# lemastero/scala_typeclassopedia

Patterns from math (Category theory, Abstract algebra) in Scala: minimal description + links to good explanations
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# Scala typeclassopedia

Abstract Algebra

Category Theory

## Category Theory

### Functor (Covariant Functor)

Abstraction for type constructor (type with "hole", type parameter) that can be mapped over.

Containers (List, Tree, Option) can apply given function to every element in the collection. Computation effects (Option - may not have value, List - may have multiple values, Either/Validated - may contain value or error) can apply function to a value inside this effect without changing the effect.

```trait Functor[F[_]] {
def map[A,B](a: F[A])(f: A => B): F[B]
}```
• Functor Laws:
1. identify: `xs.map(identity) == xs`
2. composition: `xs.map(f).map(g) == xs.map(x => g(f(x))`

If Functor satisfy fist law then it also satisfy second law: (Haskell) The second Functor law is redundant - David Luposchainsky if we don't include bottom values - (Haskell) contrexample using undefined.

• Instances can be implemented for: List, Vector, Option, Either, Validated, Tuple1, Tuple2, Function

• Counterexamples

1. Set: Twitter discussion Explanation by Mark Seemann Set is not a functor. Comments in alleycats
2. Map: Cats Issue 1831
• Derived methods of Functor:
```def lift[A, B](f: A => B): F[A] => F[B] // lift regular function to function inside container
def fproduct[A, B](fa: F[A])(f: A => B): F[(A, B)] // zip elements with result after applying f
def as[A, B](fa: F[A], b: B): F[B] // replace every element with b
def void[A](fa: F[A]): F[Unit] // clear preserving structure
def tupleLeft[A, B](fa: F[A], b: B): F[(B, A)]
def tupleRight[A, B](fa: F[A], b: B): F[(A, B)]
def widen[A, B >: A](fa: F[A]): F[B]```

### Apply

Apply is a Functor that can apply function already inside container to container of arguments.

Apply is a weaker version of Applicative that cannot put value inside effetc F.

```trait Apply[F[_]] extends Functor[F] {
def ap[A, B](ff: F[A => B])(fa: F[A]): F[B]
}```
• Explanation by John DeGoes why `ap` is usefull from Haskell perspective, but in Scala better would be to use `product` (named `zip`) (reddit)

• Derived methods

```def apply2[A, B, Z]   (fa: F[A], fb: F[B])          (ff: F[(A,B) => Z]): F[Z]
def apply3[A, B, C, Z](fa: F[A], fb: F[B], fc: F[C])(ff: F[(A,B,C) => Z]): F[Z]
// ...

def map2[A , B, Z]  (fa: F[A], fb: F[B])          (f: (A, B) => Z):    F[Z]
def map3[A, B, C, Z](fa: F[A], fb: F[B], fc: F[C])(f: (A, B, C) => Z): F[Z]
// ...

def tuple2[A, B]   (fa: F[A], fb: F[B]):           F[(A, B)]
def tuple3[A, B, C](fa: F[A], fb: F[B], fc: F[C]): F[(A, B, C)]
// ...

def product[A,B](fa: F[A], fb: F[B]): F[(A, B)]
def flip[A, B](ff: F[A => B]): F[A] => F[B]```

### Applicative (Applicative Functor)

Applicative Functor is a Functor that can:

• apply function already inside container to container of arguments (so it is Apply)
• put value into container (lift into effect)
```trait Applicative[F[_]] extends Apply[F] {
def pure[A](value: A): F[A]
}```
• Applicative Laws:
1. identify: `xs.apply(pure(identity)) == xs` apply identify function lifted inside effect does nothing
2. homomorphism: `pure(a).apply(pure(f)) == pure(f(a))` lifting value a and applying lifted function f is the same as apply function to this value and then lift result
3. interchange: `pure(a).apply(ff) == ff.apply(pure(f => f(a)))` where `ff: F[A => B]`
4. map: `fa.map(f) == fa.apply(pure(f))`
• Derived methods - see Apply

• Applicatives can be composed

• Minimal set of methods to implement Applicative (other methods can be derived from them):

• map2, pure
• apply, pure
• Resources:

• herding cats - Applicative: blog post
• FSiS 2 - Applicative type class - Michael Pilquist: video
• FSiS 3 - Monad type class - Michael Pilquist: video
• Cats: docs
• Applicative programming with effects - Conor McBride, Ross Paterson (shorter) longer
• The Essence of the Iterator Pattern - Jeremy Gibbons, Bruno C. d. S. Oliveira: (paper)
• The Essence of the Iterator Pattern - Eric Torreborre (blog post)
• Lifting - Tony Morris: blog post
• (Haskell) Abstracting with Applicatives - Gershom Bazerman (blog post)
• (Haskell) Algebras of Applicatives - Gershom Bazerman (blog post)
• Functional Patterns in C++, 2. Currying, Applicative - Bartosz Milewski video

### Selective (Selective applicative functors)

"Extend the Applicative type class with a single method that makes it possible to be selective about effects."

"handle is a selective function application: you apply a handler function of type A => B when given a value of type Left(a), but can skip the handler (along with its effects) in the case of Right(b)."

Andrey Mokhov

```trait Selective[F[_]] extends Applicative[F] {
def handle[A, B](fab: F[Either[A, B]], ff: F[A => B]): F[B]
def select[A, B, C](fab: F[Either[A, B]], fac: F[A => C], fbc: F[B => C]): F[C]
}```
• Resources:
• Selective applicative functors - Andrey Mokhov (blog post)

We add to Apply ability `flatMap` that can join two computations and use the output from previous computations to decide what computations to run next.

```trait Monad[F[_]] extends Apply[F] {
def pure[A](value: A): F[A]
def flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]
}```

1. flatmap associativity: `fa.flatMap(f).flatMap(g) == fa.flatMap(a => f(a).flatMap(b => g(b))`
2. left identity: `pure(a).flatMap(f) == f(a)`
3. right identity: `fa.flatMap(a => pure(a)) == fa`
• Minimal set of methods to implement Monad (others can be derived using them):

• pure, flatMap
• pure, flatten, map
• pure, flatten, apply
• pure, flatten, map2
• Derived methods:

