Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

OAlg.Category.Path

Contents

Path

Description

category of paths over morphisms.

Synopsis

data Path m x y where
- IdPath :: Struct (ObjectClass m) x -> Path m x x
- (:.) :: m y z -> Path m x y -> Path m x z
toOp2Path :: Morphism m => Path m x y -> Path (Op2 m) y x
fromOp2Path :: Morphism m => Path (Op2 m) x y -> Path m y x
compose :: Category m => Path m x y -> m x y
mPath :: Morphism m => m x y -> Path m x y
reverse :: (Morphism m, Morphism f) => (forall u. Struct (ObjectClass m) u -> Struct (ObjectClass f) u) -> (forall u v. m u v -> f v u) -> Path m x y -> Path f y x
pthFoldr :: (forall x y. m x y -> f x -> f y) -> f a -> Path m a b -> f b
pthLength :: Path m x y -> N

Path

data Path m x y where Source #

paths over morphisms.

Constructors

IdPath :: Struct (ObjectClass m) x -> Path m x x
(:.) :: m y z -> Path m x y -> Path m x z infixr 9

Instances

Instances details

HomAlgebraic k h => HomAlgebraic k (Path h) Source #
Instance details Defined in OAlg.Hom.Algebraic
HomVectorial k h => HomVectorial k (Path h) Source #
Instance details Defined in OAlg.Hom.Vectorial
Applicative m => Applicative (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods amap :: Path m a b -> a -> b Source #
Morphism m => Category (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods cOne :: Struct (ObjectClass (Path m)) x -> Path m x x Source # (.) :: Path m y z -> Path m x y -> Path m x z Source #
(Cayleyan2 m, EmbeddableMorphismTyp m) => Cayleyan2 (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods invert2 :: Path m x y -> Path m y x Source #
EmbeddableMorphismTyp m => EmbeddableMorphismTyp (Path m) Source #
Instance details Defined in OAlg.Category.Path
(Applicative m, Morphism m) => Functorial (Path m) Source #
Instance details Defined in OAlg.Category.Path
Morphism m => Morphism (Path m) Source #
Instance details Defined in OAlg.Category.Path Associated Types type ObjectClass (Path m) Source # Methods homomorphous :: Path m x y -> Homomorphous (ObjectClass (Path m)) x y Source # domain :: Path m x y -> Struct (ObjectClass (Path m)) x Source # range :: Path m x y -> Struct (ObjectClass (Path m)) y Source #
(EmbeddableMorphismTyp m, Eq2 m) => Eq2 (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods eq2 :: Path m x y -> Path m x y -> Bool Source #
Show2 m => Show2 (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods show2 :: Path m a b -> String Source #
(Morphism m, Validable2 m) => Validable2 (Path m) Source #
Instance details Defined in OAlg.Category.Path Methods valid2 :: Path m x y -> Statement Source #
(Entity2 h, EmbeddableMorphismTyp h) => Entity2 (Path h) Source #
Instance details Defined in OAlg.Category.Path
HomAdditive h => HomAdditive (Path h) Source #
Instance details Defined in OAlg.Hom.Additive
HomDistributive h => HomDistributive (Path h) Source #
Instance details Defined in OAlg.Hom.Distributive
FunctorialHomFibred h => FunctorialHomFibred (Path h) Source #
Instance details Defined in OAlg.Hom.Fibred
HomFibred h => HomFibred (Path h) Source #
Instance details Defined in OAlg.Hom.Fibred Methods rmap :: Path h a b -> Root a -> Root b Source #
HomFibredOriented h => HomFibredOriented (Path h) Source #
Instance details Defined in OAlg.Hom.Fibred
HomMultiplicative h => HomMultiplicative (Path h) Source #
Instance details Defined in OAlg.Hom.Multiplicative.Definition
FunctorialHomOriented h => FunctorialHomOriented (Path h) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
HomOriented h => HomOriented (Path h) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods pmap :: Path h a b -> Point a -> Point b Source #
EmbeddableMorphism m t => EmbeddableMorphism (Path m) t Source #
Instance details Defined in OAlg.Category.Path
Show2 (Path m) => Show (Path m x y) Source #
Instance details Defined in OAlg.Category.Path Methods showsPrec :: Int -> Path m x y -> ShowS # show :: Path m x y -> String # showList :: [Path m x y] -> ShowS #
Eq2 (Path m) => Eq (Path m x y) Source #
Instance details Defined in OAlg.Category.Path Methods (==) :: Path m x y -> Path m x y -> Bool # (/=) :: Path m x y -> Path m x y -> Bool #
Morphism m => Dualisable (Path m x y) Source #
Instance details Defined in OAlg.Category.Path Methods toDual :: Path m x y -> Dual (Path m x y) Source # fromDual :: Dual (Path m x y) -> Path m x y Source #
Reducible (PathHomOp s a b) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods reduce :: PathHomOp s a b -> PathHomOp s a b Source #
Validable2 (Path m) => Validable (Path m x y) Source #
Instance details Defined in OAlg.Category.Path Methods valid :: Path m x y -> Statement Source #
Reducible (PathOpMap f s a b) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods reduce :: PathOpMap f s a b -> PathOpMap f s a b Source #
type Dual (Path m x y :: Type) Source #
Instance details Defined in OAlg.Category.Path type Dual (Path m x y :: Type) = Path (Op2 m) y x
type ObjectClass (Path m) Source #
Instance details Defined in OAlg.Category.Path type ObjectClass (Path m) = ObjectClass m

toOp2Path :: Morphism m => Path m x y -> Path (Op2 m) y x Source #

the opposite path.

fromOp2Path :: Morphism m => Path (Op2 m) x y -> Path m y x Source #

from the opposite path.

compose :: Category m => Path m x y -> m x y Source #

composing the morphisms of a path.

mPath :: Morphism m => m x y -> Path m x y Source #

embedding morphisms into paths.

reverse :: (Morphism m, Morphism f) => (forall u. Struct (ObjectClass m) u -> Struct (ObjectClass f) u) -> (forall u v. m u v -> f v u) -> Path m x y -> Path f y x Source #

reversing a path given by the formal inverse function.

pthFoldr :: (forall x y. m x y -> f x -> f y) -> f a -> Path m a b -> f b Source #

folding from the right.

pthLength :: Path m x y -> N Source #

the length of a path.