Math.Categories.FinSet

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.

Function

data Function a

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

Eq a => Eq (Function a)

Defined in Math.Categories.FinSet

Methods

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

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

Show a => Show (Function a)

Defined in Math.Categories.FinSet

Methods

showsPrec :: Int -> Function a -> ShowS

show :: Function a -> String

showList :: [Function a] -> ShowS

(PrettyPrint a, Eq a) => PrettyPrint (Function a)

Defined in Math.Categories.FinSet

Methods

pprint :: Function a -> String

Eq a => Morphism (Function a) (Set a)

Defined in Math.Categories.FinSet

Methods

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

source :: Function a -> Set a

target :: Function a -> Set a

Eq a => Category (FinSet a) (Function a) (Set a)

Defined in Math.Categories.FinSet

Methods

identity :: FinSet a -> Set a -> Function a

ar :: FinSet a -> Set a -> Set a -> Set (Function a)

genAr :: FinSet a -> Set a -> Set a -> Set (Function a)

decompose :: FinSet a -> Function a -> [Function a]

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

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

FinSet

data FinSet a

FinSet is the category of finite sets.

Constructors

FinSet

Instances

Eq (FinSet a)

Defined in Math.Categories.FinSet

Methods

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

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

Show (FinSet a)

Defined in Math.Categories.FinSet

Methods

showsPrec :: Int -> FinSet a -> ShowS

show :: FinSet a -> String

showList :: [FinSet a] -> ShowS

(PrettyPrint a, Eq a) => PrettyPrint (FinSet a)

Defined in Math.Categories.FinSet

Methods

pprint :: FinSet a -> String

Eq a => Category (FinSet a) (Function a) (Set a)

Defined in Math.Categories.FinSet

Methods

identity :: FinSet a -> Set a -> Function a

ar :: FinSet a -> Set a -> Set a -> Set (Function a)

genAr :: FinSet a -> Set a -> Set a -> Set (Function a)

decompose :: FinSet a -> Function a -> [Function a]