```def flatten[A](ffa: F[F[A]]): F[A]
def sequence[G[_], A](as: G[F[A]])(implicit G: Traverse[G]): F[G[A]]
def traverse[A, G[_], B](value: G[A])(f: A => F[B])(implicit G: Traverse[G]): F[G[B]]
def replicateA[A](n: Int, fa: F[A]): F[List[A]]
def unit: F[Unit] // put under effect ()
def factor[A, B](ma: F[A], mb: F[B]): F[(A, B)]```

Wrapper around function from given type. Input type can be seen as some configuration required to produce result.

```case class Reader[-In, +R](run: In => R) {
def map[R2](f: R => R2): Reader[In, R2] =

def flatMap[R2, In2 <: In](f: R => Reader[In2, R2]): Reader[In2, R2] =
}```

### Writer

• Resources
• Monadic Logging and You - Martin Snyder (video)
• The Writer Monad using Scala (example) - Tony Morris: blog post

• Resources

• Resources

• Resources

### Contravariant (Contravariant Functor)

```trait Contravariant[F[_]] {
def contramap[A, B](f: B => A): F[A] => F[B]
}```
• Contravariant is not called Cofunctor (like Monad -> Comonad, Appy -> Coapply) because when we inverse arrows in Functor definition we just get Functor definition (with A, B swapped). More on this on SO.

• Resources

### Divide (Contravariant Apply)

```trait Divide[F[_]] extends Contravariant[F] {
def divide[A, B, C](fa: F[A], fb: F[B])(f: C => (A, B)): F[C]
}```
• Laws: let `def delta[A]: A => (A, A) = a => (a, a)`
1. composition `divide(divide(a1, a2)(delta), a3)(delta) == divide(a1, divide(a2, a3),(delta))(delta)`
• Derived methods:
```def divide1[A1, Z]    (a1: F[A1])           (f: Z => A1): F[Z] // contramap
def divide2[A1, A2, Z](a1: F[A1], a2: F[A2])(f: Z => (A1, A2)): F[Z]
// ...
def tuple2[A1, A2]    (a1: F[A1], a2: F[A2]):            F[(A1, A2)]
def tuple3[A1, A2, A3](a1: F[A1], a2: F[A2], a3: F[A3]): F[(A1, A2, A3)]
// ...
def deriving2[A1, A2, Z](f: Z => (A1, A2))(implicit a1: F[A1], a2: F[A2]): F[Z]
def deriving3[A1, A2, A3, Z](f: Z => (A1, A2, A3))(implicit a1: F[A1], a2: F[A2], a3: F[A3]): F[Z]
// ...```
• Resources
• Discrimination is Wrong: Improving Productivity - Edward Kmett (video)
• scalaz (src)

### Divisible (Contravariant Applicative)

```trait Divisible[F[_]] extends Divide[F] {
def conquer[A]: F[A]
}```
• Laws: let `def delta[A]: A => (A, A) = a => (a, a)`
1. composition `divide(divide(a1, a2)(delta), a3)(delta) == divide(a1, divide(a2, a3),(delta))(delta)`
2. right identity: `divide(fa, conquer)(delta) == fa`
3. left identity: `divide(conquer, fa)(delta) == fa`
• Resources

### Bifunctor

Abstracts over type constructor with 2 "holes". Represents two independent functors:

```trait Bifunctor[F[_, _]] {
def bimap[A, B, C, D](fab: F[A, B])(f: A => C, g: B => D): F[C, D]
}```
• Bifunctor Laws
1. identity `xs.bimap(identity, identity) == xs` bimap with two identify function does nothing
2. composition `xs.bimap(f, h).bimap(g,i) == xs.bimap(x => g(f(x), x => h(i(x))` you can bimap using f and h and then bimap using g and i or bimap once using composition Second law is automatically fulfilled if the first law holds.
• Alternatively can be specified by providing
```def leftMap[A, B, C](fab: F[A, B])(f: A => C): F[C, B]
def rightMap[A, B, D](fab: F[A, B])(g: B => D): F[A, D]```

In that case identity law must hold for both functions: 3. identity `xs.leftMap(identity) == xs` leftMap with identify function does nothing 4. identity `xs.rightMap(identity) == xs` rightMap with identify function does nothing If leftMap and rightMap and bimap are specified then additional lwa must be fullfilled: 5. `xs.bimap(f, g) == xs.leftMap(f).rightMap(g)`

• Derived methods
```def leftMap[A, B, C](fab: F[A, B])(f: A => C): F[C, B]
def rightMap[A, B, D](fab: F[A, B])(g: B => D): F[A, D]
def leftFunctor[X]: Functor[F[?, X]]
def rightFunctor[X]: Functor[F[X, ?]]
def umap[A, B](faa: F[A, A])(f: A => B): F[B, B]
def widen[A, B, C >: A, D >: B](fab: F[A, B]): F[C, D]```

### Bifunctor Join

Turn a Bifunctor with both arguments with the same type into Functor.

```newtype Join p a = Join { runJoin :: p a a }

-- instance
-- Bifunctor p => Functor (Join p)```

### Bifunctor Wrap

Functor over second argument of a Bifunctor

```newtype WrappedBifunctor p a b = WrapBifunctor { unwrapBifunctor :: p a b }

-- instance
-- Bifunctor p => Functor (WrappedBifunctor p a)```

### Bifunctor Flip

Swap arguments of Bifunctor

```newtype Flip p a b = Flip { runFlip :: p b a }

-- instance
-- Bifunctor p => Bifunctor (Flip p)```

### Bifunctor Joker

Functor over second argument of Bifunctor

```newtype Joker g a b = Joker { runJoker :: g b }

-- instance
-- Functor g => Bifunctor (Joker g :: * -> * -> *)
-- Functor g => Functor (Joker g a)```

### Bifunctor Clown

Functor over second argument of Bifunctor

```newtype Clown f a b = Clown { runClown :: f a }

-- instances
-- Functor (Clown f a :: * -> *)
-- Functor f => Bifunctor (Clown f :: * -> * -> *)```

### Bifunctor Product

Product of two Bifunctors

```data Product f g a b = Pair (f a b) (g a b)

-- instance
-- (Bifunctor f, Bifunctor g) => Bifunctor (Product f g)```

### Bifunctor Sum

Sum of two Bifunctors

```data Sum p q a b = L2 (p a b) | R2 (q a b)

-- instance
-- (Functor f, Bifunctor p) => Functor (Tannen f p a)```

### Bifunctor Tannen

Compose Functor on the outside.

