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ScalaWorld 2017 - A Pragmatic Introduction to Category Theory

ScalaWorld 2017 - A Pragmatic Introduction to Category Theory

Daniela Sfregola

September 19, 2017

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  1. AGENDA > Intro > Monoid > Functor > Applicative >

    Monad github.com/DanielaSfregola/tutorial-cat
  2. MONOID'S RULES Identity n o id == id o n

    == n Composition forall x, y => x o y Associativity x o (y o z) == (x o y) o z
  3. EXERCISES ON MONOID > Define a monoid for Int >

    Define a monoid for String sbt 'testOnly *Monoid*' github.com/DanielaSfregola/tutorial-cat
  4. CATEGORY IN A BOX > Objects are in a Box

    > All the arrows are copied
  5. FUNCTOR'S RULES Identity map(id) == id Composition map(g o f)

    == map(g) o map(f) Associativity map(h o g) o map(f) == map(h) o map(g o f)
  6. EXERCISES ON FUNCTOR > Define a functor for Maybe >

    Define a functor for ZeroOrMore sbt 'testOnly *Functor*' github.com/DanielaSfregola/tutorial-cat
  7. COMBINE BOXES TOGETHER > How to create a new box

    > How to combine their values together
  8. APPLICATIVE class Applicative[Box[_]] extends Functor[Box] { def pure[A](a: A): Box[A]

    def ap[A, B](boxF: Box[A => B])(boxA: Box[A]): Box[B] def map[A, B](boxA: Box[A])(f: A => B): Box[B] = ??? // spoiler alert }
  9. APPLICATIVE class Applicative[Box[_]] extends Functor[Box] { def pure[A](a: A): Box[A]

    def ap[A, B](boxF: Box[A => B])(value: Box[A]): Box[B] def ap2[A1, A2, B](boxF: Box[(A1, A2) => B]) (value1: Box[A1], value2: Box[A2]): Box[B] // up to 22 values! // same for map }
  10. EXERCISES ON APPLICATIVE > Define map in terms of ap

    and pure > Define an applicative for Maybe > Define an applicative for ZeroOrMore sbt 'testOnly *Applicative*' github.com/DanielaSfregola/tutorial-cat
  11. MONAD (AS FUNCTOR) class Monad[Box[_]] extends Functor[Box] { def flatten[A](bb:

    Box[Box[A]]): Box[A] def flatMap[A, B](valueA: Box[A]) (f: A => Box[B]): Box[B] = { val bb: Box[Box[B]] = map(valueA)(f) bb.flatten } }
  12. FOR-COMPREHENSION val boxA: Box[A] def toBoxB: A => Box[B] def

    toBoxC: B => Box[C] def toBoxD: C => Box[D] for { a <- boxA b <- toBoxB(a) c <- toBoxC(b) d <- toBoxD(c) } yield d
  13. MONAD (AS APPLICATIVE) trait Monad[Box[_]] extends Applicative[Box] { def flatMap[A,

    B](boxA: Box[A])(f: A => Box[B]): Box[B] // TODO - implement using flatMap def flatten[A](boxBoxA: Box[Box[A]]): Box[A] = ??? // TODO - implement using flatMap and map override def ap[A, B](boxF: Box[A => B])(boxA: Box[A]): Box[B] = ??? // TODO - implement using flatMap and pure override def map[A, B](boxA: Box[A])(f: A => B): Box[B] = ??? }
  14. EXERCISES ON MONAD (1) > Define flatten using flatMap >

    Define map using flatMap and pure > Define ap using flatMap and map github.com/DanielaSfregola/tutorial-cat
  15. EXERCISES ON MONAD (2) > Define a monad for Maybe

    > Define a monad for ZeroOrMore sbt 'testOnly *Monad*' github.com/DanielaSfregola/tutorial-cat
  16. SUMMARY CATEGORY THEORY >> how things compose MONOID >> combining

    2 values into 1 FUNCTOR >> values lifted to a context APPLICATIVE >> independent values applied to a function in a context MONAD >> ops in sequence in a context
  17. Band BoundedSemilattice CommutativeGroup CommutativeMonoid CommutativeSemigroup Eq Group Semigroup Monoid Order

    PartialOrder Semilattice Alternative Applicative ApplicativeError Apply Bifoldable Bimonad Bitraverse Cartesian CoflatMap Comonad ContravariantCartesian FlatMap Foldable Functor Inject InvariantMonoidal Monad MonadError MonoidK NotNull Reducible SemigroupK Show ApplicativeAsk Bifunctor Contravariant Invariant Profunctor Strong Traverse Arrow Category Choice Compose Cats Type Classes kernel core/functor core/arrow core The highlighted type classes are the first ones you should learn. They’re well documented and well-known so it’s easy to get help. a |+| b a === b a =!= b a |@| b a *> b a <* b a <+> b a >>> b a <<< b a > b a >= b a < b a <= b Sync Async Effect LiftIO effect Some type classes introduce symbolic operators. NonEmptyTraverse InjectK CommutativeArrow CommutativeFlatMap CommutativeMonad ApplicativeLayer FunctorLayer ApplicativeLayerFunctor FunctorLayerFunctor ApplicativeLocal FunctorEmpty FunctorListen FunctorTell FunctorRaise MonadLayer MonadLayerFunctor MonadLayerControl MonadState TraverseEmpty Functor Applicative Monad Traverse mtl MTL type classes do not extend core type classes directly, but the effect is similar; the dashed line can be read “implies”.
  18. WANNA KNOW MORE? > Category Theory for the WH by

    @PhilipWadler > Category Theory by @BartoszMilewski > Cats-Infographics by Rob Norris - @tpolecat > Cats Documentation - Type Classes