data-category-0.10: Category theory

Data.Category.Dialg

Description

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

Synopsis

# Documentation

data Dialgebra f g a where Source #

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 Source #

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 :: Type -> Type -> Type) Source # The category of (F,G)-dialgebras. Instance detailsDefined in Data.Category.Dialg Methodssrc :: Dialg f g a b -> Obj (Dialg f g) a Source #tgt :: Dialg f g a b -> Obj (Dialg f g) b Source #(.) :: Dialg f g b c -> Dialg f g a b -> Dialg f g a c Source # HasInitialObject (Dialg (Tuple1 ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) ()) (DiagProd ((->) :: Type -> Type -> Type))) Source # 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)=. Instance detailsDefined in Data.Category.Dialg Associated Typestype InitialObject (Dialg (Tuple1 (->) (->) ()) (DiagProd (->))) :: Kind k Source # MethodsinitialObject :: Obj (Dialg (Tuple1 (->) (->) ()) (DiagProd (->))) (InitialObject (Dialg (Tuple1 (->) (->) ()) (DiagProd (->)))) Source #initialize :: Obj (Dialg (Tuple1 (->) (->) ()) (DiagProd (->))) a -> Dialg (Tuple1 (->) (->) ()) (DiagProd (->)) (InitialObject (Dialg (Tuple1 (->) (->) ()) (DiagProd (->)))) a Source # type InitialObject (Dialg (Tuple1 ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) ()) (DiagProd ((->) :: Type -> Type -> Type))) Source # Instance detailsDefined in Data.Category.Dialg type InitialObject (Dialg (Tuple1 ((->) :: Type -> Type -> Type) ((->) :: Type -> Type -> Type) ()) (DiagProd ((->) :: Type -> Type -> Type))) = NatNum

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

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

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

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

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

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

type InitialFAlgebra f = InitialObject (Alg f) Source #

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

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 Source #

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) Source #

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

data NatNum Source #

Constructors

 Z () S NatNum

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

data FreeAlg m Source #

Constructors

Instances
 (Functor m, Dom m ~ k, Cod m ~ k) => Functor (FreeAlg m) Source # FreeAlg M takes x to the free algebra (M x, mu_x) of the monad M. Instance detailsDefined in Data.Category.Dialg Associated Typestype Dom (FreeAlg m) :: Type -> Type -> Type Source #type Cod (FreeAlg m) :: Type -> Type -> Type Source #type (FreeAlg m) :% a :: Type Source # Methods(%) :: FreeAlg m -> Dom (FreeAlg m) a b -> Cod (FreeAlg m) (FreeAlg m :% a) (FreeAlg m :% b) Source # type Dom (FreeAlg m) Source # Instance detailsDefined in Data.Category.Dialg type Dom (FreeAlg m) = Dom m type Cod (FreeAlg m) Source # Instance detailsDefined in Data.Category.Dialg type Cod (FreeAlg m) = Alg m type (FreeAlg m) :% a Source # Instance detailsDefined in Data.Category.Dialg type (FreeAlg m) :% a = m :% a

freeAlg :: (Functor m, Dom m ~ k, Cod m ~ k) => Monad m -> Obj (Cod m) x -> Algebra m (m :% x) Source #

data ForgetAlg m Source #

Constructors

 ForgetAlg
Instances
 (Functor m, Dom m ~ k, Cod m ~ k) => Functor (ForgetAlg m) Source # ForgetAlg m is the forgetful functor for Alg m. Instance detailsDefined in Data.Category.Dialg Associated Typestype Dom (ForgetAlg m) :: Type -> Type -> Type Source #type Cod (ForgetAlg m) :: Type -> Type -> Type Source #type (ForgetAlg m) :% a :: Type Source # Methods(%) :: ForgetAlg m -> Dom (ForgetAlg m) a b -> Cod (ForgetAlg m) (ForgetAlg m :% a) (ForgetAlg m :% b) Source # type Dom (ForgetAlg m) Source # Instance detailsDefined in Data.Category.Dialg type Dom (ForgetAlg m) = Alg m type Cod (ForgetAlg m) Source # Instance detailsDefined in Data.Category.Dialg type Cod (ForgetAlg m) = Dom m type (ForgetAlg m) :% a Source # Instance detailsDefined in Data.Category.Dialg type (ForgetAlg m) :% a = a

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