Copyright	Guillaume Sabbagh 2022
License	GPL-3
Maintainer	[email protected]
Stability	experimental
Portability	portable
Safe Haskell	Safe-Inferred
Language	Haskell2010

Math.CocompleteCategory

Contents

Colimit type
Helper typeclasses to define a CocompleteCategory
Cocomplete Category

Description

Typeclasses for Category with special properties such as being cocomplete.

A Category might have coproducts meaning that any discreteDiagram on it has a colimit.

A Category might have coequalizers meaning that any parallelDiagram on it has a colimit.

If a Category have both coproducts and coequalizers, it is cocomplete meaning that it has all small colimits.

To compute colimits in a custom FiniteCategory, see colimits in Math.ConeCategory.

Synopsis

data Colimit i t
- = Coprojection t
- | CoproductElement i t
uncoproject :: Colimit oIndex t -> t
class (Category c m o, Morphism m o, Category cLim mLim oLim, Morphism mLim oLim) => HasCoproducts c m o cLim mLim oLim oIndex | c -> m, m -> o, cLim -> mLim, mLim -> oLim, c oIndex -> cLim where
- coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex c m o -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex cLim mLim oLim
class (Category c m o, Morphism m o) => HasCoequalizers c m o | c -> m, m -> o where
- coequalize :: Diagram Parallel ParallelAr ParallelOb c m o -> Cocone Parallel ParallelAr ParallelOb c m o
colimitFromCoproductsAndCoequalizers :: (Category c m o, Morphism m o, Category cLim mLim oLim, Morphism mLim oLim, HasCoproducts c m o cLim mLim oLim oIndex, HasCoequalizers cLim mLim oLim, FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq oIndex, Eq mIndex, Eq cIndex) => (m -> mLim) -> Diagram cIndex mIndex oIndex c m o -> Cocone cIndex mIndex oIndex cLim mLim oLim
class (Category c m o, Morphism m o, Category cLim mLim oLim, Morphism mLim oLim) => CocompleteCategory c m o cLim mLim oLim cIndex mIndex oIndex | c -> m, m -> o, cLim -> mLim, mLim -> oLim, c cIndex mIndex oIndex -> cLim where
- colimit :: (FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq cIndex, Eq mIndex, Eq oIndex) => Diagram cIndex mIndex oIndex c m o -> Cocone cIndex mIndex oIndex cLim mLim oLim
- coprojectBase :: Diagram cIndex mIndex oIndex c m o -> Diagram c m o cLim mLim oLim
uncoprojectBase :: CocompleteCategory c m o cLim mLim oLim cIndex mIndex oIndex => Diagram cIndex mIndex oIndex c m o -> Diagram cLim mLim oLim c m o

Colimit type

data Colimit i t Source #

For a Category parametrized over a type t, the apex of the colimit of a diagram indexed by a category parametrized over a type i will contain Coproduct constructions. A given distinguished element can be constructed from a Coprojection at a given index.

For example, in FinSet, let's consider a discrete diagram from DiscreteTwo to (FinSet Int) which selects {1,2} and {3,4}. The nadir of the colimit is obviously {(A,1),(A,2),(B,3),(B,4)}, note that it is not an object of (Finset Int) but an object of (FinSet (DiscreteTwoOb,Int)). The Colimit type allows to construct type (FinSet (Colimit DiscreteTwo Int)) in which we can consider the original objects with Coprojection and the new distinguished elements with Coproduct. Thus, the colimit will be {Coproduct A 1,Coproduct A 2,Coproduct B 3,Coproduct B 4}. We can construct coprojections in the same category, for example along A, which will map (Coprojection 1) to Coproduct A 1.

Constructors

Coprojection t	A `Coprojection` is the parameter type of the original category to coproject them to the colimit.
CoproductElement i t	A `CoproductElement` is a couple containing an indexing object and an object of type t.