```newtype Tannen f p a b = Tannen { runTannen :: f (p a b) }

-- instances
-- (Functor f, Bifunctor p) => Bifunctor (Tannen f p)
-- (Functor f, Bifunctor p) => Functor (Tannen f p a)```

### Bifunctor Biff

Compose two Functors inside Bifunctor

```newtype Biff p f g a b = Biff { runBiff :: p (f a) (g b) }

-- instance
-- (Bifunctor p, Functor f, Functor g) => Bifunctor (Biff p f g)```

### Invariant (Invariant Functor, Exponential Functor)

Functor that can create covariant functor (by passing identity as g) or contravariant functor (by passing identity to f). It represent situation when given type constructor contains type on both positive and negative position (like function A => A).

```trait Invariant[F[_]] {
def imap[A, B](fa: F[A])(f: A => B)(g: B => A): F[B]
}```

• Resources

• Explorations in Variance - Michael Pilquist (video)
• Cats docs
• Exponential on Functor level tweeter Oleg Grenrus
• (Haskell) Rotten Bananas - Edward Kmett (blog post) (There is nice example, beware: a lot of recursion schemas)
• Category Theory 8.1: Function objects, exponentials (video)

### Distributive

Traverse are composable Distributive it require only Functor (and Traverse require Applicative

```trait Distributive[F[_]] extends Functor[F] {
def collect[G[_]:Functor,A,B](fa: G[A])(f: A => F[B]): F[G[B]]
def distribute[G[_]:Functor,A](fa: G[F[A]]): F[G[A]]
}```

Abstraction for type with one hole that allows:

• map over (extends Functor)
• get current value
• duplicate one layer of abstraction It is dual to Monad (Monad allow to put value in and collapse one layer).
```trait CoflatMap[F[_]] extends Functor[F] {
def coflatMap[A, B](fa: F[A])(f: F[A] => B): F[B]
}

def extract[A](ca: C[A]): A // counit
def duplicate[A](ca: C[A]): C[C[A]] // coflatten
def extend[A, B](ca: C[A])(f: C[A] => B): C[B] = map(duplicate(ca))(f) // coflatMap, cobind
}```

If we define extract and extend:

1. `fa.extend(_.extract) == fa`
2. `fa.extend(f).extract == f(fa)`
3. `fa.extend(f).extend(g) == fa.extend(a => g(a.extend(f)))`

If we define comonad using map, extract and duplicate: 3. `fa.duplicate.extract == fa` 4. `fa.duplicate.map(_.extract) == fa` 5. `fa.duplicate.duplicate == fa.duplicate.map(_.duplicate)`

And if we provide implementation for both duplicate and extend: 6. `fa.extend(f) == fa.duplicate.map(f)` 7. `fa.duplicate == fa.extend(identity)` 8. `fa.map(h) == fa.extend(faInner => h(faInner.extract))`

• Derived methods:
` def extend[A, B](ca: C[A])(f: C[A] => B): C[B] = map(duplicate(ca))(f) // coFlatMap`

Method extend can be use to chain oparations on comonads - this is called coKleisli composition.

• Implementations of comonad can be done for: None empty list, Rose tree, Identity

• Resources

Wrap value of type A with some context R.

```case class CoReader[R, A](extract: A, ask: R) {
}```

### Cowriter

It is like Writer monad, combines all logs (using Monid) when they are ready.

```case class Cowriter[W, A](tell: W => A)(implicit m: Monoid[W]) {
def extract: A = tell(m.empty)
def duplicate: Cowriter[W, Cowriter[W, A]] = Cowriter( w1 =>
Cowriter( w2 =>
tell(m.append(w1, w2))
)
)
def map[B](f: A => B) = Cowriter(tell andThen f)
}```
• Resources
• Scala Comonad Tutorial, Part 1 - Rúnar Bjarnason (blog post)

`trait Bimonad[T] extends Monad[T] with Comonad[T]`
• They simplify resolution of implicits for things that are Monad and Comonad

Resources:

### Foldable

Given definition of foldLeft (eager, left to right0) and foldRight (lazi, right to left) provide additional way to fold Monoid.

```trait Foldable[F[_]]  {
def foldLeft[A, B](fa: F[A], b: B)(f: (B, A) => B): B
def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B): B
}```
• Laws: no. You can define condition that foldLeft and foldRight must be consistent.
• Derived methods (are different for scalaz and Cats):
```def foldMap[A, B](fa: F[A])(f: A => B)(implicit B: Monoid[B]): B
def foldM    [G[_], A, B](fa: F[A], z: B)(f: (B, A) => G[B])(implicit G: Monad[G]): G[B] // foldRightM
def foldLeftM[G[_], A, B](fa: F[A], z: B)(f: (B, A) => G[B])(implicit G: Monad[G]): G[B]
def find[A](fa: F[A])(f: A => Boolean): Option[A] // findLeft findRight
def forall[A](fa: F[A])(p: A => Boolean): Boolean // all
def exists[A](fa: F[A])(p: A => Boolean): Boolean // any
def isEmpty[A](fa: F[A]): Boolean // empty
def get[A](fa: F[A])(idx: Long): Option[A] // index
def size[A](fa: F[A]): Long // length
def toList[A](fa: F[A]): List[A]
def intercalate[A](fa: F[A], a: A)(implicit A: Monoid[A]): A
def existsM[G[_], A](fa: F[A])(p: A => G[Boolean])(implicit G: Monad[G]): G[Boolean] // anyM
def forallM[G[_], A](fa: F[A])(p: A => G[Boolean])(implicit G: Monad[G]): G[Boolean] // allM

// Cats specific
def filter_[A](fa: F[A])(p: A => Boolean): List[A]
def takeWhile_[A](fa: F[A])(p: A => Boolean): List[A]
def dropWhile_[A](fa: F[A])(p: A => Boolean): List[A]
def nonEmpty[A](fa: F[A]): Boolean
def foldMapM[G[_], A, B](fa: F[A])(f: A => G[B])(implicit G: Monad[G], B: Monoid[B]): G[B]
def traverse_[G[_], A, B](fa: F[A])(f: A => G[B])(implicit G: Applicative[G]): G[Unit]
def sequence_[G[_]: Applicative, A](fga: F[G[A]]): G[Unit]
def foldK[G[_], A](fga: F[G[A]])(implicit G: MonoidK[G]): G[A]

