V.hs

{-# LANGUAGE MultiParamTypeClasses #-}

{-| Module  : FiniteCategories
Description : The V category contains two arrows pointing to the same object.
Copyright   : Guillaume Sabbagh 2021
License     : GPL-3
Maintainer  : guillaumesabbagh@protonmail.com
Stability   : experimental
Portability : portable

The V category contains two arrows pointing to the same object.
-}

module UsualCategories.V 
(
    VOb(..),
    VAr(..),
    V(..)
)
where
    import          FiniteCategory.FiniteCategory
    import          IO.PrettyPrint
    
    -- | Object of the V category.
    data VOb = A | B | C deriving (Eq, Show)
    
    -- | Morphism of the V category.
    data VAr = IdA | IdB | IdC | F | G deriving (Eq, Show)
    
    -- | The V category.
    data V = V deriving (Eq, Show)
    
    instance Morphism VAr VOb where
        source IdA = A
        source IdB = B
        source IdC = C
        source F = A
        source G = B
        target IdA = A
        target IdB = B
        target IdC = C
        target _ = C
        (@) IdA IdA = IdA
        (@) F IdA = F
        (@) IdB IdB = IdB
        (@) G IdB = G
        (@) IdC F = F
        (@) IdC G = G
        (@) IdC IdC = IdC
    
    instance FiniteCategory V VAr VOb where
        ob = const [A,B,C]
        identity _ A = IdA
        identity _ B = IdB
        identity _ C = IdC
        ar _ A A = [IdA]
        ar _ A C = [F]
        ar _ B B = [IdB]
        ar _ B C = [G]
        ar _ C C = [IdC]
        ar _ _ _ = []
        
    instance GeneratedFiniteCategory V VAr VOb where
        genAr = defaultGenAr
        decompose = defaultDecompose
        
    instance PrettyPrintable VOb where
        pprint = show
        
    instance PrettyPrintable VAr where
        pprint = show
    
    instance PrettyPrintable V where
        pprint = show