Math.Categories.TotalOrder

Any total (or linear) order induces a preorder category where elements are objects, there is an arrow between two objects iff the relation is satisfied.

(See Categories for the working mathematican. Saunders Mac Lane. p.11)

Documentation

data IsSmallerThan a

IsSmallerThan is the type of morphisms in a linear order, it reminds the fact that there is a morphism from a source to a target iff the source is smaller than the target.

Constructors

IsSmallerThan a a

Instances

PrettyPrint a => PrettyPrint (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

pprint :: Int -> IsSmallerThan a -> String

pprintWithIndentations :: Int -> Int -> String -> IsSmallerThan a -> String

pprintIndent :: Int -> IsSmallerThan a -> String

Simplifiable a => Simplifiable (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

simplify :: IsSmallerThan a -> IsSmallerThan a

Generic (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Associated Types

type Rep (IsSmallerThan a) :: Type -> Type

Methods

from :: IsSmallerThan a -> Rep (IsSmallerThan a) x

to :: Rep (IsSmallerThan a) x -> IsSmallerThan a

Show a => Show (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

showsPrec :: Int -> IsSmallerThan a -> ShowS

show :: IsSmallerThan a -> String

showList :: [IsSmallerThan a] -> ShowS

Eq a => Eq (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

(==) :: IsSmallerThan a -> IsSmallerThan a -> Bool

(/=) :: IsSmallerThan a -> IsSmallerThan a -> Bool

Eq a => Morphism (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

(@) :: IsSmallerThan a -> IsSmallerThan a -> IsSmallerThan a

(@?) :: IsSmallerThan a -> IsSmallerThan a -> Maybe (IsSmallerThan a)

source :: IsSmallerThan a -> a

target :: IsSmallerThan a -> a

(Enum a, Ord a) => Category (OrdinalCategory a) (IsSmallerThan a) a

Defined in Math.Categories.OrdinalCategory

Methods

identity :: OrdinalCategory a -> a -> IsSmallerThan a

ar :: OrdinalCategory a -> a -> a -> Set (IsSmallerThan a)

genAr :: OrdinalCategory a -> a -> a -> Set (IsSmallerThan a)

decompose :: OrdinalCategory a -> IsSmallerThan a -> [IsSmallerThan a]

(Eq a, Ord a) => Category (TotalOrder a) (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

identity :: TotalOrder a -> a -> IsSmallerThan a

ar :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

genAr :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

decompose :: TotalOrder a -> IsSmallerThan a -> [IsSmallerThan a]

(Enum a, Ord a) => HasCoequalizers (OrdinalCategory a) (IsSmallerThan a) a

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

Ord a => HasCoequalizers (TotalOrder a) (IsSmallerThan a) a

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

(Enum a, Ord a) => HasEqualizers (OrdinalCategory a) (IsSmallerThan a) a

Defined in Math.Categories.OrdinalCategory

Methods

equalize :: Diagram Parallel ParallelAr ParallelOb (OrdinalCategory a) (IsSmallerThan a) a -> Cone Parallel ParallelAr ParallelOb (OrdinalCategory a) (IsSmallerThan a) a

Ord a => HasEqualizers (TotalOrder a) (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

equalize :: Diagram Parallel ParallelAr ParallelOb (TotalOrder a) (IsSmallerThan a) a -> Cone Parallel ParallelAr ParallelOb (TotalOrder a) (IsSmallerThan a) a

(Enum a, Ord a, Eq oIndex) => HasCoproducts (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a oIndex

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

(Ord a, Eq oIndex) => HasCoproducts (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a oIndex

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

(Enum a, Ord a, Eq oIndex) => HasProducts (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a oIndex

Defined in Math.Categories.OrdinalCategory

Methods

product :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Cone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (OrdinalCategory a) (IsSmallerThan a) a

(Ord a, Eq oIndex) => HasProducts (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a oIndex

Defined in Math.Categories.TotalOrder

Methods

product :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (TotalOrder a) (IsSmallerThan a) a -> Cone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (TotalOrder a) (IsSmallerThan a) a

(Enum a, Ord a, Eq mIndex, Eq oIndex) => CocompleteCategory (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a cIndex mIndex oIndex

