ScalaMatsuri 2018 - A pragmatic introduction to Category Theory

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A PRAGMATIC INTRODUCTION TO CATEGORY THEORY @DANIELASFREGOLA SCALA MATSURI 2018 github.com/DanielaSfregola/tutorial-cat

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

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I AM NOT A MATHEMATICIAN

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

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

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CATEGORY THEORY DEEPER UNDERSTANDING ON OUR CODE

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HOW DO WE REASON ?

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COMPOSITION ABSTRACTION

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CATEGORY THEORY HOW THINGS COMPOSE

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ARROW THEORY CATEGORY THEORY HOW THINGS COMPOSE

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WHAT IS A CATEGORY?

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COMPOSITION LAW

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IDENTITY LAW

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COMPOSITION + ASSOCIATIVITY

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CATEGORY'S RULES > Identity > Composition > Associativity

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A PRACTICAL EXAMPLE

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CATEGORY WITH 1 OBJECT

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CATEGORY WITH 1 OBJECT = MONOID

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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

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A PRACTICAL EXAMPLE

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

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

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CATEGORY WITH 1+ OBJECT

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CATEGORY IN A BOX

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

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LIFTING: CONTEXT VS CONTENT

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

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CATEGORY IN A BOX = FUNCTOR

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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)

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LIFTING: CONTEXT VS CONTENT

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

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

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BOX FUNCTION + BOX VALUES

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COMBINE MORE BOXES = APPLICATIVE

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

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

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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 }

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BOX FUNCTION + BOX VALUES

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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 }

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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

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BOX IN A BOX

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BOX IN A BOX

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FUSE TWO BOXES = MONAD

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MONAD'S RULES > Identity > Composition > Associativity

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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 } }

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BOXES IN A SEQUENCE

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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

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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] = ??? }

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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

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

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FUNCTOR VS ENDOFUNCTOR

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MONAD IS A MONOID IN THE CATEGORY OF ENDOFUNCTORS

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MONAD MONOID => pure + flatten ENDOFUNCTORS => map

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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

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FORGET ABOUT THE DETAILS FOCUS ON HOW THINGS COMPOSE

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

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