Personal tools

Category theory/Natural transformation

From HaskellWiki

< Category theory(Difference between revisions)
Jump to: navigation, search
m (Better section structure)
(In fact, even has a more general type (Integral a => a -> Bool) than described here)
Line 16: Line 16:
   
 
:<math>\eta : \Phi \to \Psi</math>
 
:<math>\eta : \Phi \to \Psi</math>
  +
<haskell>maybeToList :: Maybe a -> [a]</haskell>
   
 
{| Border=2 CellPadding=2 CellSpacing=2 | Dia
 
{| Border=2 CellPadding=2 CellSpacing=2 | Dia
Line 48: Line 49:
   
 
=== Horizontal arrows ===
 
=== Horizontal arrows ===
  +
  +
/Note: <hask>even</hask> has a more general type (<hask>Integral a => a -> Bool</hask>) than described here/
   
 
:<math>f : X \to Y</math>
 
:<math>f : X \to Y</math>
  +
<haskell>
  +
even :: Int -> Bool
  +
</haskell>
   
 
{| Border=2 CellPadding=2 CellSpacing=2
 
{| Border=2 CellPadding=2 CellSpacing=2

Revision as of 19:21, 2 October 2006

Contents

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

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

/Note:
even
has a more general type (
Integral a => a -> Bool
) than described here/
f : X \to Y
 even :: Int -> Bool
\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]