Data.Category.Dialg

Dialg(F,G), the category of (F,G)-dialgebras and (F,G)-homomorphisms.

Documentation

data Dialgebra f g a where

Objects of Dialg(F,G) are (F,G)-dialgebras.

Constructors

Dialgebra :: (Category c, Category d, Dom f ~ c, Dom g ~ c, Cod f ~ d, Cod g ~ d, Functor f, Functor g) => Obj c a -> d (f :% a) (g :% a) -> Dialgebra f g a

data Dialg f g a b where

Arrows of Dialg(F,G) are (F,G)-homomorphisms.

Constructors

DialgA :: (Category c, Category d, Dom f ~ c, Dom g ~ c, Cod f ~ d, Cod g ~ d, Functor f, Functor g) => Dialgebra f g a -> Dialgebra f g b -> c a b -> Dialg f g a b

Instances

Category (Dialg f g)

The category of (F,G)-dialgebras.

HasInitialObject (Dialg (Tuple1 (->) (->) ()) (DiagProd (->)))

The category for defining the natural numbers and primitive recursion can be described as Dialg(F,G), with F(A)=<1,A> and G(A)=<A,A>.

dialgId :: Dialgebra f g a -> Obj (Dialg f g) a

dialgebra :: Obj (Dialg f g) a -> Dialgebra f g a

type Alg f = Dialg f (Id (Dom f))

type Algebra f a = Dialgebra f (Id (Dom f)) a

type Coalg f = Dialg (Id (Dom f)) f

type Coalgebra f a = Dialgebra (Id (Dom f)) f a

type InitialFAlgebra f = InitialObject (Alg f)

The initial F-algebra is the initial object in the category of F-algebras.

type TerminalFAlgebra f = TerminalObject (Coalg f)

The terminal F-coalgebra is the terminal object in the category of F-coalgebras.

type Cata f a = Algebra f a -> Alg f (InitialFAlgebra f) a

A catamorphism of an F-algebra is the arrow to it from the initial F-algebra.

type Ana f a = Coalgebra f a -> Coalg f a (TerminalFAlgebra f)

A anamorphism of an F-coalgebra is the arrow from it to the terminal F-coalgebra.

data NatNum

Constructors

Z ()

S NatNum

primRec :: (() -> t) -> (t -> t) -> NatNum -> t

data FreeAlg m

Constructors

FreeAlg (Monad m)

Instances

(Functor m, ~ (* -> * -> *) (Dom m) k, ~ (* -> * -> *) (Cod m) k) => Functor (FreeAlg m)

FreeAlg M takes x to the free algebra (M x, mu_x) of the monad M.

data ForgetAlg m

Constructors

ForgetAlg

Instances

(Functor m, ~ (* -> * -> *) (Dom m) k, ~ (* -> * -> *) (Cod m) k) => Functor (ForgetAlg m)

ForgetAlg m is the forgetful functor for Alg m.

eilenbergMooreAdj :: (Functor m, Dom m ~ k, Cod m ~ k) => Monad m -> Adjunction (Alg m) k (FreeAlg m) (ForgetAlg m)