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

Math.Categories.FinSet

Contents

Function
FinSet

Description

The FinSet category has finite sets as objects and functions as morphisms.

Finite sets are represented by weak sets from Data.WeakSet and functions by enriched weak maps from Data.WeakMap.

These structures are homogeneous, meaning you can only have sets containing one type of objects in a given FinSet category.

See the category PureSet for the category of sets which can be arbitrarily nested.

Synopsis

data Function a = Function {
- function :: Map a a
- codomain :: Set a
}
(||!||) :: Eq a => Function a -> a -> a
data FinSet a = FinSet

Function

data Function a Source #

A Function (finite function) is a weak map enriched with a codomain.

We have to store the codomain to retrieve the target set of a morphism in FinSet.

Constructors

Function
Fields function :: Map a a codomain :: Set a

Instances

Instances details

(PrettyPrint a, Eq a) => PrettyPrint (Function a) Source #
Instance details Defined in Math.Categories.FinSet Methods pprint :: Int -> Function a -> String Source # pprintWithIndentations :: Int -> Int -> String -> Function a -> String Source # pprintIndent :: Int -> Function a -> String Source #
(Simplifiable a, Eq a) => Simplifiable (Function a) Source #
Instance details Defined in Math.Categories.FinSet Methods simplify :: Function a -> Function a #
Generic (Function a) Source #
Instance details Defined in Math.Categories.FinSet Associated Types type Rep (Function a) :: Type -> Type Methods from :: Function a -> Rep (Function a) x to :: Rep (Function a) x -> Function a
Show a => Show (Function a) Source #
Instance details Defined in Math.Categories.FinSet Methods showsPrec :: Int -> Function a -> ShowS show :: Function a -> String showList :: [Function a] -> ShowS
Eq a => Eq (Function a) Source #
Instance details Defined in Math.Categories.FinSet Methods (==) :: Function a -> Function a -> Bool (/=) :: Function a -> Function a -> Bool
Eq a => Morphism (Function a) (Set a) Source #
Instance details Defined in Math.Categories.FinSet Methods (@) :: Function a -> Function a -> Function a Source # (@?) :: Function a -> Function a -> Maybe (Function a) Source # source :: Function a -> Set a Source # target :: Function a -> Set a Source #
Eq a => Category (FinSet a) (Function a) (Set a) Source #
Instance details Defined in Math.Categories.FinSet Methods identity :: FinSet a -> Set a -> Function a Source # ar :: FinSet a -> Set a -> Set a -> Set (Function a) Source # genAr :: FinSet a -> Set a -> Set a -> Set (Function a) Source # decompose :: FinSet a -> Function a -> [Function 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 a => HasEqualizers (FinSet a) (Function a) (Set a) Source #
Instance details Defined in Math.Categories.FinSet Methods equalize :: Diagram Parallel ParallelAr ParallelOb (FinSet a) (Function a) (Set a) -> Cone Parallel ParallelAr ParallelOb (FinSet a) (Function a) (Set 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 a, Eq oIndex) => HasProducts (FinSet a) (Function a) (Set a) (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods product :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet a) (Function a) (Set a) -> Cone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit 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 #
(Eq a, Eq mIndex, Eq oIndex) => CompleteCategory (FinSet a) (Function a) (Set a) (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods limit :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Cone cIndex mIndex oIndex (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) Source # projectBase :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Diagram (FinSet a) (Function a) (Set a) (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) Source #
Eq a => CartesianClosedCategory (FinSet a) (Function a) (Set a) (FinSet (TwoProduct a)) (Function (TwoProduct a)) (Set (TwoProduct a)) (FinSet (Cartesian a)) (Function (Cartesian a)) (Set (Cartesian a)) Source #
Instance details Defined in Math.Categories.FinSet Methods internalHom :: TwoBase (FinSet a) (Function a) (Set a) -> Tripod (FinSet (Cartesian a)) (Function (Cartesian a)) (Set (Cartesian a)) Source #
type Rep (Function a) Source #
Instance details Defined in Math.Categories.FinSet type Rep (Function a) = D1 ('MetaData "Function" "Math.Categories.FinSet" "FiniteCategories-0.6.5.1-inplace" 'False) (C1 ('MetaCons "Function" 'PrefixI 'True) (S1 ('MetaSel ('Just "function") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map a a)) :*: S1 ('MetaSel ('Just "codomain") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Set a))))

(||!||) :: Eq a => Function a -> a -> a Source #

A function to apply a Function to an object in the domain of the Function.

