ConeCategory.ConeCategory

A cone is an object in the comma category (D|1_F) where D is the diagonal functor and 1_F is the diagram that selects the diagram of interest in the functor category.

A cocone is an object in the comma category (1_F|D).

Cone related functions and types.

type Cone c1 m1 o1 c2 m2 o2 = CommaObject o2 One (NaturalTransformation c1 m1 o1 c2 m2 o2)

A Cone is a CommaObject in the CommaCategory (D|1_F).

type ConeMorphism c1 m1 o1 c2 m2 o2 = CommaMorphism o2 One m2 One (NaturalTransformation c1 m1 o1 c2 m2 o2)

A ConeMorphism is a morphism between cones.

type ConeCategory c1 m1 o1 c2 m2 o2 = CommaCategory c2 m2 o2 One One One (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)

ConeCategory is the type of the cone category, it is a CommaCategory (D|1_F).

apex :: Cone c1 m1 o1 c2 m2 o2 -> o2

Returns the apex of a Cone.

coneToNaturalTransformation :: Cone c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2

Returns the Cone as a NaturalTransformation.

naturalTransformationToCone . coneToNaturalTransformation = id

coneToNaturalTransformation . naturalTransformationToCone = id

naturalTransformationToCone :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2) => NaturalTransformation c1 m1 o1 c2 m2 o2 -> Cone c1 m1 o1 c2 m2 o2

Returns a NaturalTransformation as a Cone.

naturalTransformationToCone . coneToNaturalTransformation = id

coneToNaturalTransformation . naturalTransformationToCone = id

mkConeCategory :: (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) => Diagram c1 m1 o1 c2 m2 o2 -> ConeCategory c1 m1 o1 c2 m2 o2

Constructs the category of cones of a diagram. Objects of the category are CommaObject objects with the apex of the cone in the indexSrc field and the natural transformation in the arrow field.

conesOfApex :: (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) => Diagram c1 m1 o1 c2 m2 o2 -> o2 -> [Cone c1 m1 o1 c2 m2 o2]

Returns all cones of a given apex.

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

Returns the list of terminal objects in a category.

limits :: (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) => Diagram c1 m1 o1 c2 m2 o2 -> [Cone c1 m1 o1 c2 m2 o2]

Returns limits of a diagram (terminal cones).

Cocone related functions and types.

type Cocone c1 m1 o1 c2 m2 o2 = CommaObject One o2 (NaturalTransformation c1 m1 o1 c2 m2 o2)

A Cocone is a CommaObject in the CommaCategory (1_F|D).

type CoconeMorphism c1 m1 o1 c2 m2 o2 = CommaMorphism One o2 One m2 (NaturalTransformation c1 m1 o1 c2 m2 o2)

A CoconeMorphism is a morphism between cocones.

type CoconeCategory c1 m1 o1 c2 m2 o2 = CommaCategory One One One c2 m2 o2 (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)

CoconeCategory is the type of the cocone category, it is a CommaCategory (1_F|D).

nadir :: Cocone c1 m1 o1 c2 m2 o2 -> o2

Returns the nadir of a Cocone.

coconeToNaturalTransformation :: Cocone c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2

Returns the Cocone as a NaturalTransformation.

naturalTransformationToCocone :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2) => NaturalTransformation c1 m1 o1 c2 m2 o2 -> Cocone c1 m1 o1 c2 m2 o2

Returns a NaturalTransformation as a Cocone.

naturalTransformationToCocone . coconeToNaturalTransformation = id

coconeToNaturalTransformation . naturalTransformationToCocone = id

mkCoconeCategory :: (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) => Diagram c1 m1 o1 c2 m2 o2 -> CoconeCategory c1 m1 o1 c2 m2 o2

Constructs the category of cocones of a diagram. Objects of the category are CommaObject objects with the nadir of the cone in the indexTgt field and the natural transformation in the arrow field.

coconesOfNadir :: (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) => Diagram c1 m1 o1 c2 m2 o2 -> o2 -> [Cocone c1 m1 o1 c2 m2 o2]

Returns all cocones of a given nadir.

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

Returns the list of intial objects in a category.

colimits :: (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) => Diagram c1 m1 o1 c2 m2 o2 -> [Cocone c1 m1 o1 c2 m2 o2]

Returns colimits of a diagram (initial cocones).