Its All About Morphisms Its All About Morphisms Uberto Barbini - - PowerPoint PPT Presentation

It’s All About Morphisms It’s All About Morphisms

Uberto Barbini @ramtop https://medium.com/@ramtop

#VoxxedVienna #morphisms #morphisms @ramtop

Ab About m

• ut me

progr grammer er OO OOP TDD Kot

• tli

lin Agi gile le Funct nctio iona nal P PRo Rogr gramming ing Blog: https://medium.com/ ramtop @ Twitter: ramtop @ Fina nance nce Ind ndust stry

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Map Map of th

• f this p

s prese sentation ntation

Ca Catego egory Mono noid id Funct nctor

• r

Applic icative Mona nad Natural T l Transf nsformation

• n

Morphism sms a s all th the w e way ay dow down...

Yone neda da

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I d I don’t

• n’t care

e about

• ut

Mona Monads ds, w why sh y shoul

• uld I?

I?

Precisely

Neith Neither do er do I what I ca care a abou

• ut is

t is to defin to define s sys ystem em beh behaviou aviour

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This p presentati ntation w

• n will

ll b be a s a succ ccess i if mos most of t of you w you will ll not f not fall all a asleep This p presentati ntation w

• n will

ll b be a s a succ ccess i if mos most of t of you w you will ll not f not fall all a asleep

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You will consider that Functional Programming is about transformations and preserving properties. Not (only) lambdas and flatmap This p presentati ntation w

• n will

ll b be a s a succ ccess i if

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Wh What is at is t this is C Cat ateg egory thi

• ry thingy

ngy?

Invented in 1940s “with the goal of understanding the processes that preserve mathematical st struc ucture.” “Category Theory is about rel elat ation between things” “General abstract no nonsense nsense”

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Once upon a time there was a Category of Stuffed Toys and a Category of Tigers...

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1) A collection 1) A collection

• f Objects
• f Objects

A Cat Categ egory is is defin efined ed in in 5 5 step eps:

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2) A c 2) A collection

• llection
• f Arrows
• f Arrows

A Cat Categ egory is is defin efined ed in in 5 5 step eps:

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3) Each A ) Each Arrow rrow works

• rks on
• n 2

2 Objects Objects

A Cat Categ egory is is defin efined ed in in 5 5 step eps:

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4) Arrows can 4) Arrows can be combined be combined

A Cat Categ egory is is defin efined ed in in 5 5 step eps:

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5) Each Object 5) Each Object has an Identity has an Identity

(an arro (an arrow po pointing ting t to it

• itsel

elf)

A Cat Categ egory is is defin efined ed in in 5 5 step eps:

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A Category Exa A Category Examp mple

Tube Map: Objects stations

• Arrows

travel routes

• Each Arrows connect 2 stations

Arrow composition is travelling along the line Identity Arrow is staying in the same station

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London London Tub ube Category Category

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This is pr pres esentation entation is a is a Categor tegory y as as well! ell!

Ca Catego egory Monoi noid Funct nctor

• r

Appli lica cativ ive Mona nad Natural T l Transf nsformatio ion Yoned neda

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Eve Event nt Source Ca Source Cate tegor

• ry

New ewOr Order Rea Ready dy Ca Cance ncell lled ed Ret Returned ed Dispatch ched Closed ed

https://skillsmatter.com/skillscasts/11486-functional-cqrs AddItem AddItem Cancel Cancel Dispatch Dispatch Return Return Close Close

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Func Functional Pro ional Program ramming ming is is als also a C

• a Cat

ateg egory ry

Each programming language has a Category: Types pes are the objects and Fun unct ctions are the morphisms. Part rtial fu l functions don’t have a defined return for all inputs. In reality all programming functions are partial: they can raise Exceptions or never end. They always have an hidden return of Bottom Type ( ) ⊥* ⊥*

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Change of mindset!

Ob Objec ect O Oriented ented Fu Funct ctional nal Living Bacter ng Bacteria Gea ears and and P Pipes es Op Opaqu aque Trans anspar arent nt Hi Hidden St n State ate Immutabl table Stat State Int nterfaces ces Type C Clas asses

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Kotl

• tlin f

n for f

• r functi

unctional p

• nal program

rammi ming ng

• Nullable and not-nullable types
• Type Aliases
• Class extensions
• Tail recursion
• Pattern matching (when)
• Arrow-kt bindinds with coroutines
• Arrow-kt Typeclasses (?)

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arrow-kt.io

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KEEP-87 TypeClasses

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Pur Purity ity and and Imm Immutab utability ility

For the Category morphisms to work in programming we need Puri rity and Im Immutabi abili lity. But they are not a goal per se, only a necessity for the main goal: co compo posi sition and transf sform rmation. Ultimately everything is converted in assem assembly bly which is neither pure nor immutable. We need those quality

• nly for expo

posed sed code

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Monoid Monoid

What about the category of morphisms of a category? Are they composable? It’s a Category with a single Object and lots of Morphisms A Ca Categor egory y wi with th only one

• ne Obj

Objec ect t is a Mo Monoi noid

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Progr Programming ing wi with th Monoids

• noids

A type class with two methods combine monoid append

• empty

neutral element

• (x <> y) <> z = x <> (y <> z) -- associativity

empty <> x = x -- left identity x <> empty = x -- right identity

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Gener neric ics Type Contructors ype Contructors

List<A> is just an abstract type to build List<Int> List<String> List<User> etc.

