Personal tools

Category theory/Natural transformation

From HaskellWiki

(Difference between revisions)
Jump to: navigation, search
m (Commutative diagram is in D category)
(Categorizing under Category:Theoretical foundations)
Line 142: Line 142:
* [http://haskell.org/hawiki/CategoryTheory_2fNaturalTransformation?action=highlight&value=natural+transformation The corresponding HaWiki article] is not migrated here yet, so You can see it for more information.
* [http://haskell.org/hawiki/CategoryTheory_2fNaturalTransformation?action=highlight&value=natural+transformation The corresponding HaWiki article] is not migrated here yet, so You can see it for more information.
* Wikipedia's [http://en.wikipedia.org/wiki/Natural_transformation Natural transformation] article
* Wikipedia's [http://en.wikipedia.org/wiki/Natural_transformation Natural transformation] article
 +
 +
[[Category:Theoretical foundations]]

Revision as of 16:23, 3 October 2006

Contents

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 \mathcal C, \mathcal D denote categories.
  • Let \Phi, \Psi : \mathcal C \to \mathcal D be functors.
  • Let X, Y \in \mathbf{Ob}(\mathcal C). Let f \in \mathrm{Hom}_{\mathcal C}(X, Y).

Let us define the \eta : \Phi \to \Psi natural transformation. It associates to each object of \mathcal{C} a morphism of \mathcal{D} in the following way (usually, not sets are discussed here, but proper classes, so I do not use term “function” for this \mathbf{Ob}(\mathcal C) \to \mathbf{Mor}(\mathcal D) mapping):

  • \forall A \in \mathbf{Ob}(\mathcal C) \longmapsto \eta_A \in \mathrm{Hom}_{\mathcal D}(\Phi(A), \Psi(A)). We call ηA the component of η at A.
  • \eta_Y \cdot \Phi(f) = \Psi(f) \cdot \eta_X

Thus, the following diagram commutes (in \mathcal D):

1.2 Vertical arrows: sides of objects

… showing how the natural transformation works.

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

1.2.1 Left: side of X object

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

1.2.2 Right: side of Y object

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

1.3 Horizontal arrows: sides of functors

f : X \to Y 
even :: Int -> Bool

1.3.1 Side of Φ functor

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

1.3.2 Side of Ψ functor

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

1.4 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 . fmap even
Nothing
[]
[]
Just 0
[True]
[True]
Just 1
[False]
[False]

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

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

Category: Theoretical foundations