FiniteCategory.FiniteCategory

The goal of this module is to represent small finite categories in order to make usual constructions automatically on them (e.g. (co)limits, (co)completion, etc.) There is a package that deals with categories as a data type (Data.Category) but it does not provide Ar and Ob operators because the package uses the fact that the language itself is a category to represent categories (Hask is (almost) a category, Data.Category benefit from type check at compilation time). The counterpart is that to use Data.Category, you need to create yourself the usual constructions on your category. To construct automatically (co)limits and other constructions, we require that the category can enumerate every morphisms between two objects. We also require that the objects are not types because we want to create objects at runtime (that is why we abandon type check at compilation time for the structure of our categories). Lastly, we require that morphisms can be equated for usual constructions purposes.

For example, see the FinSet category and the Composition Graph.

Morphism typeclass and related functions

class Morphism m o | m -> o where

A morphism can be composed with the (@) operator, it has a source and a target.

The (@) operator should not be confused with the as-pattern. When using the operator, surround the @ symbol with spaces.

Morphism is a multiparametrized type class where m is the type of the morphism and o the type of the objects source and target. Source and target are the same type of objects, we distinguish objects not by their type but instead by their values.

Methods

(@) :: m -> m -> m

The composition g@f should throw an error if source g != target f. This is a consequence of loosing type check at compilation time, we defer the exception handling to execution time.

Composition is associative :

f @ (g @ h) = (f @ g) @ h

source :: m -> o

target :: m -> o

Instances

Morphism Zero Zero

Defined in UsualCategories.Zero

Methods

(@) :: Zero -> Zero -> Zero

source :: Zero -> Zero

target :: Zero -> Zero

Morphism VAr VOb

Defined in UsualCategories.V

Methods

(@) :: VAr -> VAr -> VAr

source :: VAr -> VOb

target :: VAr -> VOb

Morphism TwoAr TwoOb

Defined in UsualCategories.Two

Methods

(@) :: TwoAr -> TwoAr -> TwoAr

source :: TwoAr -> TwoOb

target :: TwoAr -> TwoOb

Morphism ThreeAr ThreeOb

Defined in UsualCategories.Three

Methods

(@) :: ThreeAr -> ThreeAr -> ThreeAr

source :: ThreeAr -> ThreeOb

target :: ThreeAr -> ThreeOb

Morphism SquareAr SquareOb

Defined in UsualCategories.Square

Methods

(@) :: SquareAr -> SquareAr -> SquareAr

source :: SquareAr -> SquareOb

target :: SquareAr -> SquareOb

Morphism ParallelAr ParallelOb

Defined in UsualCategories.Parallel

Methods

(@) :: ParallelAr -> ParallelAr -> ParallelAr

source :: ParallelAr -> ParallelOb

target :: ParallelAr -> ParallelOb

Morphism One One

Defined in UsualCategories.One

Methods

(@) :: One -> One -> One

source :: One -> One

target :: One -> One

Morphism HatAr HatOb

Defined in UsualCategories.Hat

Methods

(@) :: HatAr -> HatAr -> HatAr

source :: HatAr -> HatOb

target :: HatAr -> HatOb

Eq a => Morphism (DiscreteIdentity a) (DiscreteObject a)

Defined in UsualCategories.DiscreteCategory

Methods

(@) :: DiscreteIdentity a -> DiscreteIdentity a -> DiscreteIdentity a

source :: DiscreteIdentity a -> DiscreteObject a

target :: DiscreteIdentity a -> DiscreteObject a

Ord a => Morphism (FinOrdMap a) (Set a)

Defined in Set.FinOrdSet

Methods

(@) :: FinOrdMap a -> FinOrdMap a -> FinOrdMap a

source :: FinOrdMap a -> Set a

target :: FinOrdMap a -> Set a

Eq a => Morphism (FinMap a) (FinSet a)