FinSet

data FinSet a Source #

FinSet is the category of finite sets.

Constructors

FinSet

Instances

Instances details

PrettyPrint (FinSet a) Source #
Instance details Defined in Math.Categories.FinSet Methods pprint :: Int -> FinSet a -> String Source # pprintWithIndentations :: Int -> Int -> String -> FinSet a -> String Source # pprintIndent :: Int -> FinSet a -> String Source #
Simplifiable (FinSet a) Source #
Instance details Defined in Math.Categories.FinSet Methods simplify :: FinSet a -> FinSet a #
Generic (FinSet a) Source #
Instance details Defined in Math.Categories.FinSet Associated Types type Rep (FinSet a) :: Type -> Type Methods from :: FinSet a -> Rep (FinSet a) x to :: Rep (FinSet a) x -> FinSet a
Show (FinSet a) Source #
Instance details Defined in Math.Categories.FinSet Methods showsPrec :: Int -> FinSet a -> ShowS show :: FinSet a -> String showList :: [FinSet a] -> ShowS
Eq (FinSet a) Source #
Instance details Defined in Math.Categories.FinSet Methods (==) :: FinSet a -> FinSet a -> Bool (/=) :: FinSet a -> FinSet a -> Bool
Eq a => Category (FinSet a) (Function a) (Set a) Source #
Instance details Defined in Math.Categories.FinSet Methods identity :: FinSet a -> Set a -> Function a Source # ar :: FinSet a -> Set a -> Set a -> Set (Function a) Source # genAr :: FinSet a -> Set a -> Set a -> Set (Function a) Source # decompose :: FinSet a -> Function a -> [Function 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 a => HasEqualizers (FinSet a) (Function a) (Set a) Source #
Instance details Defined in Math.Categories.FinSet Methods equalize :: Diagram Parallel ParallelAr ParallelOb (FinSet a) (Function a) (Set a) -> Cone Parallel ParallelAr ParallelOb (FinSet a) (Function a) (Set 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 a, Eq oIndex) => HasProducts (FinSet a) (Function a) (Set a) (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods product :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet a) (Function a) (Set a) -> Cone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit 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 #
(Eq a, Eq mIndex, Eq oIndex) => CompleteCategory (FinSet a) (Function a) (Set a) (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) cIndex mIndex oIndex Source #
Instance details Defined in Math.Categories.FinSet Methods limit :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Cone cIndex mIndex oIndex (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) Source # projectBase :: Diagram cIndex mIndex oIndex (FinSet a) (Function a) (Set a) -> Diagram (FinSet a) (Function a) (Set a) (FinSet (Limit oIndex a)) (Function (Limit oIndex a)) (Set (Limit oIndex a)) Source #
Eq a => CartesianClosedCategory (FinSet a) (Function a) (Set a) (FinSet (TwoProduct a)) (Function (TwoProduct a)) (Set (TwoProduct a)) (FinSet (Cartesian a)) (Function (Cartesian a)) (Set (Cartesian a)) Source #
Instance details Defined in Math.Categories.FinSet Methods internalHom :: TwoBase (FinSet a) (Function a) (Set a) -> Tripod (FinSet (Cartesian a)) (Function (Cartesian a)) (Set (Cartesian a)) Source #
type Rep (FinSet a) Source #
Instance details Defined in Math.Categories.FinSet type Rep (FinSet a) = D1 ('MetaData "FinSet" "Math.Categories.FinSet" "FiniteCategories-0.6.5.1-inplace" 'False) (C1 ('MetaCons "FinSet" 'PrefixI 'False) (U1 :: Type -> Type))