scalaz

Category

object Category extends AnyRef

Source
Category.scala
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Type Members

  1. trait <*> [F[_], G[_]] extends AnyRef

  2. case class <= [A, B] (value: (B) ⇒ A) extends NewType[(B) ⇒ A] with Product with Serializable

    The flipped Function1 type

  3. type <=> [A, B] = Iso[Function1, A, B]

    Set isomorphism

  4. type <~> [F[_], G[_]] = Iso2[~>, F, G]

    Natural isomorphism between functors

  5. type <~~> [F[_, _], G[_, _]] = Iso3[~~>, F, G]

    Isomorphism natural in both sides of a bifunctor

  6. type Adjunction [F[_], U[_]] = Iso3[~~>, [A, B](F[A]) ⇒ B, [A, B](A) ⇒ U[B]]

  7. type Alpha [Arr[_, _], X, Y] = ~>[[α]Arr[α, X], [α]Arr[α, Y]]

  8. case class Compose [F[_], G[_], Arr[_, _], X] (value: F[G[X]]) extends NewType[F[G[X]]] with Product with Serializable

    Functor composition

  9. case class Discrete [X, A, B] (value: (X) ⇒ X) extends NewType[(X) ⇒ X] with Product with Serializable

  10. type GeneralAdjunction [P[_, _], Q[_, _], F[_], U[_]] = Iso3[~~>, [A, B]P[F[A], B], [A, B]Q[A, U[B]]]

  11. trait GeneralizedContravariant [C[_, _], D[_, _], F[_]] extends AnyRef

  12. trait GeneralizedFunctor [C[_, _], D[_, _], F[_]] extends AnyRef

    A Functor that is not necessarily an endofunctor in the Scala category.

  13. case class Iso [Arr[_, _], A, B] (to: Arr[A, B], from: Arr[B, A]) extends Product with Serializable

    Isomorphism for arrows of kind * -> * -> *

  14. case class Iso2 [Arr[_[_], _[_]], F[_], G[_]] (to: Arr[F, G], from: Arr[G, F]) extends Product with Serializable

    Isomorphism for arrows of kind (* -> *) -> (* -> *) -> *

  15. case class Iso3 [Arr[_[_, _], _[_, _]], F[_, _], G[_, _]] (to: Arr[F, G], from: Arr[G, F]) extends Product with Serializable

    Isomorphism for arrows of kind (* -> * -> *) -> (* -> * -> *) -> *

  16. class MonoidCategory [M] extends GeneralizedCategory with Hom

    Attributes
    sealed

  17. trait Nat [Arr[_, _], F[_], G[_]] extends AnyRef

    Generalized natural transformations

  18. class Ord2 [X, A, B] extends AnyRef

    Attributes
    sealed

  19. trait P [+IX, +IY] extends AnyRef

    Index for a product category

  20. case class ProductCategory [UX <: Hom, UY <: Hom] (_1: GeneralizedCategory { type U = UX }, _2: GeneralizedCategory { type U = UY }) extends GeneralizedCategory with Hom with Product with Serializable

  21. trait Reader [R] extends AnyRef

  22. trait Writer [R] extends AnyRef

Value Members

  1. def != (arg0: AnyRef): Boolean

    Attributes
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  2. def != (arg0: Any): Boolean

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  3. def ## (): Int

    Attributes
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    AnyRef → Any

  4. implicit val <:<_Category : Category[<:<]

    Attributes
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  5. implicit val =:=_Category : Category[=:=]

    Attributes
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  6. def == (arg0: AnyRef): Boolean

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  7. def == (arg0: Any): Boolean

    Attributes
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  8. implicit def CokleisliCategory [M[_]] (implicit arg0: Comonad[M]): Category[[α, β]Cokleisli[M, α, β]]

    Attributes
    implicit

  9. implicit def ComposeContravariants [F[_], G[_]] (implicit arg0: Contravariant[F], arg1: Contravariant[G]): Functor[[X]Compose[F, G, <=, X]]

    Compose contravariants

    Compose contravariants

    Attributes
    implicit

  10. implicit def ComposeFunctors [F[_], G[_]] (implicit arg0: Functor[F], arg1: Functor[G]): Functor[[X]Compose[F, G, Function1, X]]

    Compose functors

    Compose functors

    Attributes
    implicit

  11. implicit def DiscreteCategory [X] : Category[[α, β]Discrete[X, α, β]]

    Discrete categories, whose only morphism is the identity function.