Defined in Set.FinSet

Methods

(@) :: FinMap a -> FinMap a -> FinMap a

source :: FinMap a -> FinSet a

target :: FinMap a -> FinSet a

Morphism m o => Morphism (OppositeMorphism m o) o

Defined in OppositeCategory.OppositeCategory

Methods

(@) :: OppositeMorphism m o -> OppositeMorphism m o -> OppositeMorphism m o

source :: OppositeMorphism m o -> o

target :: OppositeMorphism m o -> o

(Eq a, Eq b) => Morphism (CGMorphism a b) a

Defined in CompositionGraph.CompositionGraph

Methods

(@) :: CGMorphism a b -> CGMorphism a b -> CGMorphism a b

source :: CGMorphism a b -> a

target :: CGMorphism a b -> a

(Eq a, Eq b) => Morphism (SCGMorphism a b) a

Defined in CompositionGraph.SafeCompositionGraph

Methods

(@) :: SCGMorphism a b -> SCGMorphism a b -> SCGMorphism a b

source :: SCGMorphism a b -> a

target :: SCGMorphism a b -> a

(Eq c, Eq m, Eq o) => Morphism (FinFunctor c m o) c

Defined in Cat.FinCat

Methods

(@) :: FinFunctor c m o -> FinFunctor c m o -> FinFunctor c m o

source :: FinFunctor c m o -> c

target :: FinFunctor c m o -> c

(Eq c, Eq m, Eq o) => Morphism (PartialFunctor c m o) c

Defined in Cat.PartialFinCat

Methods

(@) :: PartialFunctor c m o -> PartialFunctor c m o -> PartialFunctor c m o

source :: PartialFunctor c m o -> c

target :: PartialFunctor c m o -> c

(Morphism m1 o1, Morphism m2 o2) => Morphism (ProductMorphism m1 o1 m2 o2) (ProductObject o1 o2)

Defined in ProductCategory.ProductCategory

Methods

(@) :: ProductMorphism m1 o1 m2 o2 -> ProductMorphism m1 o1 m2 o2 -> ProductMorphism m1 o1 m2 o2

source :: ProductMorphism m1 o1 m2 o2 -> ProductObject o1 o2

target :: ProductMorphism m1 o1 m2 o2 -> ProductObject o1 o2

(Morphism m1 o1, Morphism m2 o2, Eq o1, Eq o2, Eq m3) => Morphism (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

(@) :: CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3

source :: CommaMorphism o1 o2 m1 m2 m3 -> CommaObject o1 o2 m3

target :: CommaMorphism o1 o2 m1 m2 m3 -> CommaObject o1 o2 m3

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => Morphism (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)

Defined in FunctorCategory.FunctorCategory

Methods

(@) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2

source :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2

target :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2

compose :: Morphism m o => [m] -> m

Returns the composition of the list of morphisms.

For example : compose [f,g,h] = f @ g @ h

FiniteCategory typeclass

class FiniteCategory c m o | c -> m, m -> o where

A category is multiparametrized by the type of its morphisms and the type of its objects.

Minimal complete definition

ob, identity, ar

Methods

ob :: c -> [o]

ob should return a list of unique objects :

List.nub (ob c) = ob c

identity :: Morphism m o => c -> o -> m

identity should return the identity associated to the object o in the category c.

The identity morphism is a morphism such that the two following properties are verified :

f @ identity c (source f) = f

identity c (target g) @ g = g

ar

Arguments

:: Morphism m o

=> c

The category

-> o

The source of the morphisms

-> o

The target of the morphisms

-> [m]

The list of morphisms in the category c between source and target

ar should return the list of all unique arrows between a source and a target :

List.nub (ar c s t) = ar c s t

Arrows with different source or target should not be equal :

(s1 = s2 && t1 = t2) || List.intersect (ar c s1 t1) (ar c s2 t2) = []

arrows :: (FiniteCategory c m o, Morphism m o) => c -> [m]

arrows returns the list of all unique morphisms of a category.

