? wiki.haskell.org/Typeclassopedia, Edward Kmett libs 2) Describe github.com/lemastero/scala_typeclassopedia 3) In Scala 3 4) Talk using new vocabulary! 73
- 74. Good general theory does not search for the maximum generality, but for the right generality. Saunders Mac Lane Choose the right level of abstraction, to see the problem more clearly Eugenia Cheng, Category in Life https://www.youtube.com/watch?v=ho7oagHeqNc Thank you :) 74
- 75. Thank you :) 75