Adjunction.MD

Representable & Adjunctions

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
}

• Implementations: Scalaz 7, Cats, Haskell, Agda, UniMath, nLab

• Resources:

  • (Haskell) Representing Applicatives - Gershom Bazerman (blog post)
  • (Category Theory, Haskell) Representable Functors - Bartosz Milewski (blog post)
  • (Category Theory, Haskell) Category Theory II 4.1: Representable Functors - Bartosz Milewski (video) Scala code translation
  • (Haskell) Zippers Using Representable And Cofree - Chris Penner (blog post):
  • Reasoning with representable functors - Adelbert Chang (blog post)
  • (Haskell) Radix Sort, Trie Trees, And Maps From Representable Functors - Chris Penner (blog post)
  • (Haskell) Monad.Representable.Reader, Monad.Representable.State, Comonad.Representable.Store
  • Moore for Less - Edward Kmett (blog post)
  • Representable Functors - Danny Gratzer (blog post)
  • Which Haskell Functors are equivalent to the Reader functor - pigworker SO
  • Representable Functor isomorphic to (Bool -> a) SO
  • masaeedu/representable (different formulation of Representable functors that carries the domain around)
  • usage of Representable in old hanshoglund/music-suite
  • Java Mojang/DataFixerUpper Representable

Corepresentable

• Implementations: Haskell

Adjunction

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
}

• Implementations Scalaz 7, Haskell, Purescript, UniMath, nLab

Adjunction can be defined between Reader monad and Coreader comonad.

• Resources:
  • Adjunctions And Battleship - Chris Penner (blog post)
  • Scala Comonad Tutorial, Part 2 - Rúnar Bjarnason (blog post)
  • Adjunctions in Everyday Life - Rúnar Bjarnason (video Scala) (video Haskell)
  • Scalaz docs
  • Haskell libraries using Adjunctions
  • usage in ekmett/representable-tries
  • (Haskell) Representing Adjunctions - Edward Kmett (blog post)
  • (Haskell) Zapping Adjunctions - Edward Kmett (blog post)
  • Adjunctions - TheCatsters (vide playlist)
  • State monad using Adjunctions kaifransson/adjoint-stacks
  • (Haskell) Free And Forgetful Functors - Chris Penner (blog post)
  • Adjunctions - M.M. Fokkinga, Lambert Meertens
  • Generic Programming with Adjunctions - Ralf Hinze
  • Relational Algebra by Way of Adjunctions - Jeremy Gibbons, Fritz, Henglein, Ralf Hinze, Nicolas Wu
  • (Haskell) Monad.Trans.Adjoint, Comonad.Trans.Adjoint, Monad.Trans.Contravariant.Adjoint
  • (Haskell) SO - Monads as adjunctions
  • (Haskell) SO - Which common monads arise from adjunctions in Hask?
  • (Haskell) SO - How to Factorize Continuation Monad into Left & Right Adjoints?
  • (Haskell) SO - What are the adjoint functor pairs corresponding to common monads in Haskell?
  • (Unison) Adjunction used as template of types
  • Adjunction for Garbage Collection with Application to Graph Rewriting - D. Duval, R. Echahed, F. Prost

Adjoint Triples

• Resources:
  • (Haskell) Adjoint Triples - Dan Doel (blog post)
  • (Haskell) adjunctions/Control.Comonad.Trans.Adjoint
  • Adjoint functors and triples - Samuel Eilenberg and John C. Moore
  • Fancy Algebra (Graduate Topics Course) - Drew Armstrong
  • nLab adjoint triple