Instances

FiniteCategory Zero Zero Zero

Defined in UsualCategories.Zero

Methods

ob :: Zero -> [Zero]

identity :: Zero -> Zero -> Zero

ar :: Zero -> Zero -> Zero -> [Zero]

arrows :: Zero -> [Zero]

FiniteCategory V VAr VOb

Defined in UsualCategories.V

Methods

ob :: V -> [VOb]

identity :: V -> VOb -> VAr

ar :: V -> VOb -> VOb -> [VAr]

arrows :: V -> [VAr]

FiniteCategory Two TwoAr TwoOb

Defined in UsualCategories.Two

Methods

ob :: Two -> [TwoOb]

identity :: Two -> TwoOb -> TwoAr

ar :: Two -> TwoOb -> TwoOb -> [TwoAr]

arrows :: Two -> [TwoAr]

FiniteCategory Three ThreeAr ThreeOb

Defined in UsualCategories.Three

Methods

ob :: Three -> [ThreeOb]

identity :: Three -> ThreeOb -> ThreeAr

ar :: Three -> ThreeOb -> ThreeOb -> [ThreeAr]

arrows :: Three -> [ThreeAr]

FiniteCategory Square SquareAr SquareOb

Defined in UsualCategories.Square

Methods

ob :: Square -> [SquareOb]

identity :: Square -> SquareOb -> SquareAr

ar :: Square -> SquareOb -> SquareOb -> [SquareAr]

arrows :: Square -> [SquareAr]

FiniteCategory Parallel ParallelAr ParallelOb

Defined in UsualCategories.Parallel

Methods

ob :: Parallel -> [ParallelOb]

identity :: Parallel -> ParallelOb -> ParallelAr

ar :: Parallel -> ParallelOb -> ParallelOb -> [ParallelAr]

arrows :: Parallel -> [ParallelAr]

FiniteCategory One One One

Defined in UsualCategories.One

Methods

ob :: One -> [One]

identity :: One -> One -> One

ar :: One -> One -> One -> [One]

arrows :: One -> [One]

FiniteCategory Hat HatAr HatOb

Defined in UsualCategories.Hat

Methods

ob :: Hat -> [HatOb]

identity :: Hat -> HatOb -> HatAr

ar :: Hat -> HatOb -> HatOb -> [HatAr]

arrows :: Hat -> [HatAr]

Eq a => FiniteCategory (DiscreteCategory a) (DiscreteIdentity a) (DiscreteObject a)

Defined in UsualCategories.DiscreteCategory

Methods

ob :: DiscreteCategory a -> [DiscreteObject a]

identity :: DiscreteCategory a -> DiscreteObject a -> DiscreteIdentity a

ar :: DiscreteCategory a -> DiscreteObject a -> DiscreteObject a -> [DiscreteIdentity a]

arrows :: DiscreteCategory a -> [DiscreteIdentity a]

Ord a => FiniteCategory (FinOrdSet a) (FinOrdMap a) (Set a)

Defined in Set.FinOrdSet

Methods

ob :: FinOrdSet a -> [Set a]

identity :: FinOrdSet a -> Set a -> FinOrdMap a

ar :: FinOrdSet a -> Set a -> Set a -> [FinOrdMap a]

arrows :: FinOrdSet a -> [FinOrdMap a]

Eq a => FiniteCategory (FinSetCat a) (FinMap a) (FinSet a)

Defined in Set.FinSet

Methods

ob :: FinSetCat a -> [FinSet a]

identity :: FinSetCat a -> FinSet a -> FinMap a

ar :: FinSetCat a -> FinSet a -> FinSet a -> [FinMap a]

arrows :: FinSetCat a -> [FinMap a]

(Eq a, Eq b) => FiniteCategory (CompositionGraph a b) (CGMorphism a b) a

Defined in CompositionGraph.CompositionGraph

Methods

ob :: CompositionGraph a b -> [a]

identity :: CompositionGraph a b -> a -> CGMorphism a b

ar :: CompositionGraph a b -> a -> a -> [CGMorphism a b]

arrows :: CompositionGraph a b -> [CGMorphism a b]