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

coprojectBase :: Diagram cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Diagram (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a

(Ord a, Eq mIndex, Eq oIndex) => CocompleteCategory (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a cIndex mIndex oIndex

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

coprojectBase :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Diagram (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a

(Enum a, Ord a, Eq mIndex, Eq oIndex) => CompleteCategory (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a cIndex mIndex oIndex

Defined in Math.Categories.OrdinalCategory

Methods

limit :: Diagram cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Cone cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a

projectBase :: Diagram cIndex mIndex oIndex (OrdinalCategory a) (IsSmallerThan a) a -> Diagram (OrdinalCategory a) (IsSmallerThan a) a (OrdinalCategory a) (IsSmallerThan a) a

(Ord a, Eq mIndex, Eq oIndex) => CompleteCategory (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a cIndex mIndex oIndex

Defined in Math.Categories.TotalOrder

Methods

limit :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Cone cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a

projectBase :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Diagram (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a

type Rep (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

type Rep (IsSmallerThan a) = D1 ('MetaData "IsSmallerThan" "Math.Categories.TotalOrder" "FiniteCategories-0.6.4.0-inplace" 'False) (C1 ('MetaCons "IsSmallerThan" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)))

data TotalOrder a

A TotalOrder category is the category induced by a total order.

(See Categories for the working mathematican. Saunders Mac Lane. p.11)

Constructors

TotalOrder

Instances

PrettyPrint (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

pprint :: Int -> TotalOrder a -> String

pprintWithIndentations :: Int -> Int -> String -> TotalOrder a -> String

pprintIndent :: Int -> TotalOrder a -> String

Simplifiable (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

simplify :: TotalOrder a -> TotalOrder a

Generic (TotalOrder a)

Defined in Math.Categories.TotalOrder

Associated Types

type Rep (TotalOrder a) :: Type -> Type

Methods

from :: TotalOrder a -> Rep (TotalOrder a) x

to :: Rep (TotalOrder a) x -> TotalOrder a

Show (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

showsPrec :: Int -> TotalOrder a -> ShowS

show :: TotalOrder a -> String

showList :: [TotalOrder a] -> ShowS

Eq (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

(==) :: TotalOrder a -> TotalOrder a -> Bool

(/=) :: TotalOrder a -> TotalOrder a -> Bool

(Eq a, Ord a) => Category (TotalOrder a) (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

identity :: TotalOrder a -> a -> IsSmallerThan a

ar :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

genAr :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

decompose :: TotalOrder a -> IsSmallerThan a -> [IsSmallerThan a]

Ord a => HasCoequalizers (TotalOrder a) (IsSmallerThan a) a

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

Ord a => HasEqualizers (TotalOrder a) (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

equalize :: Diagram Parallel ParallelAr ParallelOb (TotalOrder a) (IsSmallerThan a) a -> Cone Parallel ParallelAr ParallelOb (TotalOrder a) (IsSmallerThan a) a

(Ord a, Eq oIndex) => HasCoproducts (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a oIndex

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

(Ord a, Eq oIndex) => HasProducts (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a oIndex

Defined in Math.Categories.TotalOrder

Methods

product :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (TotalOrder a) (IsSmallerThan a) a -> Cone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (TotalOrder a) (IsSmallerThan a) a

(Ord a, Eq mIndex, Eq oIndex) => CocompleteCategory (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a cIndex mIndex oIndex

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

coprojectBase :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Diagram (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a

(Ord a, Eq mIndex, Eq oIndex) => CompleteCategory (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a cIndex mIndex oIndex

Defined in Math.Categories.TotalOrder

Methods

limit :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Cone cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a

projectBase :: Diagram cIndex mIndex oIndex (TotalOrder a) (IsSmallerThan a) a -> Diagram (TotalOrder a) (IsSmallerThan a) a (TotalOrder a) (IsSmallerThan a) a

type Rep (TotalOrder a)

Defined in Math.Categories.TotalOrder

type Rep (TotalOrder a) = D1 ('MetaData "TotalOrder" "Math.Categories.TotalOrder" "FiniteCategories-0.6.4.0-inplace" 'False) (C1 ('MetaCons "TotalOrder" 'PrefixI 'False) (U1 :: Type -> Type))