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Type Cla e Class ss vs vs Inter nterfa face ce

• Interfaces “unify” different types:

Cat and Dogs can be treated as Animals

• Type Classes “group” types with similar behaviour, wi

witho hout hi hiding t g thei eir t r type pes: Cats and Dogs can both form couples but cats can mate

• nly with cats and dogs with dogs.

You cannot represent that with interfaces.

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Typecl Typeclas ass Ins Instanc ances es

• Typeclasses work with instances (like a singleton)
• List is not a Monoid nor a Functor nor a Monad but it

has an (or more) instance of Monoid one of Functor and

• ne of Monad
• Technically we implement instances as an interface with

a singletons specific implementation.

• We can have different implementation for difference in

evaluation, for example because of concurrency

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Enough h talk, , let let’s ’s see ee the he co code! e!

Mono noid id TypeCla Class Inst nstances. nces... ...Give u e us the co combi bine e ne extens nsion

• n

funct nctio ion

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The f e fut utur ure (?) (?)

extension interface Monoid<T> { infix fun T.add(t: T): T } extension object IntMonoid: Monoid<Int> { inline fun Int.add(t: Int) = this + t } inline fun <T> sum(t1: T, t2: T, t3: T, with Monoid<T>) = t1 add t2 add t3 fun main() { sum(1, 2, 3) //no boxing because of inlining }

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Trans ransform

• rmers

ers a a.k.a. F .a. Functors

• rs

Very Important!!

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Functors unctors

Functors can map both object ects (types) and morph rphism (functions) between two categories Functors map must preserve the structure and some properties But can also work inside the same category (End Endofu funct nctors rs)

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Functor Laws unctor Laws

Functor is a TypeClass with a Map function that works like this. Id is the identity function. map id x = x map (g <> f) = map g <> map f) Passing the ID function must return the original value Map must honour associativity of two functions

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Try Functor ry Functor

Fail ilure w without E Excep ceptio ion Kee eep a a cont ntex ext and M nd Map o

• per

eratio ion n

• n `it

n `it

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Functo Functors rs are al are also Fo Forklif ifts ts

val lifted = Try.functor().lift {x:String -> x.toInt()} lifted(Try.Success("42")) //Try.Success(42) Lift a function from A --> B to F<A> --> F<B>

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Yoneda Yoneda L Lem emma ma

Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set. If C is a locally small category (i.e. the hom- sets are actual sets and not proper classes), then each object A of C gives rise to a natural functor to Set called a hom-functor.

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Yoneda Yoneda L Lem emma ma

F<?>.map(A→B): F<B> (? must be A) The actual implementation is useful to compose of mappings without executing until we decide co combine bine a all l the m maps in s in

• ne

ne a and nd t then en apply ly i it. No co

• copie

ies o

• f L

List ist

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Na Natural ural T Trans ransfo forma rmations ions

A nat atural t ral tran ransf sform rmat ation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Transforming Data → Functions Data → Functions Transforming Functions → Functors Functions → Functors Transforming Functors → Natural Functors → Natural Transformations Transformations

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Natural Tr Transfo forma mations

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Natur Natural al Trans Transform

• rmati

tions

• ns

val list = Try {"3".toInt()}.toOption().toList() //[3] val fail = Try {"xyz".toInt()}.toOption().toList() //[]

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Com

• mbini

ining Functors ng Functors

We can imagine 2 ways to combine 2 functors F + F = F F<f> map F<a> = F<f(a)>

• r

F * F = F flatmap (a → F<a → F<a>>) = F a

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Applic Applicative ative Functors unctors

• We can combine a value inside a Functor with a fu

funct ction inside another Functor

• If the function want more than 1 param, it will return a

function with x-1 params.

• Applicative can apply

pply:

• F<A> applied to F<A->B> to create F<B>

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Try A y Appli plicat cative F Fun uncto tor

Sil Silly ly e exa xample le of

• f f

funct nctio ion n that ca can raise a ise an E n Excep ceptio ion Exce ceptio ion r n raised! d!

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The M wor word... ...

• What about a category of (endo)functors?
• Some functors have a monoid instance, others not.
• How can we call the Category of Endofunctors with a

Monoid instance?

A Monad is just a Monoid in the category of Endofunctors, what's the problem?