// scalaz specific
def filterLength[A](fa: F[A])(f: A => Boolean): Int
def maximum[A: Order](fa: F[A]): Option[A]
def maximumOf[A, B: Order](fa: F[A])(f: A => B): Option[B]
def minimum[A: Order](fa: F[A]): Option[A]
def minimumOf[A, B: Order](fa: F[A])(f: A => B): Option[B]
def splitWith[A](fa: F[A])(p: A => Boolean): List[NonEmptyList[A]]
def splitBy[A, B: Equal](fa: F[A])(f: A => B): IList[(B, NonEmptyList[A])]
def selectSplit[A](fa: F[A])(p: A => Boolean): List[NonEmptyList[A]]
def distinct[A](fa: F[A])(implicit A: Order[A]): IList[A]```

### Traverse

Functor with method traverse and folding functions from Foldable.

```trait Traverse[F[_]] extends Functor[F] with Foldable[F] {
def traverse[G[_]: Applicative, A, B](fa: F[A])(f: A => G[B]): G[F[B]]
}```
```def sequence[G[_]:Applicative,A](fga: F[G[A]]): G[F[A]]
def zipWithIndex[A](fa: F[A]): F[(A, Int)] // indexed
// ... other helper functions are different for scalaz and cats```

### SemigroupK (Plus)

Semigroup that abstracts over type constructor F. For any proper type A can produce Semigroup for F[A].

```trait SemigroupK[F[_]] {
def combineK[A](x: F[A], y: F[A]): F[A]  // plus
def algebra[A]: Semigroup[F[A]] //  semigroup
}```
• SemigroupK can compose

• Resources:

• Scalaz (src)
• Cats docs src
• FSiS 6 - SemigroupK, MonoidK, MonadFilter, MonadCombine - Michael Pilquist (video)

### MonoidK (PlusEmpty)

Monoid that abstract over type constructor `F`. For any proper type `A` can produce Monoid for `F[A]`.

```trait MonoidK[F[_]] extends SemigroupK[F] {
def empty[A]: F[A]
override def algebra[A]: Monoid[F[A]] // monoid
}```
• MonoidK can compose

• Resources:

### TraverseFilter

• Implementations

## Monads not compose - solutions

"Monad transformers just aren’t practical in Scala." John A De Goes

• Resources

## Natural transformation (FunctionK)

Represent mappings between two functors.

```trait NaturalTransf[F[_], G[_]] {
def apply[A](fa: F[A]): G[A]
}```

## Free constructions

abstraction free construction
Monoid List, Vector
Functor Yoneda, Coyoneda, Density, Codensity, Right Kan Extension, Left Kan Extension, Day Convolution
Applicative FreeApplicative
Alternative Free Alternative
Profunctor Profunctor CoYoneda, Profunctor Yoneda, Tambara, Pastro, Cotambara, Copastro, TambaraSum, PastroSum, CotambaraSum, CopastroSum, Closure, Environment, CofreeTraversing, FreeTraversing, Traversing
ProfunctorFunctor Profunctor CoYoneda, Profunctor Yoneda, Tambara, Pastro, Cotambara, Copastro, TambaraSum, PastroSum, CotambaraSum, CopastroSum, Closure, Environment, CofreeTraversing, FreeTraversing
ProfunctorMonad Pastro, Copastro, PastroSum, CopastroSum, Environment, FreeTraversing
ProfunctorComonad Tambara, Cotambara, TambaraSum, CotambaraSum, Closure, CofreeTraversing
Strong Tambara, Pastro, Traversing
Costrong Cotambara, Copastro
Choice TambaraSum, PastroSum
Cochoice CotambaraSum, CopastroSum, Traversing
Closed Closure, Environment
Traversing CofreeTraversing, FreeTraversing
Arrow Free Arrow

### Free Applicative

ADT (sometimes implemented using Fix point data type) that form a Monad without any other conditions:

```sealed trait Free[F[_],A]
case class Return[F[_],A](a: A) extends Free[F,A]
case class Suspend[F[_],A](s: F[Free[F,A]]) extends Free[F,A]```

### Cofree

Create comonad for any given type A. It is based on rose tree (multiple nodes, value in each node) where List is replaced with any Functor F. Functor F dedicdes how Cofree comonad is branching.

```case class Cofree[A, F[_]](extract: A, sub: F[Cofree[A, F]])(implicit functor: Functor[F]) {
def map[B](f: A => B): Cofree[B, F] = Cofree(f(extract), functor.map(sub)(_.map(f)))
def duplicate: Cofree[Cofree[A, F], F] = Cofree(this, functor.map(sub)(_.duplicate))
def extend[B](f: Cofree[A, F] => B): Cofree[B, F] = duplicate.map(f) // coKleisi composition
}```

### Representable

```// TODO Haskell extends Distrivutive, Scalaz require F to be Functor
trait Representable[F[_], Rep] {
def tabulate[X](f: Rep => X): F[X]
def index[X](fx: F[X])(f: Rep): X
}```

Adjunction[F,B] spacify relation between two Functors (There is natural transformation between composition of those two functors and identity.) We say that F is left adjoint to G.

```trait Adjunction[F[_], G[_]] {
def left[A, B](f: F[A] => B): A => G[B]
def right[A, B](f: A => G[B]): F[A] => B
}```

## (Co)Yoneda & (Co)Density & Kan Extensions

### Yoneda

Construction that abstract over type constructor and allow to effectively stack computations.

In Category Theory

Yoneda Lemma states that: `[C,Set](C(a,-),F) ~ Fa` Set of natural transformations from `C` to `Set` of the Hom functor `C(a,-)` to Functor `F: C -> Set` is isomorphic to `Fa`

It is possible to formulate Yoneda Lemma in terms of Ends, and we get Ninja Yoneda Lemma: ∫ `Set(C(a,x),F(x)) ~ Fa`

That corresponds to:

`def yoneda[R](cax: A => X, fx F[X]) ~ F[A]`

