category-extras-0.52.1: Various modules and constructs inspired by category theory Source code Contents Index

Control.Functor.Algebra Portability non-portable (rank-2 polymorphism) Stability experimental Maintainer Edward Kmett <ekmett@gmail.com>

Description

Algebras, Coalgebras, Bialgebras, and Dialgebras and their (co)monadic variants

Synopsis

type Dialgebra f g a = f a -> g a type GDialgebra f g w m a = f (w a) -> g (m a) type Bialgebra f g a = (Algebra f a, Coalgebra g a) type GBialgebra f g w m a = (GAlgebra f w a, GCoalgebra g m a) type Algebra f a = f a -> a type GAlgebra f w a = f (w a) -> a type Coalgebra f a = a -> f a type GCoalgebra f m a = a -> f (m a) liftAlgebra :: (Functor f, Comonad w) => Algebra f :~> GAlgebra f w liftCoalgebra :: (Functor f, Monad m) => Coalgebra f :~> GCoalgebra f m liftDialgebra :: (Functor g, Functor f, Comonad w, Monad m) => Dialgebra f g :~> GDialgebra f g w m fromCoalgebra :: Coalgebra f :~> Dialgebra Identity f fromAlgebra :: Algebra f :~> Dialgebra f Identity fromBialgebra :: Bialgebra f g :~> Dialgebra (f :*: Identity) (Identity :*: g)

Documentation

type Dialgebra f g a = f a -> g a Source

F,G-dialgebras generalize algebras and coalgebraas

type GDialgebra f g w m a = f (w a) -> g (m a) Source

type Bialgebra f g a = (Algebra f a, Coalgebra g a) Source

F-G-bialgebras are representable by DiAlg (f :+: Identity) (Identity :+: g) a and so add no expressive power, but are a lot more convenient.

type GBialgebra f g w m a = (GAlgebra f w a, GCoalgebra g m a) Source

type Algebra f a = f a -> a Source

F-Algebras

type GAlgebra f w a = f (w a) -> a Source

F-W-Comonadic Algebras for a given comonad W

type Coalgebra f a = a -> f a Source

F-Coalgebras

type GCoalgebra f m a = a -> f (m a) Source

F-M-Monadic Coalgebras for a given monad M

liftAlgebra :: (Functor f, Comonad w) => Algebra f :~> GAlgebra f w Source

Turn an F-algebra into a F-W-algebra by throwing away the comonad

liftCoalgebra :: (Functor f, Monad m) => Coalgebra f :~> GCoalgebra f m Source

Turn a F-coalgebra into a F-M-coalgebra by returning into a monad

liftDialgebra :: (Functor g, Functor f, Comonad w, Monad m) => Dialgebra f g :~> GDialgebra f g w m Source

fromCoalgebra :: Coalgebra f :~> Dialgebra Identity f Source

fromAlgebra :: Algebra f :~> Dialgebra f Identity Source

fromBialgebra :: Bialgebra f g :~> Dialgebra (f :*: Identity) (Identity :*: g) Source

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