A PRAGMATIC INTRODUCTION TO CATEGORY THEORY @DANIELASFREGOLA SCALA WORLD 2017 github.com/DanielaSfregola/tutorial-cat 

AGENDA > Intro > Monoid > Functor > Applicative > Monad github.com/DanielaSfregola/tutorial-cat 

HELLOOOOO > ex Java Developer > OOP background > I am not a mathematician ! 

I AM NOT A MATHEMATICIAN 

YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE SCALA CODE 

YOU DO NOT NEED TO KNOW CATEGORY THEORY TO WRITE FUNCTIONAL CODE 

CATEGORY THEORY DEEPER UNDERSTANDING ON OUR CODE 

HOW DO WE REASON ? 

COMPOSITION ABSTRACTION 

OBJECT ORIENTED PROGRAMMING (OOP) > composition of objects > abstraction over the behaviour of each object 

FUNCTIONAL PROGRAMMING (FP) > composition of functions > abstraction over mathematical laws 

CATEGORY THEORY HOW THINGS COMPOSE 

ARROW THEORY CATEGORY THEORY HOW THINGS COMPOSE 

WHAT IS A CATEGORY? 

COMPOSITION LAW 

IDENTITY LAW 

COMPOSITION + ASSOCIATIVITY 

CATEGORY'S RULES > Identity > Composition > Associativity 

A PRACTICAL EXAMPLE 

CATEGORY WITH 1 OBJECT 

CATEGORY WITH 1 OBJECT = MONOID 

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 

A PRACTICAL EXAMPLE 

MONOID trait Monoid[A] { def identity: A def compose(x: A, y: A): A } 

EXERCISES ON MONOID > Define a monoid for Int > Define a monoid for String sbt 'testOnly *Monoid*' github.com/DanielaSfregola/tutorial-cat 

CATEGORY WITH 1+ OBJECT 

CATEGORY IN A BOX 

CATEGORY IN A BOX > Objects are in a Box > All the arrows are copied 

LIFTING: CONTEXT VS CONTENT 

EXAMPLE OF BOXES > Option > Future > Try > List > Either 

CATEGORY IN A BOX = FUNCTOR 

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) 

LIFTING: CONTEXT VS CONTENT 

CHANGE THE CONTENT OF A BOX > How to change the context 

FUNCTOR class Functor[Box[_]] { def map[A, B](boxA: Box[A]) (f: A => B): Box[B] } 

EXERCISES ON FUNCTOR > Define a functor for Maybe > Define a functor for ZeroOrMore sbt 'testOnly *Functor*' github.com/DanielaSfregola/tutorial-cat 

BOX FUNCTION + BOX VALUES 

COMBINE MORE BOXES INTO ONE = APPLICATIVE 

APPLICATIVE'S RULES > Identity > Associativity > Homorphism > Interchange ...and more! 

COMBINE BOXES TOGETHER > How to create a new box > How to combine their values together 

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 } 

BOX FUNCTION + BOX VALUES 

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 } 

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 

BOX IN A BOX 

BOX IN A BOX 

FUSE TWO BOXES TOGETHER = MONAD 

MONAD'S RULES > Left Identity > Right Identity > Associativity 

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 } } 

BOXES IN A SEQUENCE 

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 

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] = ??? } 

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 

EXERCISES ON MONAD (2) > Define a monad for Maybe > Define a monad for ZeroOrMore sbt 'testOnly *Monad*' github.com/DanielaSfregola/tutorial-cat 

MONAD IS A MONOID IN THE CATEGORY OF ENDOFUNCTORS 

MONAD MONOID => pure + flatten ENDOFUNCTORS => map 

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 

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”. 

FORGET ABOUT THE DETAILS FOCUS ON HOW THINGS COMPOSE 

WANNA KNOW MORE? > Category Theory for the WH by @PhilipWadler > Category Theory by @BartoszMilewski > Cats-Infographics by Rob Norris - @tpolecat > Cats Documentation - Type Classes 

THANK YOU! > Twitter: @DanielaSfregola > Blog: danielasfregola.com github.com/DanielaSfregola/tutorial-cat