```trait Yoneda[F[_], A] {
def run[R](f: A => R): F[R]
}```
• we need Functor instance for F to create instance of Yoned for F
```def liftYoneda[F[_], A](fa: F[A])(implicit FunctorF: Functor[F]): Yoneda[F, A] =
new Yoneda[F, A] {
def run[R2](f: A => R2): F[R2] = FunctorF.map(fa)(f)
}```
• we don't need the fact that F is a Functor to go back to F
`def lowerYoneda[F[_], A](y: Yoneda[F, A]): F[A] = y.run(identity[A])`
• we can define Functor instance without any requirement on F:
```def yonedaFunctor[F[_]]: Functor[Yoneda[F, ?]] =
new Functor[Yoneda[F, ?]] {
def map[A, B](fa: Yoneda[F, A])(f: A => B): Yoneda[F, B] =
new Yoneda[F, B] {
def run[C](f2: B => C): F[C] = fa.run(f andThen f2)
}
}```

### Coyoneda

Rúnar in Free Monads and the Yoneda Lemma describe this type as a proof that: "if we have a type B, a function of type (B => A) for some type A, and a value of type F[B] for some functor F, then we certainly have a value of type F[A]"

This result from Category Theory allow us to perform `Coyoneda Trick`:

If we have following type:

```trait Coyoneda[F[_], A] {
type B
def f: B => A
def fb: F[B]
}```

then type constructor F can be lifted to Coyoneda

`def liftCoyoneda[F[_], A](fa: F[A]): Coyoneda[F, A]`

we can map over lifted constructor F without any requirements on F. So Coyoneda is a Free Functor:

```implicit def coyoFunctor[F[_]]: Functor[Coyoneda[F, ?]] = new Functor[Coyoneda[F, ?]] {
def map[A, AA](fa: Coyoneda[F, A])(ff: A => AA): Coyoneda[F, AA] = new Coyoneda[F, AA] {
type B = fa.B
def f: B => AA = fa.f andThen ff
def fb: F[B] = fa.fb
}
}```

We even can change the oryginal type of F

```def hoistCoyoneda[F[_], G[_], A, C](fab : NaturalTransf[F,G])(coyo: Coyoneda[F, A]): Coyoneda[G, A] =
new Coyoneda[G, A] {
type B = coyo.B
def f: B => A = coyo.f
def fb: G[B] = fab(coyo.fb)
}```

Finally to get back from Coyoneda fantazy land to reality of F, we need a proof that it is a Functor:

`def lowerCoyoneda(implicit fun: Functor[F]): F[A]`

### Right Kan extension

```trait Ran[G[_], H[_], A] {
def runRan[B](f: A => G[B]): H[B]
}```
• We can create functor for Ran without any requirements on G, H
```def ranFunctor[G[_], H[_]]: Functor[Ran[G, H, ?]] =
new Functor[Ran[G, H, ?]] {

def map[A, B](fa: Ran[G, H, A])(f: A => B): Ran[G, H, B] =
new Ran[G, H, B] {
def runRan[C](f2: B => G[C]): H[C] =
fa.runRan(f andThen f2)
}
}```
• We can define Monad for Ran without any requirements on G, H. Monad generated by Ran is Codensity.
```def codensityMonad[F[_], A](ran: Ran[F, F, A]): Codensity[F, A] =
new Codensity[F, A] {
def run[B](f: A => F[B]): F[B] = ran.runRan(f)
}```

### Left Kan Extension

```trait Lan[F[_], H[_], A] {
type B
val hb: H[B]
def f: F[B] => A
}```
• we can define Functor for it
```def lanFunctor[F[_], H[_]]: Functor[Lan[F, H, ?]] = new Functor[Lan[F, H, ?]]() {
def map[A, X](x: Lan[F, H, A])(fax: A => X): Lan[F, H, X] = {
new Lan[F, H, X] {
type B = x.B
val hb: H[B] = x.hb
def f: F[B] => X = x.f andThen fax
}
}
}```

Density is a Comonad that is simpler that Left Kan Extension. More precisely it is comonad formed by left Kan extension of a Functor along itself.)

```trait Density[F[_], Y] { self =>
type X
val fb: F[X]
def f: F[X] => Y

def densityToLan: Lan[F,F,Y] = new Lan[F,F,Y] {
type B = X
val hb: F[B] = fb
def f: F[B] => Y = self.f
}
}

object Density {
def apply[F[_], A, B](kba: F[B] => A, kb: F[B]): Density[F, A] = new Density[F, A] {
type X = B
val fb: F[X] = kb
def f: F[X] => A = kba
}
}```

Density form a Functor without any conditions of F so it is a Free Functor. Similar like Lan.

```def functorInstance[K[_]]: Functor[Density[K, ?]] = new Functor[Density[K, ?]] {
def map[A, B](x: Density[K, A])(fab: A => B): Density[K, B] = Density[K,B,x.X](x.f andThen fab, x.fb)
}```

Density is a Comonad without any conditions of F so it is a Free Comonad.

```def comonadInstance[K[_]]: Comonad[Density[K, ?]] = new Comonad[Density[K, ?]] {
def extract[A](w: Density[K, A]): A = w.f(w.fb)
def duplicate[A](wa: Density[K, A]): Density[K, Density[K, A]] =
Density[K, Density[K, A], wa.X](kx => Density[K, A, wa.X](wa.f, kx), wa.fb)
def map[A, B](x: Density[K, A])(f: A => B): Density[K, B] = x.map(f)
}```

### Codensity

Interface with flatMap'ish method:

```trait Codensity[F[_], A] {
def run[B](f: A => F[B]): F[B]
}```

that gives us monad (without any requirement on F):

```implicit def codensityMonad[G[_]]: Monad[Codensity[G, ?]] =
def map[A, B](fa: Codensity[G, A])(f: A => B): Codensity[G, B] =
new Codensity[G, B] {
def run[C](f2: B => G[C]): G[C] = fa.run(f andThen f2)
}

def unit[A](a: A): Codensity[G, A] =
new Codensity[G, A] {
def run[B](f: A => G[B]): G[B] = f(a)
}

def flatMap[A, B](c: Codensity[G, A])(f: A => Codensity[G, B]): Codensity[G, B] =
new Codensity[G, B] {
def run[C](f2: B => G[C]): G[C] = c.run(a => f(a).run(f2))
}
}```

### Functor Functor (FFunctor)

Functor that works on natural transformations rather than on regular types

```trait FFunctor[FF[_]] {
def ffmap[F[_],G[_]](nat: NaturalTransf[F,G]): FF[F] => FF[G]
}```
• Laws:

• identity: `ffmap id == id`
• composition: `ffmap (eta . phi) = ffmap eta . ffmap phi`
• Resources