(Eq a, Eq b) => FiniteCategory (SafeCompositionGraph a b) (SCGMorphism a b) a

Defined in CompositionGraph.SafeCompositionGraph

Methods

ob :: SafeCompositionGraph a b -> [a]

identity :: SafeCompositionGraph a b -> a -> SCGMorphism a b

ar :: SafeCompositionGraph a b -> a -> a -> [SCGMorphism a b]

arrows :: SafeCompositionGraph a b -> [SCGMorphism a b]

(FiniteCategory c m o, Morphism m o, Eq m, Eq o) => FiniteCategory (FullSubcategory c m o) m o

Defined in Subcategories.FullSubcategory

Methods

ob :: FullSubcategory c m o -> [o]

identity :: FullSubcategory c m o -> o -> m

ar :: FullSubcategory c m o -> o -> o -> [m]

arrows :: FullSubcategory c m o -> [m]

(FiniteCategory c m o, Morphism m o, Eq m, Eq o) => FiniteCategory (FreeSubcategory c m o) m o

Defined in Subcategories.FreeSubcategory

Methods

ob :: FreeSubcategory c m o -> [o]

identity :: FreeSubcategory c m o -> o -> m

ar :: FreeSubcategory c m o -> o -> o -> [m]

arrows :: FreeSubcategory c m o -> [m]

(FiniteCategory c m o, Morphism m o) => FiniteCategory (OppositeCategory c m o) (OppositeMorphism m o) o

Defined in OppositeCategory.OppositeCategory

Methods

ob :: OppositeCategory c m o -> [o]

identity :: OppositeCategory c m o -> o -> OppositeMorphism m o

ar :: OppositeCategory c m o -> o -> o -> [OppositeMorphism m o]

arrows :: OppositeCategory c m o -> [OppositeMorphism m o]

(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o) => FiniteCategory (FinCat c m o) (FinFunctor c m o) c

Defined in Cat.FinCat

Methods

ob :: FinCat c m o -> [c]

identity :: FinCat c m o -> c -> FinFunctor c m o

ar :: FinCat c m o -> c -> c -> [FinFunctor c m o]

arrows :: FinCat c m o -> [FinFunctor c m o]

(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o) => FiniteCategory (PartialFinCat c m o) (PartialFunctor c m o) c

Defined in Cat.PartialFinCat

Methods

ob :: PartialFinCat c m o -> [c]

identity :: PartialFinCat c m o -> c -> PartialFunctor c m o

ar :: PartialFinCat c m o -> c -> c -> [PartialFunctor c m o]

arrows :: PartialFinCat c m o -> [PartialFunctor c m o]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => FiniteCategory (Subcategory c1 m1 o1 c2 m2 o2) m2 o2

Defined in Subcategories.Subcategory

Methods

ob :: Subcategory c1 m1 o1 c2 m2 o2 -> [o2]

identity :: Subcategory c1 m1 o1 c2 m2 o2 -> o2 -> m2

ar :: Subcategory c1 m1 o1 c2 m2 o2 -> o2 -> o2 -> [m2]

arrows :: Subcategory c1 m1 o1 c2 m2 o2 -> [m2]

(FiniteCategory c1 m1 o1, Morphism m1 o1, FiniteCategory c2 m2 o2, Morphism m2 o2) => FiniteCategory (ProductCategory c1 m1 o1 c2 m2 o2) (ProductMorphism m1 o1 m2 o2) (ProductObject o1 o2)

Defined in ProductCategory.ProductCategory

Methods

ob :: ProductCategory c1 m1 o1 c2 m2 o2 -> [ProductObject o1 o2]

identity :: ProductCategory c1 m1 o1 c2 m2 o2 -> ProductObject o1 o2 -> ProductMorphism m1 o1 m2 o2

ar :: ProductCategory c1 m1 o1 c2 m2 o2 -> ProductObject o1 o2 -> ProductObject o1 o2 -> [ProductMorphism m1 o1 m2 o2]

arrows :: ProductCategory c1 m1 o1 c2 m2 o2 -> [ProductMorphism m1 o1 m2 o2]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => FiniteCategory (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)