Instances

Instances details

(Eq a, Eq oIndex) => HasCoproducts (FinSet a) (Function a) (Set a) (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet a) (Function a) (Set a) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) Source #
(Eq a, Eq mIndex, Eq oIndex) => CocompleteCategory (FinSet a) (Function a) (Set a) (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods colimit :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Cocone cIndex mIndex oIndex (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Diagram (FinSet a) (Function a) (Set a) (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) Source #
(PrettyPrint i, PrettyPrint t, Eq i) => PrettyPrint (Colimit i t) Source #
Instance details Defined in Math.CocompleteCategory Methods pprint :: Int -> Colimit i t -> String Source # pprintWithIndentations :: Int -> Int -> String -> Colimit i t -> String Source # pprintIndent :: Int -> Colimit i t -> String Source #
(Simplifiable t, Simplifiable i) => Simplifiable (Colimit i t) Source #
Instance details Defined in Math.CocompleteCategory Methods simplify :: Colimit i t -> Colimit i t #
Generic (Colimit i t) Source #
Instance details Defined in Math.CocompleteCategory Associated Types type Rep (Colimit i t) :: Type -> Type Methods from :: Colimit i t -> Rep (Colimit i t) x to :: Rep (Colimit i t) x -> Colimit i t
(Show i, Show t) => Show (Colimit i t) Source #
Instance details Defined in Math.CocompleteCategory Methods showsPrec :: Int -> Colimit i t -> ShowS show :: Colimit i t -> String showList :: [Colimit i t] -> ShowS
(Eq i, Eq t) => Eq (Colimit i t) Source #
Instance details Defined in Math.CocompleteCategory Methods (==) :: Colimit i t -> Colimit i t -> Bool (/=) :: Colimit i t -> Colimit i t -> Bool
(Eq n, Eq e, Eq oIndex) => HasCoproducts (FinGrph n e) (GraphHomomorphism n e) (Graph n e) (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) oIndex Source #
Instance details Defined in Math.Categories.FinGrph Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) Source #
(Eq e, Eq n, Eq mIndex, Eq oIndex) => CocompleteCategory (FinGrph n e) (GraphHomomorphism n e) (Graph n e) (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinGrph Methods colimit :: Diagram cIndex mIndex oIndex (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Cocone cIndex mIndex oIndex (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Diagram (FinGrph n e) (GraphHomomorphism n e) (Graph n e) (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) Source #
(Eq e, Eq n, Eq oIndex, Eq mIndex, Morphism mIndex oIndex, FiniteCategory cIndex mIndex oIndex) => CocompleteCategory (FinSketch n e) (SketchMorphism n e) (Sketch n e) (FinSketch (Colimit oIndex n) (Colimit oIndex e)) (SketchMorphism (Colimit oIndex n) (Colimit oIndex e)) (Sketch (Colimit oIndex n) (Colimit oIndex e)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinSketch Methods colimit :: Diagram cIndex mIndex oIndex (FinSketch n e) (SketchMorphism n e) (Sketch n e) -> Cocone cIndex mIndex oIndex (FinSketch (Colimit oIndex n) (Colimit oIndex e)) (SketchMorphism (Colimit oIndex n) (Colimit oIndex e)) (Sketch (Colimit oIndex n) (Colimit oIndex e)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinSketch n e) (SketchMorphism n e) (Sketch n e) -> Diagram (FinSketch n e) (SketchMorphism n e) (Sketch n e) (FinSketch (Colimit oIndex n) (Colimit oIndex e)) (SketchMorphism (Colimit oIndex n) (Colimit oIndex e)) (Sketch (Colimit oIndex n) (Colimit oIndex e)) Source #
(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o, FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq oIndex, Eq mIndex) => CocompleteCategory (FinCat c m o) (FinFunctor c m o) c (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) cIndex mIndex oIndex Source #	Note that computing a `ColimitCategory` is MUCH slower than computing a `LimitCategory` as coequalizers of categories are trickier than equalizers of categories. We can only compute colimits of `CompositionGraph`s as coequalizing might create new formal morphisms when gluing two objects. Therefore `colimit` transforms all given categories into `CompositionGraph`s. If you already have a `CompositionGraph`, consider using `colimitOfCompositionGraphs` instead of `colimit`.
Instance details Defined in Math.FiniteCategories.ColimitCategory Methods colimit :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Cocone cIndex mIndex oIndex (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Diagram (FinCat c m o) (FinFunctor c m o) c (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) Source #
(Eq e, Eq n, Eq oIndex) => HasCoproducts (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) (FinCat (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (FinFunctor (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) oIndex Source #
Instance details Defined in Math.FiniteCategories.ColimitCategory Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinCat (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (FinFunctor (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) Source #
type Rep (Colimit i t) Source #
Instance details Defined in Math.CocompleteCategory type Rep (Colimit i t) = D1 ('MetaData "Colimit" "Math.CocompleteCategory" "FiniteCategories-0.6.5.1-inplace" 'False) (C1 ('MetaCons "Coprojection" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 t)) :+: C1 ('MetaCons "CoproductElement" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 i) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 t)))

uncoproject :: Colimit oIndex t -> t Source #

Remove the constructor Coprojection from an original object t, throws an error if a CoproductElement is given.

Helper typeclasses to define a `CocompleteCategory`

class (Category c m o, Morphism m o, Category cLim mLim oLim, Morphism mLim oLim) => HasCoproducts c m o cLim mLim oLim oIndex | c -> m, m -> o, cLim -> mLim, mLim -> oLim, c oIndex -> cLim where Source #

The typeclass of categories which have all coproducts.

