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Category theory/Natural transformation

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1 Example:
maybeToList

 map even $ maybeToList $ Just 5

yields the same as

 maybeToList $ map even $ Just 5

yields: both yield

 [False]

1.1 Vertical arrows: sides of objects

… showing the operation of the natural transformation.

\eta : \Phi \to \Psi
maybeToList :: Maybe a -> [a]

1.1.1 Left: side of X object

\eta_X : \Phi(X) \to \Psi(X)
maybeToList :: Maybe Int -> [Int]
Nothing
[]
Just 0
[0]
Just 1
[1]

1.1.2 Right: side of Y object

\eta_Y : \Phi(Y) \to \Psi(Y)
maybeToList :: Maybe Bool -> [Bool]
Nothing
[]
Just True
[True]
Just False
[False]

1.2 Horizontal arrows: sides of functors

f : X \to Y
 even :: Int -> Bool

1.2.1 Side of Φ functor

\Phi(f) : \Phi(X) \to \Phi(Y)
map even:: Maybe Int -> Maybe Bool
Nothing
Nothing
Just 0
Just True
Just 1
Just False

1.2.2 Side of Ψ functor

\Psi(f) : \Psi(X) \to \Psi(Y)
map even:: [Int] -> [Bool]
[]
[]
[0]
[True]
[1]
[False]

1.3 Commutativity of the diagram

\Psi(f) \cdot \eta_X = \eta_Y \cdot \Phi(f)

both paths span between

\Phi(X) \to \Psi(Y)
Maybe Int -> [Bool]
map even . maybeToList
maybeToList . map even
Nothing
[]
[]
Just 0
[True]
[True]
Just 1
[False]
[False]

1.4 Remarks

  • even
    has a more general type (
    Integral a => a -> Bool
    ) than described here
  • Words “side”, “horizontal”, “vertical”, “left”, “right” serve here only to point to the discussed parts of a diagram, thus, they are not part of the scientific terminology.