Limits.MD

ADT (Algebra of types)

• Resources
  • Haskell wiki ADT
  • 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

• Implementations scala.Unit purescript-prelude/Data.Unit, UniMath, nLab

• Resources

  • Category Theory 4.1: Terminal and initial objects video Scala translation
  • (Category Theory) Limits and colimits - TheCatsters video playlist

Void

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

• Implementations scala.Nothing purescript-prelude/Data.Void Idris prelude/Prelude/Uninhabited, UniMath, nLab

• Resources

  • Category Theory 4.1: Terminal and initial objects video Scala translation
  • (Category Theory) Limits and colimits - TheCatsters video playlist

Sum (Coproduct)

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

• Implementations scala.util.Either purescript-either/Data.Either, UniMath, nLab

Product

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

• Implementations scala.Product2 scala.Tuple2 purescript-tuples/Data.Tuple, UniMath, nLab

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

Implementations Scalaz Haskell

Limits

Cone

• Resources
  • (Haskell) category-extras Cone
  • (Haskell) data-category Cone

Cocone

• Resources
  • (Haskell) category-extras Cocone
  • (Haskell) data-category Cocone

Diagonal Functor

• Resources
  • (Haskell) category-extras Diagonal Functor

Limit

• Resources
  • Category Theory II 1.2: Limits - Bartosz Milewski (video)
  • Category Theory II 2.1: Limits, Higher order functors - Bartosz Milewski (video)
  • Category Theory II 2.2: Limits, Naturality - Bartosz Milewski (video)
  • Category Theory II 3.1: Examples of Limits and Colimits (video)
  • Limits and Colimits - Bartosz Milewski (blog post)
  • Understanding Limits - Bartosz Milewski (blog post)
  • TheCatsters - Limits and colimits (video playlist)
  • (Haskell) category-extras Limit
  • (Haskell) data-category Limit
  • Boolean Limits statebox/idris-ct

Colimit

• Resources
  • (Haskell) category-extras Colimit
  • (Haskell) data-category Colimit
  • Boolean CoLimits statebox/idris-ct