### Monoidal Categories, Monoid Object

In Category Theory a Monoidal Category is a Category with a Bifuctor and morphisms that satisfy some laws (see gist for details).

```trait MonoidalCategory[M[_, _], I] {
val tensor: Bifunctor[M]
val mcId: I

def rho[A]    (mai: M[A,I]): A
def rho_inv[A](a:   A):      M[A, I]

def lambda[A]      (mia: M[I,A]): A
def lambda_inv[A,B](a: A):        M[I, A]

def alpha[A,B,C](    mabc: M[M[A,B], C]): M[A, M[B,C]]
def alpha_inv[A,B,C](mabc: M[A, M[B,C]]): M[M[A,B], C]
}```

We can create monoidal category where product (Tuple) is a bifunctor or an coproduct (Either).

Monoidal Categories are usefull if we consider category of endofunctors. If we develop concept of Monoid Object then it is possible to define Monads as Monoid Object in Monoidal Category of Endofunctors with Product as Bifunctor Applicative as Monoid Object in Monoidal Category of Endofunctors with Day convolution as Bifunctor

In category of Profunctors with Profunctor Product as Bifunctor the Monoid Ojbect is Arrow.

### Day Convolution

Monads are monoids in a monoidal category of endofunctors. Applicative functors are also monoids in a monoidal category of endofunctors but as a tensor is used Day convolution.

There is nice intuition for Day convolution as generalization of one of Applicative Functor methods.

```data Day f g a where
Day :: forall x y. (x -> y -> a) -> f x -> g y -> Day f g a```
• Scala
```trait DayConvolution[F[_], G[_], A] {
type X
type Y
val fx: F[X]
val gy: G[Y]
def xya: (X, Y) => A
}```
• There is various ways to create Day Convolution:
```def day[F[_], G[_], A, B](fab: F[A => B], ga: G[A]): Day[F, G, B]
def intro1[F[_], A](fa: F[A]): Day[Id, F, A]
def intro2[F[_], A](fa: F[A]): Day[F, Id, A]```
• Day convolution can be transformed by mapping over last argument, applying natural transformation to one of type constructors, or swapping them
```def map[B](f: A => B): Day[F, G, B]
def trans1[H[_]](nat: NaturalTransf[F, H]): Day[H, G, A]
def trans2[H[_]](nat: NaturalTransf[G, H]): Day[F, H, A]
def swapped: Day[G, F, A] = new Day[G, F, A]```
• There is various ways to collapse Day convolution into value in type constructor:
```def elim1[F[_], A](d: Day[Id, F, A])(implicit FunF: Functor[F]): F[A]
def elim2[F[_], A](d: Day[F, Id, A])(implicit FunF: Functor[F]): F[A]
def dap[F[_], A](d: Day[F, F, A])(implicit AF: Applicative[F]): F[A]```
• We can define Functor instance without any conditions on type constructors (so it forms Functor for free like Coyoneda):
```def functorDay[F[_], G[_]]: Functor[DayConvolution[F, G, ?]] = new Functor[DayConvolution[F, G, ?]] {
def map[C, D](d: DayConvolution[F, G, C])(f: C => D): DayConvolution[F, G, D] =
new DayConvolution[F, G, D] {
type X = d.X
type Y = d.Y
val fx: F[X] = d.fx
val gy: G[Y] = d.gy

def xya: X => Y => D = x => y => f(d.xya(x)(y))
}
}```

### Profunctor

Profunctor abstract over

• type constructor with two holes `P[_,_]`
• operation `def dimap(preA: NewA => A, postB: B => NewB): P[A, B] => P[NewA, NewB]` that given `P[A,B]` and two functions
• apply first `preA` before first type of `P` (ast as contravariant functor)
• apply second `postB` after second type of `P` (act as functor)

Alternatively we can define Profunctor not using dimap but using two separate functions:

• def lmap(f: AA => A): P[A,C] => P[AA,C] = dimap(f,identity[C])
• def rmap(f: B => BB): P[A,B] => P[A,BB] = dimap(identity[A], f)

Profunctors in Haskell were explored by sifpe at blog A Neighborhood of Infinity in post Profunctors in Haskell Implemented in Haskell: ekmett/profunctors

```trait Profunctor[F[_, _]] {
def dimap[A, B, C, D](fab: F[A, B])(f: C => A)(g: B => D): F[C, D]
}```
• Alternatively we can define functor using:
```def lmap[A, B, C](fab: F[A, B])(f: C => A): F[C, B]
def rmap[A, B, C](fab: F[A, B])(f: B => C): F[A, C]```
• Most popular is instance for Function with 1 argument:
```trait Profunctor[Function1] {
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
}```

Becasue Profunctors can be used as base to define Arrows therefore there are instances for Arrow like constructions like `Kleisli`

• In Category Theroy: When we have Category `C` and `D` and `D'` the opposite category to D, then a Profunctor `P` is a Functor `D' x C -> Set` We write `D -> C` In category of types and functions we use only one category, so Profunctor P is `C' x C => C`

• Laws:

• if we define Profunctor using dimap:

• `dimap id id == id`
• `dimap (f . g) (h . i) == dimap g h . dimap f i` Second law we get for free by parametricity.
• if specify lmap or rmap

• `lmap id == id`
• `rmap id == id`
• `lmap (f . g) == lmap g . lmap f`
• `rmap (f . g) == rmap f . rmap g`

• if specify both (in addition to law for dimap and laws for lmap:
• `dimap f g == lmap f . rmap g`

### Star

Lift Functor into Profunctor "forward"

`case class Star[F[_],D,C](runStar: D => F[C])`

If `F` is a Functor then `Star[F, ?, ?]` is a Profunctor:

```def profunctor[F[_]](implicit FF: Functor[F]): Profunctor[Star[F, ?,?]] = new Profunctor[Star[F, ?, ?]] {
def dimap[X, Y, Z, W](ab: X => Y, cd: Z => W): Star[F, Y, Z] => Star[F, X, W] = bfc =>
Star[F,X, W]{ x =>
val f: Y => F[Z] = bfc.runStar
val fz: F[Z] = f(ab(x))
FF.map(fz)(cd)
}
}```

### CoStar

Lift Functor into Profunctor "backwards"

`case class Costar[F[_],D,C](runCostar: F[D] => C)`

If `F` is a Functor then `Costar[F, ?, ?]` is a Profunctor

```def profunctor[F[_]](FF: Functor[F]): Profunctor[Costar[F, ?, ?]] = new Profunctor[Costar[F, ?, ?]] {
def dimap[A, B, C, D](ab: A => B, cd: C => D): Costar[F, B, C] => Costar[F, A, D] = fbc =>
Costar{ fa =>
val v: F[B] = FF.map(fa)(ab)
val c: C = fbc.runCostar(v)
cd(c)
}
}```

### Strong Profunctor

Profunctor with additional method `first` that lift profunctor so it can run on first element of tuple.