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Mo Monad Rec nad Recipe ipe

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Monads Allow Sequences

• f Instructions

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Monads Monads Bi Binding nding

From here:

val university: IO<University> = getStudent("John Smith").flatMap { student -> getUniversity(student.universityId).flatMap { university -> getDean(university.deanId) } }

To here:

val university: IO<University> = IO.monad().binding { val student = getStudent("John Smith").bind() val university = getUniversity(student.universityId).bind() val dean = getDean(university.deanId).bind() dean }

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Functiona nctional l Pr Progr

• gramm

amming ing Di Dilemm lemma: a:

Per Perfec ectly Pur ly Pure e Pr Progr

• gram

ams s are e Per erfec ectly ly U Useless seless

E n t e r E f f e c t s

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De Depende dency In y Inje jecti tion

• n

cou

• urtesy

sy of

• f Reade

eader Mon

• nad

d

What’s h happen if n if u user ca canno nnot be be f fetch ched ed? Runs h ns here

fun getUser(userId:String):Reader Reader<Context, User> { fun getCommonFriends(u1: User, u2: User): Reader Reader<Context, List<User>>

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Monads

• nads Zoo

Zoo

All Monads are also Functors and Applicative, but the opposite is not true. Each Monad as it’s specific logic on top

• f the Monads Laws

These are already available in many libraries but you can extend and create your own.

• Option
• Try
• List
• Either
• Reader
• Writer
• State
• IO
• Free

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Why s y studyi ying C ng Categories ategories?

Morphisms allows you to work at compile time with your Do Domain K Knowled wledge ge Learn how to co compo pose functions and pres preserve properties Monads and other typeclasses should em emerge erge from your code, not the other way round Learn a common t term erminolo logy gy and the reasons behind that

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How How to us to use mor

• rphisms

phisms in n real l world

• rld pr

program

• grams?

s?

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Real World Ex eal World Exam amples ples

Web Se Server a er as a f func uncti tion github.com/http4k/http4k Func uncti tiona nal D Doma

• main D

n Design gn github.com/uberto/anticapizzeria

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To l learn m earn more: re:

Category Theory for Programmers bartoszmilewski.com Cats in scala typelevel.org/cats Arrow in Kotlin arrow-kt.io

If yo f you enj enjoyed yed pleas please f e follo llow w me o e on t n twit witter and medi ter and medium ram ramtop @ramtoq @ramtoq

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Endofunc ndofunctor tors

• In programming we only work with Endofunctors in the

Category of our types

• Endofunctor map a => F a where F a is a type

constructor in the same category of a (the type system category)

• Endofunctors are very important because we can compose

them!

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Higher Higher Kinded inded Type ypes

In Java we cannot abstract at more than one level like List<A<B>>

• r ignoring the second level List<A<?>>

But in Functional programming makes sense generalise over the container or context like: X<A> + f(A)->B = X<B>

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Mona Monads ds Laws ws

"The he diagra gram co commutes es" means that the map produced by following any path through the diagram is the same.

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Li List A App ppli licati cative

Can y you gu guess ess h how

• w it

it w wor

• rk?

Transf nsform a st standa ndard L d List ist i in a n a Arrow

• w L

List ist ( (Lis istK)

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List Ma Map2

Exact ctly ly s same r e resu esult lt

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Read eader: er: A A Func Functor on f r on functio unctions ns

Val userId = 123 val reader = Reader{dbUrl:String -> DbConn(dbUrl)} val name = reader.run("myDbConn") .map{it.getUser(userId) } .map{it.name} .value() //"Joe"

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Pu Purit ity Bubbles les + + Eve vents = Act ctors

#VoxxedVienna #morphisms #morphisms @ramtop "Rather than thinking about function pu puri rity as a go goal, you have to think about which pro propert perties es you want your program to pres preserv erve." @raulraja

Perf erfect ectly ly Pu Pure Pro re Progra rams ms are re Per Perfect ectly ly Us Useless eless

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London Tu Tube e Map is is a Mo Monoid!

(if you squint hard enough)

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Ok… bu but w t why using ng a O a Obj bject O ct Ori riented ted La Lang ngua uage f for

• r F

Functi tion

• nal

Prog

• gra

ramm mming at at al all? l?

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Conclusions: Conclusions:

why fu function ctional l prog rogram rammin ing?

More composition Less boilerplate Less repetitions Less bugs Less need for tests Easier concurrency Less sugar Different thinking Performance issues Complex state More precision Different testing style

VS VS

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Monads Transformers

• Monads don’t compose

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Validated example

• Better than Try

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Klei leisli li Arrows

• ws

Original problem: Monads don’t allow for concurrency Arrows can be useful in Reactive Functional Programming Kleisli is a type of Arrow for a Monadic context It’s old ReaderT actually

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Arrows

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CandyDispenser

• Code example
• Let’s bring all together
• We have a function Seed

(Seed, Candy)

• Another one (State, Input)

State

• How can we combine them?
• We want

(Dispenser, [Input]) (Dispenser, [Candy])

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Adjoint Adjoint Func Functor tors

Functor F from category D to C Functor G from C to D Every adjunction 〈 F, G, , ε, θ η 〉 gives rise to an associated monad 〈 T, , η μ 〉 in the category D.