data-category-0.6.2: Category theory

LicenseBSD-style (see the file LICENSE)
Maintainersjoerd@w3future.com
Stabilityexperimental
Portabilitynon-portable
Safe HaskellSafe
LanguageHaskell2010

Data.Category.Boolean

Description

2, or the Boolean category. It contains 2 objects, one for true and one for false. It contains 3 arrows, 2 identity arrows and one from false to true.

Synopsis

Documentation

data Fls Source #

Instances

data Tru Source #

Instances

data Boolean a b where Source #

Constructors

Fls :: Boolean Fls Fls 
F2T :: Boolean Fls Tru 
Tru :: Boolean Tru Tru 

Instances

Category Boolean Source #

Boolean is the category with true and false as objects, and an arrow from false to true.

Methods

src :: Boolean a b -> Obj Boolean a Source #

tgt :: Boolean a b -> Obj Boolean b Source #

(.) :: Boolean b c -> Boolean a b -> Boolean a c Source #

HasBinaryCoproducts Boolean Source #

Disjunction is the binary coproduct in the Boolean category.

Associated Types

type BinaryCoproduct (Boolean :: * -> * -> *) x y :: * Source #

Methods

inj1 :: Obj Boolean x -> Obj Boolean y -> Boolean x (BinaryCoproduct Boolean x y) Source #

inj2 :: Obj Boolean x -> Obj Boolean y -> Boolean y (BinaryCoproduct Boolean x y) Source #

(|||) :: Boolean x a -> Boolean y a -> Boolean (BinaryCoproduct Boolean x y) a Source #

(+++) :: Boolean a1 b1 -> Boolean a2 b2 -> Boolean (BinaryCoproduct Boolean a1 a2) (BinaryCoproduct Boolean b1 b2) Source #

HasBinaryProducts Boolean Source #

Conjunction is the binary product in the Boolean category.

Associated Types

type BinaryProduct (Boolean :: * -> * -> *) x y :: * Source #

Methods

proj1 :: Obj Boolean x -> Obj Boolean y -> Boolean (BinaryProduct Boolean x y) x Source #

proj2 :: Obj Boolean x -> Obj Boolean y -> Boolean (BinaryProduct Boolean x y) y Source #

(&&&) :: Boolean a x -> Boolean a y -> Boolean a (BinaryProduct Boolean x y) Source #

(***) :: Boolean a1 b1 -> Boolean a2 b2 -> Boolean (BinaryProduct Boolean a1 a2) (BinaryProduct Boolean b1 b2) Source #

HasInitialObject Boolean Source #

False is the initial object in the Boolean category.

HasTerminalObject Boolean Source #

True is the terminal object in the Boolean category.

CartesianClosed Boolean Source #

Implication makes the Boolean category cartesian closed.

Associated Types

type Exponential (Boolean :: * -> * -> *) y z :: * Source #

Methods

apply :: Obj Boolean y -> Obj Boolean z -> Boolean (BinaryProduct Boolean (Exponential Boolean y z) y) z Source #

tuple :: Obj Boolean y -> Obj Boolean z -> Boolean z (Exponential Boolean y (BinaryProduct Boolean z y)) Source #

(^^^) :: Boolean z1 z2 -> Boolean y2 y1 -> Boolean (Exponential Boolean y1 z1) (Exponential Boolean y2 z2) Source #

type InitialObject Boolean Source # 
type TerminalObject Boolean Source # 
type BinaryCoproduct Boolean Tru Tru Source # 
type BinaryCoproduct Boolean Tru Fls Source # 
type BinaryCoproduct Boolean Fls Tru Source # 
type BinaryCoproduct Boolean Fls Fls Source # 
type BinaryProduct Boolean Tru Tru Source # 
type BinaryProduct Boolean Tru Fls Source # 
type BinaryProduct Boolean Fls Tru Source # 
type BinaryProduct Boolean Fls Fls Source # 
type Exponential Boolean Tru Tru Source # 
type Exponential Boolean Tru Fls Source # 
type Exponential Boolean Fls Tru Source # 
type Exponential Boolean Fls Fls Source # 

trueProductMonoid :: MonoidObject (ProductFunctor Boolean) Tru Source #

falseCoproductComonoid :: ComonoidObject (CoproductFunctor Boolean) Fls Source #

trueProductComonoid :: ComonoidObject (ProductFunctor Boolean) Tru Source #

falseCoproductMonoid :: MonoidObject (CoproductFunctor Boolean) Fls Source #

trueCoproductMonoid :: MonoidObject (CoproductFunctor Boolean) Tru Source #

falseProductComonoid :: ComonoidObject (ProductFunctor Boolean) Fls Source #