For Profunctor of functions from A to B this operation just apply function to first element of tuple.

```trait StrongProfunctor[P[_, _]] extends Profunctor[P] {
def first[X,Y,Z](pab: P[X, Y]): P[(X, Z), (Y, Z)]
}```
• Laws in Haskell implementation of Strong Profunctor
1. `first == dimap(swap, swap) andThen second`
2. `lmap(_.1) == rmap(_.1) andThen first`
3. `lmap(second f) andThen first == rmap(second f) andThen first`
4. `first . first ≡ dimap assoc unassoc . first`
5. `second ≡ dimap swap swap . first`
6. `lmap snd ≡ rmap snd . second`
7. `lmap (first f) . second ≡ rmap (first f) . second`
8. `second . second ≡ dimap unassoc assoc . second`

where

```assoc ((a,b),c) = (a,(b,c))
unassoc (a,(b,c)) = ((a,b),c)```

In Notions of Computation as Monoids by Exequiel Rivas and Mauro Jaskelioff in 7.1 there are following laws:

1. `dimap identity pi (first a) = dimap pi id a`
2. `first (first a) = dimap alphaInv alpha (first a)`
3. `dimap (id × f) id (first a) = dimap id (id × f) (first a)`
• Derived methods:
```def second[X,Y,Z](pab: P[X, Y]): P[(Z, X), (Z, Y)]
def uncurryStrong[P[_,_],A,B,C](pa: P[A, B => C])(S: Strong[P]): P[(A,B),C]```

In Purescript implementation of Strong there are some more helper methods that use Category constraint for P.

• Most common instance is Function with one argument:
```val Function1Strong = new Strong[Function1] with Function1Profunctor {
def first[X, Y, Z](f: Function1[X, Y]): Function1[(X,Z), (Y, Z)] = { case (x,z) => (f(x), z) }
}```

it is possible to define instance for Kleisli arrow

### Tambara

```trait Tambara[P[_,_],A,B]{
def runTambara[C]: P[(A,C),(B,C)]
}```

Tambara is a Profunctor:

```trait Profunctor[Tambara[P, ?, ?]] {
def PP: Profunctor[P]

def dimap[X, Y, Z, W](f: X => Y, g: Z => W): Tambara[P, Y, Z] => Tambara[P, X, W] = (tp : Tambara[P, Y, Z]) => new Tambara[P, X, W]{

def runTambara[C]: P[(X, C), (W, C)] = {
val fp: P[(Y,C),(Z,C)] => P[(X, C), (W, C)] = PP.dimap(
Function1Strong.first[X, Y, C](f),
Function1Strong.first[Z, W, C](g)
)
val p: P[(Y,C),(Z,C)] = tp.runTambara[C]
fp(p)
}
}
}```

It is also FunctorProfunctor:

```def promap[P[_, _], Q[_, _]](f: DinaturalTransformation[P, Q])(implicit PP: Profunctor[P]): DinaturalTransformation[Lambda[(A,B) => Tambara[P, A, B]], Lambda[(A,B) => Tambara[Q, A, B]]] = {
new DinaturalTransformation[Lambda[(A,B) => Tambara[P, A, B]], Lambda[(A,B) => Tambara[Q, A, B]]] {
def dinat[X, Y](ppp: Tambara[P, X, Y]): Tambara[Q, X, Y] = new Tambara[Q, X, Y] {
def runTambara[C]: Q[(X, C), (Y, C)] = {
val p: P[(X,C), (Y,C)] = ppp.runTambara
f.dinat[(X,C), (Y,C)](ppp.runTambara)
}
}
}
}```

# Profunctor Costrong

```trait Costrong[F[_,_]] extends Profunctor[F] {
def unfirst[A,B,D](fa: F[(A,D), (B, D)]): F[A,B]
def unsecond[A,B,D](fa: F[(D,A),(D,B)]): F[A,B]
}```

### Choice Profunctor

Profunctor with additional method left that wrap both types inside Either.

```trait ProChoice[P[_, _]] extends Profunctor[P] {
def left[A,B,C](pab: P[A, B]):  P[Either[A, C], Either[B, C]]
}```
• derived method
`def right[A,B,C](pab: P[A, B]): P[Either[C, A], Either[C, B]]`
• Resources

### Extranatural Transformation

```trait ExtranaturalTransformation[P[_,_],Q[_,_]]{
def exnat[A,B](p: P[A,B]): Q[A,B]
}```

### Profunctor Functor

Functor (endofunctor) between two Profunctors.

It is different than regualar Functor: Functor lifts regular function to function working on type constructor: def map[A, B](f: A => B): F[A] => F[B] Profunctor lifts two regular functions to work on type constructor with two holed.

And ProfunctorFunctor lifts dinatural transformation of two Profunctors P[,] => Q[,]

operates on type constructor with one hole (F[A] => F[B]) and ProfunctorFunctor and ProfunctorFunctor map P[A,B] => Q[A,B]

in Scala 2.12 we cannot express type constructor that have hole with shape that is not sepcified)

```trait ProfunctorFunctor[T[_]] {
def promap[P[_,_], Q[_,_]](dt: DinaturalTransformation[P,Q])(implicit PP: Profunctor[P]): DinaturalTransformation[Lambda[(A,B) => T[P[A,B]]], Lambda[(A,B) => T[Q[A,B]]]]
}```