Defined in FunctorCategory.FunctorCategory

Methods

ob :: FunctorCategory c1 m1 o1 c2 m2 o2 -> [Diagram c1 m1 o1 c2 m2 o2]

identity :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2

ar :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2]

arrows :: FunctorCategory c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => FiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

ob :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaObject o1 o2 m3]

identity :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaMorphism o1 o2 m1 m2 m3

ar :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

arrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

FiniteCategory functions

arFrom :: (FiniteCategory c m o, Morphism m o) => c -> o -> [m]

arFrom returns the list of unique morphisms going from a specified source.

arFrom2 :: (FiniteCategory c m o, Morphism m o) => c -> [o] -> [m]

arFrom2 same as arFrom but for multiple sources.

arTo :: (FiniteCategory c m o, Morphism m o) => c -> o -> [m]

arTo returns the list of unique morphisms going to a specified target.

arTo2 :: (FiniteCategory c m o, Morphism m o) => c -> [o] -> [m]

arTo2 same as arTo but for multiple targets.

identities :: (FiniteCategory c m o, Morphism m o) => c -> [m]

identities returns all the identities of a category.

isIdentity :: (FiniteCategory c m o, Morphism m o, Eq m, Eq o) => c -> m -> Bool

Returns wether a morphism is an identity in a category.

isNotIdentity :: (FiniteCategory c m o, Morphism m o, Eq m, Eq o) => c -> m -> Bool

Returns wether a morphism is not an identity.

isTerminal :: (FiniteCategory c m o, Morphism m o) => c -> o -> Bool

Returns wether an object is terminal in the category.

isInitial :: (FiniteCategory c m o, Morphism m o) => c -> o -> Bool

Returns wether an object is initial in the category.

terminalObjects :: (FiniteCategory c m o, Morphism m o) => c -> [o]

Returns the list of terminal objects in a category.

initialObjects :: (FiniteCategory c m o, Morphism m o) => c -> [o]

Returns the list of intial objects in a category.

GeneratedFiniteCategory typeclass and related functions

class FiniteCategory c m o => GeneratedFiniteCategory c m o where

GeneratedFiniteCategory is a FiniteCategory where some morphisms are selected as generators. The full category is generated by the generating arrows, which means that every morphism can be written as a composition of several generators. Some algorithms are simplified because they only need to deal with generators, the rest of the properties are deduced by composition. Every FiniteCategory has at least one set of generators : the set of all of its morphisms.

Minimal complete definition

genAr, decompose

Methods

genAr :: Morphism m o => c -> o -> o -> [m]

Same as ar but only returns the generators. genAr c s t should be included in ar c s t.

decompose :: Morphism m o => c -> m -> [m]

decompose decomposes a morphism into a list of generators (according to composition) :

m = compose (decompose c m)

genArrows :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> [m]

Same as arrows but only returns the generators. genArrows c should be included in arrows c.

Instances

GeneratedFiniteCategory Zero Zero Zero

Defined in UsualCategories.Zero

Methods

genAr :: Zero -> Zero -> Zero -> [Zero]

decompose :: Zero -> Zero -> [Zero]

genArrows :: Zero -> [Zero]

GeneratedFiniteCategory V VAr VOb

Defined in UsualCategories.V

Methods

genAr :: V -> VOb -> VOb -> [VAr]

decompose :: V -> VAr -> [VAr]

genArrows :: V -> [VAr]

GeneratedFiniteCategory Two TwoAr TwoOb

Defined in UsualCategories.Two

Methods

genAr :: Two -> TwoOb -> TwoOb -> [TwoAr]

decompose :: Two -> TwoAr -> [TwoAr]

genArrows :: Two -> [TwoAr]

GeneratedFiniteCategory Three ThreeAr ThreeOb

Defined in UsualCategories.Three

Methods

genAr :: Three -> ThreeOb -> ThreeOb -> [ThreeAr]

decompose :: Three -> ThreeAr -> [ThreeAr]

genArrows :: Three -> [ThreeAr]

GeneratedFiniteCategory Square SquareAr SquareOb

Defined in UsualCategories.Square

Methods

genAr :: Square -> SquareOb -> SquareOb -> [SquareAr]

decompose :: Square -> SquareAr -> [SquareAr]

genArrows :: Square -> [SquareAr]

