Category theory/Natural transformation

From HaskellWiki

1 Example:

maybeToList

map even $ maybeToList $ Just 5

yields the same as

maybeToList $ fmap even $ Just 5

yields: both yield

[False]

1.1 Commutative diagram

Let , denote categories. Let be functors. Let us define the natural transformation.

............

1.2 Vertical arrows: sides of objects

… showing how the natural transformation works.

maybeToList :: Maybe a -> [a]

1.2.1 Left: side of X object

	maybeToList :: Maybe Int -> [Int]
Nothing	[]
Just 0	[0]
Just 1	[1]

1.2.2 Right: side of Y object

	maybeToList :: Maybe Bool -> [Bool]
Nothing	[]
Just True	[True]
Just False	[False]

1.3 Horizontal arrows: sides of functors

even :: Int -> Bool

1.3.1 Side of Φ functor

	fmap even:: Maybe Int -> Maybe Bool
Nothing	Nothing
Just 0	Just True
Just 1	Just False

1.3.2 Side of Ψ functor

	map even:: [Int] -> [Bool]
[]	[]
[0]	[True]
[1]	[False]

1.4 Commutativity of the diagram

both paths span between

1.5 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.
• If You want to modifiy the #Commutative diagram, see its source code (in LaTeX using amscd).

1.6 External links

• The corresponding HaWiki article is not migrated here yet, so You can see it for more information.
• Wikipedia's Natural transformation article