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Category theory/Natural transformation

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spouse :: Parser (Maybe String)
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:<math>\Lambda\eta : \Lambda\Phi \to \Lambda\Psi</math>
:<math>\Lambda\eta : \Lambda\Phi \to \Lambda\Psi</math>
:<math>(\Lambda\eta)_X = \Lambda(\eta_X)</math>
:<math>(\Lambda\eta)_X = \Lambda(\eta_X)</math>
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=== External links ===
=== External links ===
* [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.

Revision as of 09:12, 4 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).

2 Operations

2.1 Functor and natural transformation

Let us imagine a parser library, which contains functions for parsing a form. There are two kinds of cells:

  • containing data which are optional (e.g. name of spouse)
  • containing data which consist of an enumaration of items (e.g. names of acquired languages) 
spouse :: Parser (Maybe String)
 languages :: Parser [String]

Let us imagine we have any processing (storing, archiving etc.) function which processes lists (or any other reason which forces us to convert our results tu list fomrat instead of lists). (Maybe all this example is unparactical and exaggerated!)

We can convert
Maybe
to list with
maybeToList
But if we want to do something similar with a parser on Maybe's to achieve a parser on list, then
maybeToList
is not enough alone, we must
fmap
it. E.g. if we want to convert a parser like
spouse
to be of the same type as
languages
: 
fmap maybeToList spouse

Let us see the types: We start with 

spouse :: Parser (Maybe String)
Λ(Φ(X))

or using notion of composing functors

(ΛΦ)(X)

We want to achieve 

fmap maybeToList spouse :: Parser [String]
Λ(Ψ(X))
(ΛΨ)(X)

thus we can infer 

fmap maybeToList :: Parser (Maybe [String]) -> Parser [String]
\mathrm{Hom}{(\Lambda\Phi)(X),\;(\Lambda\Psi)(X)}

In fact, we have a new "datatype converter": converting not Maybe's to lists, but parser on Maybe to Parser on list. Let us notate the corresponding natural transformation with Λη:

To each X \in \mathbf{Ob}(\mathcal C) we associate (\Lambda\eta)_X \in \mathrm{Hom}_{\mathcal D}(\Lambda\Phi)(X),\;(\Lambda\Psi)(X))
\Lambda\eta : \Lambda\Phi \to \Lambda\Psi
(Λη)X = Λ(ηX)

2.2 External links

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

Category: Theoretical foundations