Methods

coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex c m o -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex cLim mLim oLim Source #

Given a discreteDiagram selecting objects, return the coproduct of the selected objects as a Cocone.

Instances

Instances details

(Enum a, Ord a, Eq oIndex) => HasCoproducts (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a oIndex Source #
Instance details Defined in Math.Categories.OrdinalCategory Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (OrdinalCategory a) (IsSmallerThan a) a Source #
(Ord a, Eq oIndex) => HasCoproducts (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a oIndex Source #
Instance details Defined in Math.Categories.TotalOrder Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (TotalOrder a) (IsSmallerThan a) a -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (TotalOrder a) (IsSmallerThan a) a Source #
(Eq a, Eq oIndex) => HasCoproducts (FinSet a) (Function a) (Set a) (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet a) (Function a) (Set a) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) Source #
(Eq n, Eq e, Eq oIndex) => HasCoproducts (FinGrph n e) (GraphHomomorphism n e) (Graph n e) (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) oIndex Source #
Instance details Defined in Math.Categories.FinGrph Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) Source #
(Eq e, Eq n, Eq oIndex) => HasCoproducts (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) (FinCat (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (FinFunctor (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) oIndex Source #
Instance details Defined in Math.FiniteCategories.ColimitCategory Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinCat (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (FinFunctor (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) Source #

class (Category c m o, Morphism m o) => HasCoequalizers c m o | c -> m, m -> o where Source #

The typeclass of categories which have all coequalizers.

Methods

coequalize :: Diagram Parallel ParallelAr ParallelOb c m o -> Cocone Parallel ParallelAr ParallelOb c m o Source #

Given a parallelDiagram selecting arrows, return the coequalizer of the selected arrows as a Cocone.

Instances

Instances details

(Enum a, Ord a) => HasCoequalizers (OrdinalCategory a) (IsSmallerThan a) a Source #
Instance details Defined in Math.Categories.OrdinalCategory Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (OrdinalCategory a) (IsSmallerThan a) a -> Cocone Parallel ParallelAr ParallelOb (OrdinalCategory a) (IsSmallerThan a) a Source #
Ord a => HasCoequalizers (TotalOrder a) (IsSmallerThan a) a Source #
Instance details Defined in Math.Categories.TotalOrder Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (TotalOrder a) (IsSmallerThan a) a -> Cocone Parallel ParallelAr ParallelOb (TotalOrder a) (IsSmallerThan a) a Source #
Eq a => HasCoequalizers (FinSet a) (Function a) (Set a) Source #
Instance details Defined in Math.Categories.FinSet Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (FinSet a) (Function a) (Set a) -> Cocone Parallel ParallelAr ParallelOb (FinSet a) (Function a) (Set a) Source #
(Eq e, Eq n) => HasCoequalizers (FinGrph n e) (GraphHomomorphism n e) (Graph n e) Source #	BEWARE, for the coequalizer to be correct, ALL arrow labels should be different (two arrows with different source and target might have the same source and target after the coequalization process).
Instance details Defined in Math.Categories.FinGrph Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Cocone Parallel ParallelAr ParallelOb (FinGrph n e) (GraphHomomorphism n e) (Graph n e) Source #
(Eq e, Eq n) => HasCoequalizers (FinSketch n e) (SketchMorphism n e) (Sketch n e) Source #
Instance details Defined in Math.Categories.FinSketch Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (FinSketch n e) (SketchMorphism n e) (Sketch n e) -> Cocone Parallel ParallelAr ParallelOb (FinSketch n e) (SketchMorphism n e) (Sketch n e) Source #
(Eq n, Eq e) => HasCoequalizers (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) Source #
Instance details Defined in Math.FiniteCategories.ColimitCategory Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone Parallel ParallelAr ParallelOb (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) Source #

colimitFromCoproductsAndCoequalizers :: (Category c m o, Morphism m o, Category cLim mLim oLim, Morphism mLim oLim, HasCoproducts c m o cLim mLim oLim oIndex, HasCoequalizers cLim mLim oLim, FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq oIndex, Eq mIndex, Eq cIndex) => (m -> mLim) -> Diagram cIndex mIndex oIndex c m o -> Cocone cIndex mIndex oIndex cLim mLim oLim Source #

Computes efficiently colimits thanks to coproducts and coequalizers. Can be used to instantiate CocompleteCategory.

The first argument is a function to transform an object of the original category into a colimit object.

Most of the time, the original category takes one type parameter and the function uses Coprojection, when the category does not take any type parameter it is id.

For example, for FinSet, the function has to transform a function {1 -> 2} to the function {Coprojection 1 -> Coprojection 2}.

Cocomplete Category

class (Category c m o, Morphism m o, Category cLim mLim oLim, Morphism mLim oLim) => CocompleteCategory c m o cLim mLim oLim cIndex mIndex oIndex | c -> m, m -> o, cLim -> mLim, mLim -> oLim, c cIndex mIndex oIndex -> cLim where Source #

The typeclass of categories which have all colimits.