GeneratedFiniteCategory Parallel ParallelAr ParallelOb

Defined in UsualCategories.Parallel

Methods

genAr :: Parallel -> ParallelOb -> ParallelOb -> [ParallelAr]

decompose :: Parallel -> ParallelAr -> [ParallelAr]

genArrows :: Parallel -> [ParallelAr]

GeneratedFiniteCategory One One One

Defined in UsualCategories.One

Methods

genAr :: One -> One -> One -> [One]

decompose :: One -> One -> [One]

genArrows :: One -> [One]

GeneratedFiniteCategory Hat HatAr HatOb

Defined in UsualCategories.Hat

Methods

genAr :: Hat -> HatOb -> HatOb -> [HatAr]

decompose :: Hat -> HatAr -> [HatAr]

genArrows :: Hat -> [HatAr]

Eq a => GeneratedFiniteCategory (DiscreteCategory a) (DiscreteIdentity a) (DiscreteObject a)

Defined in UsualCategories.DiscreteCategory

Methods

genAr :: DiscreteCategory a -> DiscreteObject a -> DiscreteObject a -> [DiscreteIdentity a]

decompose :: DiscreteCategory a -> DiscreteIdentity a -> [DiscreteIdentity a]

genArrows :: DiscreteCategory a -> [DiscreteIdentity a]

Ord a => GeneratedFiniteCategory (FinOrdSet a) (FinOrdMap a) (Set a)

Defined in Set.FinOrdSet

Methods

genAr :: FinOrdSet a -> Set a -> Set a -> [FinOrdMap a]

decompose :: FinOrdSet a -> FinOrdMap a -> [FinOrdMap a]

genArrows :: FinOrdSet a -> [FinOrdMap a]

Eq a => GeneratedFiniteCategory (FinSetCat a) (FinMap a) (FinSet a)

Defined in Set.FinSet

Methods

genAr :: FinSetCat a -> FinSet a -> FinSet a -> [FinMap a]

decompose :: FinSetCat a -> FinMap a -> [FinMap a]

genArrows :: FinSetCat a -> [FinMap a]

(Eq a, Eq b) => GeneratedFiniteCategory (CompositionGraph a b) (CGMorphism a b) a

Defined in CompositionGraph.CompositionGraph

Methods

genAr :: CompositionGraph a b -> a -> a -> [CGMorphism a b]

decompose :: CompositionGraph a b -> CGMorphism a b -> [CGMorphism a b]

genArrows :: CompositionGraph a b -> [CGMorphism a b]

(Eq a, Eq b) => GeneratedFiniteCategory (SafeCompositionGraph a b) (SCGMorphism a b) a

Defined in CompositionGraph.SafeCompositionGraph

Methods

genAr :: SafeCompositionGraph a b -> a -> a -> [SCGMorphism a b]

decompose :: SafeCompositionGraph a b -> SCGMorphism a b -> [SCGMorphism a b]

genArrows :: SafeCompositionGraph a b -> [SCGMorphism a b]

(FiniteCategory c m o, Morphism m o, Eq m, Eq o) => GeneratedFiniteCategory (FullSubcategory c m o) m o

Defined in Subcategories.FullSubcategory

Methods

genAr :: FullSubcategory c m o -> o -> o -> [m]

decompose :: FullSubcategory c m o -> m -> [m]

genArrows :: FullSubcategory c m o -> [m]

(FiniteCategory c m o, Morphism m o, Eq m, Eq o) => GeneratedFiniteCategory (FreeSubcategory c m o) m o

Defined in Subcategories.FreeSubcategory

Methods

genAr :: FreeSubcategory c m o -> o -> o -> [m]

decompose :: FreeSubcategory c m o -> m -> [m]

genArrows :: FreeSubcategory c m o -> [m]

(GeneratedFiniteCategory c m o, Morphism m o) => GeneratedFiniteCategory (OppositeCategory c m o) (OppositeMorphism m o) o