```trait ProfunctorMonad[T[_]] extends ProfunctorFunctor[T] {
def proreturn[P[_,_]](implicit P: Profunctor[P]): DinaturalTransformation[P, Lambda[(A,B) => T[P[A,B]]]]
def projoin[P[_,_]](implicit P: Profunctor[P]): DinaturalTransformation[Lambda[(A,B) => T[T[P[A,B]]]], Lambda[(A,B) => T[P[A,B]]]]
}```
• Laws:
• `promap f . proreturn == proreturn . f`
• `projoin . proreturn == id`
• `projoin . promap proreturn == id`
• `projoin . projoin == projoin . promap projoin`

```trait ProfunctorComonad[T[_]] extends ProfunctorFunctor[T] {
def proextract[P[_,_]](implicit P: Profunctor[P]): DinaturalTransformation[Lambda[(A,B) => T[P[A,B]]], P]
def produplicate[P[_,_]](implicit P: Profunctor[P]): DinaturalTransformation[Lambda[(A,B) => T[P[A,B]]], Lambda[(A,B) => T[T[P[A,B]]]]]
}```
• Laws
• `proextract . promap f == f . proextract`
• `proextract . produplicate == id`
• `promap proextract . produplicate == id`
• `produplicate . produplicate == promap produplicate . produplicate`

### Profunctor Yoneda

```trait ProfunctorYoneda[P[_,_],A,B] {
def runYoneda[X,Y](f: X => A, g: B => Y): P[X,Y]
}```

```def dimap[AA, BB](l: AA => A, r: B => BB): ProfunctorYoneda[P, AA, BB] = new ProfunctorYoneda[P, AA, BB] {
def runYoneda[X, Y](l2: X => AA, r2: BB => Y): P[X, Y] = {
val f1: X => A = l compose l2
val f2: B => Y = r2 compose r
self.runYoneda(f1, f2)
}
}```

### Profunctor CoYoneda

```trait ProfunctorCoyoneda[P[_,_],A,B] {
type X
type Y
def f1: A => X
def f2: Y => B
def pxy: P[X,Y]
}```

helper constructor:

```def apply[XX,YY,P[_,_],A,B](ax: A => XX, yb: YY => B, p: P[XX,YY]): ProfunctorCoyoneda[P,A,B] = new ProfunctorCoyoneda[P,A,B] {
type X = XX
type Y = YY
def f1: A => X = ax
def f2: Y => B = yb
def pxy: P[X,Y] = p
}```

```def dimap[C, W](l: C => A, r: B => W): ProfunctorCoyoneda[P, C, W] =
ProfunctorCoyoneda[X, Y, P, C, W](f1 compose l, r compose f2, pxy)```

### Procompose

In general Profunctors should have straightforward way to compose them as we have the same category in definition. But to be faithfull with Category Theory definition, Profunctor Composition is defined using exitential types:

```trait Procompose[P[_,_],Q[_,_],D,C] {
type X
val p: P[X,C]
val q: Q[D,X]
}```

## Arrows

### Category

Abstraction for operations that can be composed and that provide no-op (id).

```trait Compose[F[_, _]] {
def compose[A, B, C](f: F[B, C], g: F[A, B]): F[A, C] // alias <<<
}

trait Category[F[_, _]] extends Compose[F] {
def id[A]: F[A, A]
}```

• Resources

• Resources

• Resources

• Resources

• Cats

• Resources:

### Dinatural Transformation

Dinatural Transformation is a function that change one Profunctor P into another one Q without modifying the content. It is equivalent to Natural Transformation between two Functors (but for Profunctors).

```trait DinaturalTransformation[P[_,_],Q[_,_]]{
def dinat[A](p: P[A,A]): Q[A,A]
}```
• Laws:

• `rmap f . dinat . lmap f == lmap f . dinat . rmap f`
• Resources

### Ends & Coends

Ends can be seen as infinite product. End corresponds to forall so polymorphic function:

```// P is Profunctor

trait End[P[_,_]] {
def run[A]: P[A,A]
}```

Coend can be seen as infinite coproduct (sum). Coends corresponds to exists

`data Coend p = forall x. Coend p x x`

### Align

• Resources
• Simple Algebraic Data Types - Bartosz Milewski blog post
• Category Theory 5.2: Algebraic data types - Bartosz Milewski video
• Counting type inhabitants - Alexander Konovalov blog post

### Unit

Type that has only one element

### Void

Type that has no elements. In Category Theory - Initial Object

## Sum (Coproduct)

Type represents either one or another element. In set theory: disjoint union in Category theory: coproduct (sum).

## Product

Type represents combination of two types. In Set theory cartesian product, in Category Theory product.

### These

Type that represents both sum and product (Non exclusive two values):

Tuple(a,b) => a * b Eiter(a,b) => a + b These(a,b) => (a + b) + a*b

```sealed trait These[A,B]
case class This[A, B](a: A) extends These[A,B]
case class That[A,B](b: B) extends These[A,B]
case class Those[A,B](a: A, b: B) extends These[A,B]```
• There is many abstractions that can be implemented for this data type

Resources:

Resources:

## Cayley representations

"The Cayley representation theorem (CRT): every group is isomorphic to a group of permutations" Can be extended to monoids and defined monoidal category of endofunctor

In FP the CRT is optimisation by change of representation:

• CRT for List monoid - difference lists
• CRT for applicatives - NoCaM 5.4
• CRT for arrows - ???

Resources:

• Notions of Computation as Monoids (extended version) - Exequiel Rivas, Mauro Jaskelioff (paper)

### Difference lists

• List with concatenation and empty list is monoid. W optimize list concatenation (that is slow) by representing list as function (difference list):
`type Elist[A] = List[A] => List[A]`

EList is isomorphic to List:

```def rep[A](xs: List[A]): Elist[A] = ys => xs ++ ys
def abs[A](xs: Elist[A]): List[A] = xs(Nil)```

We can concatenate EList's effectively and at the end get to the List back.

• Cayley Theorem can be define for general Monoid:
```trait CayleyTheoremForMonoid[M[_]] extends MonoidK[M] {
type MonoidEndomorphism[A] = M[A] => M[A]
def rep[A](xs: M[A]): MonoidEndomorphism[A] = ys => combineK(xs, ys)
def abs[A](xs: MonoidEndomorphism[A]): M[A] = xs(empty)
}```

Resources:

• (Haskell) Using Difference Lists - geophf blog post
• (Haskell) keepEquals with Difference Lists - geophf blog post
• (Haskell) A novel representation of lists and its application to the function "reverse" - John Hughes

### Double Cayley Representation

"optimises both left-nested sums and left-nested products"

Resources:

• A Unified View of Monadic and Applicative Non-determinism - Exequiel Rivas, Mauro Jaskeliof, Tom Schrijvers (paper)