Methods

colimit :: (FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq cIndex, Eq mIndex, Eq oIndex) => Diagram cIndex mIndex oIndex c m o -> Cocone cIndex mIndex oIndex cLim mLim oLim Source #

Return the colimit of a Diagram.

coprojectBase :: Diagram cIndex mIndex oIndex c m o -> Diagram c m o cLim mLim oLim Source #

A partial Diagram to coproject objects and morphisms of a base like a call of colimit would do.

(coconeBase (colimit diag)) should equal ((coprojectBase diag) <-@<- diag)

Instances

Instances details

(Enum a, Ord a, Eq mIndex, Eq oIndex) => CocompleteCategory (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.OrdinalCategory Methods colimit :: Diagram cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Cocone cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a Source # coprojectBase :: Diagram cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Diagram (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a Source #
(Ord a, Eq mIndex, Eq oIndex) => CocompleteCategory (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.TotalOrder Methods colimit :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Cocone cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a Source # coprojectBase :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Diagram (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a Source #
(Eq a, Eq mIndex, Eq oIndex) => CocompleteCategory (FinSet a) (Function a) (Set a) (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods colimit :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Cocone cIndex mIndex oIndex (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Diagram (FinSet a) (Function a) (Set a) (FinSet (Colimit oIndex a)) (Function (Colimit oIndex a)) (Set (Colimit oIndex a)) Source #
(Eq e, Eq n, Eq mIndex, Eq oIndex) => CocompleteCategory (FinGrph n e) (GraphHomomorphism n e) (Graph n e) (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinGrph Methods colimit :: Diagram cIndex mIndex oIndex (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Cocone cIndex mIndex oIndex (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinGrph n e) (GraphHomomorphism n e) (Graph n e) -> Diagram (FinGrph n e) (GraphHomomorphism n e) (Graph n e) (FinGrph (Colimit oIndex n) (Colimit oIndex e)) (GraphHomomorphism (Colimit oIndex n) (Colimit oIndex e)) (Graph (Colimit oIndex n) (Colimit oIndex e)) Source #
(Eq e, Eq n, Eq oIndex, Eq mIndex, Morphism mIndex oIndex, FiniteCategory cIndex mIndex oIndex) => CocompleteCategory (FinSketch n e) (SketchMorphism n e) (Sketch n e) (FinSketch (Colimit oIndex n) (Colimit oIndex e)) (SketchMorphism (Colimit oIndex n) (Colimit oIndex e)) (Sketch (Colimit oIndex n) (Colimit oIndex e)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinSketch Methods colimit :: Diagram cIndex mIndex oIndex (FinSketch n e) (SketchMorphism n e) (Sketch n e) -> Cocone cIndex mIndex oIndex (FinSketch (Colimit oIndex n) (Colimit oIndex e)) (SketchMorphism (Colimit oIndex n) (Colimit oIndex e)) (Sketch (Colimit oIndex n) (Colimit oIndex e)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinSketch n e) (SketchMorphism n e) (Sketch n e) -> Diagram (FinSketch n e) (SketchMorphism n e) (Sketch n e) (FinSketch (Colimit oIndex n) (Colimit oIndex e)) (SketchMorphism (Colimit oIndex n) (Colimit oIndex e)) (Sketch (Colimit oIndex n) (Colimit oIndex e)) Source #
(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o, FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq oIndex, Eq mIndex) => CocompleteCategory (FinCat c m o) (FinFunctor c m o) c (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) cIndex mIndex oIndex Source #	Note that computing a `ColimitCategory` is MUCH slower than computing a `LimitCategory` as coequalizers of categories are trickier than equalizers of categories. We can only compute colimits of `CompositionGraph`s as coequalizing might create new formal morphisms when gluing two objects. Therefore `colimit` transforms all given categories into `CompositionGraph`s. If you already have a `CompositionGraph`, consider using `colimitOfCompositionGraphs` instead of `colimit`.
Instance details Defined in Math.FiniteCategories.ColimitCategory Methods colimit :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Cocone cIndex mIndex oIndex (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Diagram (FinCat c m o) (FinFunctor c m o) c (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) Source #

uncoprojectBase :: CocompleteCategory c m o cLim mLim oLim cIndex mIndex oIndex => Diagram cIndex mIndex oIndex c m o -> Diagram cLim mLim oLim c m o Source #

A partial Diagram to uncoproject objects and morphisms of the base in a colimit cocone.

uncoprojectBase and coprojectBase returned Diagrams are inverses.

Colimit type

Instances

Helper typeclasses to define a CocompleteCategory

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Cocomplete Category

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Helper typeclasses to define a `CocompleteCategory`