Defined in OppositeCategory.OppositeCategory

Methods

genAr :: OppositeCategory c m o -> o -> o -> [OppositeMorphism m o]

decompose :: OppositeCategory c m o -> OppositeMorphism m o -> [OppositeMorphism m o]

genArrows :: OppositeCategory c m o -> [OppositeMorphism m o]

(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o) => GeneratedFiniteCategory (FinCat c m o) (FinFunctor c m o) c

Defined in Cat.FinCat

Methods

genAr :: FinCat c m o -> c -> c -> [FinFunctor c m o]

decompose :: FinCat c m o -> FinFunctor c m o -> [FinFunctor c m o]

genArrows :: FinCat c m o -> [FinFunctor c m o]

(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o) => GeneratedFiniteCategory (PartialFinCat c m o) (PartialFunctor c m o) c

Defined in Cat.PartialFinCat

Methods

genAr :: PartialFinCat c m o -> c -> c -> [PartialFunctor c m o]

decompose :: PartialFinCat c m o -> PartialFunctor c m o -> [PartialFunctor c m o]

genArrows :: PartialFinCat c m o -> [PartialFunctor c m o]

(GeneratedFiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, GeneratedFiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => GeneratedFiniteCategory (Subcategory c1 m1 o1 c2 m2 o2) m2 o2

Defined in Subcategories.Subcategory

Methods

genAr :: Subcategory c1 m1 o1 c2 m2 o2 -> o2 -> o2 -> [m2]

decompose :: Subcategory c1 m1 o1 c2 m2 o2 -> m2 -> [m2]

genArrows :: Subcategory c1 m1 o1 c2 m2 o2 -> [m2]

(GeneratedFiniteCategory c1 m1 o1, Morphism m1 o1, GeneratedFiniteCategory c2 m2 o2, Morphism m2 o2) => GeneratedFiniteCategory (ProductCategory c1 m1 o1 c2 m2 o2) (ProductMorphism m1 o1 m2 o2) (ProductObject o1 o2)

Defined in ProductCategory.ProductCategory

Methods

genAr :: ProductCategory c1 m1 o1 c2 m2 o2 -> ProductObject o1 o2 -> ProductObject o1 o2 -> [ProductMorphism m1 o1 m2 o2]

decompose :: ProductCategory c1 m1 o1 c2 m2 o2 -> ProductMorphism m1 o1 m2 o2 -> [ProductMorphism m1 o1 m2 o2]

genArrows :: ProductCategory c1 m1 o1 c2 m2 o2 -> [ProductMorphism m1 o1 m2 o2]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => GeneratedFiniteCategory (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)

Defined in FunctorCategory.FunctorCategory

Methods

genAr :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2]

decompose :: FunctorCategory c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2]

genArrows :: FunctorCategory c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => GeneratedFiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

genAr :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

decompose :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaMorphism o1 o2 m1 m2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

genArrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

genArFrom :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> o -> [m]

Same as arFrom but only returns the generators. genArFrom c s should be included in arFrom c s.

genArFrom2 :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> [o] -> [m]

Same as arFrom2 but only returns the generators. genArFrom2 c [s] should be included in arFrom2 c [t].

genArTo :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> o -> [m]

Same as arTo but only returns the generators. genArTo c t should be included in arTo c t.

genArTo2 :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> [o] -> [m]

Same as arTo2 but only returns the generators. genArTo2 c [t] should be included in arTo2 c [t].

defaultGenAr :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> o -> o -> [m]

Every FiniteCategory has at least one set of generators : the set of all of its morphisms.

defaultGenAr is a default method for genAr in order to transform any FiniteCategory into a GeneratedFiniteCategory.

Use defaultGenAr in conjonction with defaultDecompose.

defaultDecompose :: (GeneratedFiniteCategory c m o, Morphism m o) => c -> m -> [m]

Every FiniteCategory has at least one set of generators : the set of all of its morphisms.

defaultDecompose is a default method for decompose in order to transform any FiniteCategory into a GeneratedFiniteCategory.