    Discrete categories, whose only morphism is the identity function.

    Attributes
    implicit

  12. implicit val Function1Category : Category[Function1]

    The Set category

    The Set category

    Attributes
    implicit

  13. implicit def KleisliCategory [M[_]] (implicit arg0: Monad[M]): Category[[α, β]Kleisli[M, α, β]]

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  14. implicit def MorphismToObject [A, B, C] (a: Const2[A, B, C]): A

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  15. implicit def ObjectToMorphism [A, B, C] (a: A): Const2[A, Unit, Unit]

    Attributes
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  16. implicit val OpCategory : Category[<=]

    The opposite category of the Set category.

    The opposite category of the Set category.

    Attributes
    implicit

  17. implicit def PartialFunctionCategory : Category[PartialFunction]

    Attributes
    implicit

  18. implicit def PosetCategory [X] (implicit arg0: Order[X]): Category[[α, β]Ord2[X, α, β]]

    Every partial order gives rise to a category

    Every partial order gives rise to a category

    Attributes
    implicit

  19. def asInstanceOf [T0] : T0

    Attributes
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  20. def clone (): AnyRef

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    protected[lang]
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    @throws()

  21. def contravariantInScala [F[_]] (f: Contravariant[F]): GeneralizedFunctor[<=, Function1, F]

  22. def endoFunctorInScala [F[_]] (f: Functor[F]): GeneralizedFunctor[Function1, Function1, F]

  23. def eq (arg0: AnyRef): Boolean

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  24. def equals (arg0: Any): Boolean

    Definition Classes
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  25. def finalize (): Unit

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    @throws()

  26. implicit def flipFunctorIso [F[_], G[_]] (implicit i: $less$tilde$greater[F, G]): $less$tilde$greater[G, F]

    Natural isomorphism is commutative

    Natural isomorphism is commutative

    Attributes
    implicit

  27. implicit def flipIso [A, B] (implicit i: $less$eq$greater[A, B]): $less$eq$greater[B, A]

    Set isomorphism is commutative

    Set isomorphism is commutative

    Attributes
    implicit

  28. def getClass (): java.lang.Class[_]

    Attributes
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  29. def hashCode (): Int

    Definition Classes
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  30. def isInstanceOf [T0] : Boolean

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  31. implicit def monoidCategory [M] (implicit arg0: Monoid[M]): MonoidCategory[M]

    Attributes
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  32. def ne (arg0: AnyRef): Boolean

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  33. implicit def newTypeIso [A, B <: NewType[A]] (implicit c: (A) ⇒ B): $less$eq$greater[A, B]

    Every NewType is isomorphic to its underlying type.

    Every NewType is isomorphic to its underlying type. If its constructor is made implicit, we get an implicit isomorphism.

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  34. def notify (): Unit

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  35. def notifyAll (): Unit

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  36. implicit def opContravariant [R] : Contravariant[[α]<=[R, α]]

    Attributes
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  37. implicit def productCategory [UX <: Hom, UY <: Hom] (implicit x: GeneralizedCategory { type U = UX }, y: GeneralizedCategory { type U = UY }): ProductCategory[UX, UY]

    Attributes
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  38. implicit def reflFunctorIso [F[_]] : $less$tilde$greater[F, F]

    Natural isomorphism is reflexive

    Natural isomorphism is reflexive

    Attributes
    implicit

  39. def reflectIso [A1[_, _], A2[_, _], F[_], A, B] (implicit c1: Category[A1], c2: Category[A2], f: GeneralizedFunctor[A2, A1, F]): ($less$tilde$tilde$greater[[A, B]A1[F[A], F[B]], A2]) ⇒ (Iso[A1, F[A], F[B]]) ⇒ Iso[A2, A, B]

    Fully faithful functors reflect isomorphisms

  40. def stateAdjunction [S] : Adjunction[[S](S, S), [S](S) ⇒ S]

    The adjunction induced by curry and uncurry being isomorphic

  41. def synchronized [T0] (arg0: ⇒ T0): T0

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  42. def toString (): String

    Definition Classes
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  43. implicit def transFunctorIso [F[_], G[_], H[_]] (implicit fg: $less$tilde$greater[F, G], gh: $less$tilde$greater[G, H]): $less$tilde$greater[F, H]

    Natural isomorphism is transitive

    Natural isomorphism is transitive

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  44. def wait (): Unit

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    @throws()

  45. def wait (arg0: Long, arg1: Int): Unit

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  46. def wait (arg0: Long): Unit

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