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Category theory/Natural transformation

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1 Example

1 Example

map even $ maybeToList $ Just 5

yields the same as

maybeToList $ map even $ Just 5

yields: both yield

[False]

1.1 Vertical arrows

$\eta : \Phi \to \Psi$

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

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

1.2 Horizontal arrows

$f : X \to Y$

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

$\Psi(f) : \Psi(X) \to \Psi(Y)$
map even:: [Int] -> [Bool]
[]	[]
[0]	[T]rue
[1]	[F]alse

1.3 Commutativity of diagram

$\eta_Y \cdot \Phi(f) = \Psi(f) \cdot \eta_X$
	map even . maybeToList	maybeToList . map even
Nothing	[]	[]
Just 0	[True]	[True]
Just 1	[False]	[False]

Retrieved from "http://www.haskell.org/haskellwiki/Category_theory/Natural_transformation"

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