Use defaultDecompose in conjonction with defaultGenAr.

bruteForceDecompose :: (GeneratedFiniteCategory c m o, Morphism m o, Eq m) => c -> m -> [m]

If genAr is implemented, we can find the decomposition of a morphism by bruteforce search (we compose every arrow until we get the morphism we want).

This method is meant to be used temporarly until a proper decompose method is implemented. (It is very slow.)

isGenerator :: (GeneratedFiniteCategory c m o, Morphism m o, Eq m) => c -> m -> Bool

Returns if a morphism is a generating morphism. It can be overloaded to speed it up (for a morphism f, it computes every generators between the source and the target of f and checks if f is in the list.)

isComposite :: (GeneratedFiniteCategory c m o, Morphism m o, Eq m) => c -> m -> Bool

Opposite of isGenerator, i.e. returns if the morphism is composite.

Check for category correctness

data FiniteCategoryError m o

A data type to represent an incoherence inside a category.

Constructors

CompositionNotAssociative

(h@g)@f /= h@(g@f)

Fields

• f :: m
• g :: m
• h :: m
• fg_h :: m
• f_gh :: m

ObjectsNotUnique

dupObj was found multiple times in ob c.

Fields

• dupObj :: o

MorphismsNotUnique

dupMorph was found multiple times in the category.

Fields

• dupMorph :: m

ArrowsNotExhaustive

missingAr is an arrow found by calling ar c but is not in arrows c.

Fields

• missingAr :: m

ArrowBetweenUnknownObjects

f was found in the category but its source s or target t is not in ob c.

Fields

• f :: m
• s :: o
• t :: o

WrongSource

f was found by using ar c s t where s /= source f.

Fields

• f :: m
• realSource :: o

WrongTarget

f was found by using ar c s t where t /= target f.

Fields

• f :: m
• realTarget :: o

IdentityNotLeftNeutral

idL is not a valid identity : f @ idL /= f.

Fields

• idL :: m
• f :: m
• foidL :: m

IdentityNotRightNeutral

idR is not a valid identity : idR @ f /= f.

Fields

• f :: m
• idR :: m
• idRof :: m

MorphismsShouldNotBeEqual

f == g even though they don'y share the same source or target.

Fields

• f :: m
• g :: m

NotTransitive

f@g is not an element of ar c (source g) (target g).

Fields

• f :: m
• g :: m

GeneratorIsNotAMorphism

f is a generator but not a morphism.

Fields

• f :: m

MorphismDoesntDecomposesIntoGenerators

The morphism f decomposes into decomp where notGen is a non generating morphism.

Fields

• f :: m
• decomp :: [m]
• notGen :: m

WrongDecomposition

compose (decompose c f) /= f.

Fields

• f :: m
• decomp :: [m]
• comp :: m

Instances

(Eq m, Eq o) => Eq (FiniteCategoryError m o)

Defined in FiniteCategory.FiniteCategory

Methods

(==) :: FiniteCategoryError m o -> FiniteCategoryError m o -> Bool

(/=) :: FiniteCategoryError m o -> FiniteCategoryError m o -> Bool

(Show m, Show o) => Show (FiniteCategoryError m o)

Defined in FiniteCategory.FiniteCategory

Methods

showsPrec :: Int -> FiniteCategoryError m o -> ShowS

show :: FiniteCategoryError m o -> String

showList :: [FiniteCategoryError m o] -> ShowS

(Show m, Show o) => PrettyPrintable (FiniteCategoryError m o)

Defined in FiniteCategory.FiniteCategory

Methods

pprint :: FiniteCategoryError m o -> String

checkFiniteCategoryProperties :: (FiniteCategory c m o, Morphism m o, Eq m, Eq o) => c -> Maybe (FiniteCategoryError m o)

Checks every properties listed above for a category.

If an error is found in the category, just a FiniteCategoryError is returned. Otherwise, nothing is returned.

checkGeneratedFiniteCategoryProperties :: (GeneratedFiniteCategory c m o, Morphism m o, Eq m, Eq o) => c -> Maybe (FiniteCategoryError m o)

Same as checkFiniteCategoryProperties but for GeneratedFiniteCategory.