# Category theory/Natural transformation

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− | == Example == | + | <haskell> |

+ | map even $ maybeToList $ Just 5 | ||

+ | </haskell> | ||

+ | yields the same as | ||

+ | <haskell> | ||

+ | maybeToList $ fmap even $ Just 5 | ||

+ | </haskell> | ||

+ | yields: both yield | ||

+ | <haskell> | ||

+ | [False] | ||

+ | </haskell> | ||

+ | |||

+ | In the followings, this example will be used to illustrate the notion of natural transformation. If the examples are exaggerated and/or the definitions are incomprehensible, try [[#External links]]. | ||

+ | |||

+ | == Definition == | ||

+ | |||

+ | * Let <math>\mathcal C</math>, <math>\mathcal D</math> denote categories. | ||

+ | * Let <math>\Phi, \Psi : \mathcal C \to \mathcal D</math> be functors. | ||

+ | * Let <math>X, Y \in \mathbf{Ob}(\mathcal C)</math>. Let <math>f \in \mathrm{Hom}_{\mathcal C}(X, Y)</math>. | ||

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

+ | * <math>\forall A \in \mathbf{Ob}(\mathcal C) \longmapsto \eta_A \in \mathrm{Hom}_{\mathcal D}(\Phi(A), \Psi(A))</math>. We call <math>\eta_A</math> the component of <math>\eta</math> at ''A''. | ||

+ | * <math>\eta_Y \cdot \Phi(f) = \Psi(f) \cdot \eta_X</math> | ||

+ | Thus, the following diagram commutes (in <math>\mathcal D</math>): | ||

+ | |||

+ | [[Image:natural_transformation.png|center]] | ||

+ | |||

+ | == Example: <hask>maybeToList</hask> == | ||

+ | As already mentioned | ||

<haskell> | <haskell> | ||

map even $ maybeToList $ Just 5 | map even $ maybeToList $ Just 5 | ||

Line 6: | Line 33: | ||

yields the same as | yields the same as | ||

<haskell> | <haskell> | ||

− | maybeToList $ | + | maybeToList $ fmap even $ Just 5 |

</haskell> | </haskell> | ||

yields: both yield | yields: both yield | ||

Line 12: | Line 39: | ||

[False] | [False] | ||

</haskell> | </haskell> | ||

+ | This example will be shown in the light of the above definition in the followings. | ||

− | === Vertical arrows === | + | === Vertical arrows: sides of objects === |

+ | |||

+ | … showing how the natural transformation works. | ||

:<math>\eta : \Phi \to \Psi</math> | :<math>\eta : \Phi \to \Psi</math> | ||

+ | <haskell>maybeToList :: Maybe a -> [a]</haskell> | ||

+ | |||

+ | ==== Left: side of ''X'' object ==== | ||

{| Border=2 CellPadding=2 CellSpacing=2 | Dia | {| Border=2 CellPadding=2 CellSpacing=2 | Dia | ||

Line 31: | Line 64: | ||

| <hask>[1]</hask> | | <hask>[1]</hask> | ||

|} | |} | ||

+ | |||

+ | ==== Right: side of ''Y'' object ==== | ||

{| Border=2 CellPadding=2 CellSpacing=2 | Dia | {| Border=2 CellPadding=2 CellSpacing=2 | Dia | ||

Line 47: | Line 82: | ||

|} | |} | ||

− | === Horizontal arrows === | + | === Horizontal arrows: sides of functors === |

:<math>f : X \to Y</math> | :<math>f : X \to Y</math> | ||

+ | <haskell> | ||

+ | even :: Int -> Bool | ||

+ | </haskell> | ||

+ | |||

+ | ==== Side of <math>\Phi</math> functor ==== | ||

{| Border=2 CellPadding=2 CellSpacing=2 | {| Border=2 CellPadding=2 CellSpacing=2 | ||

|+ <math>\Phi(f) : \Phi(X) \to \Phi(Y)</math> | |+ <math>\Phi(f) : \Phi(X) \to \Phi(Y)</math> | ||

| | | | ||

− | | <hask> | + | | <hask>fmap even:: Maybe Int -> Maybe Bool</hask> |

|- | |- | ||

| <hask>Nothing</hask> | | <hask>Nothing</hask> | ||

Line 65: | Line 105: | ||

| <hask>Just False</hask> | | <hask>Just False</hask> | ||

|} | |} | ||

+ | |||

+ | ==== Side of <math>\Psi</math> functor ==== | ||

{| Border=2 CellPadding=2 CellSpacing=2 | {| Border=2 CellPadding=2 CellSpacing=2 | ||

|+ <math>\Psi(f) : \Psi(X) \to \Psi(Y)</math> | |+ <math>\Psi(f) : \Psi(X) \to \Psi(Y)</math> | ||

+ | | | ||

| <hask>map even:: [Int] -> [Bool]</hask> | | <hask>map even:: [Int] -> [Bool]</hask> | ||

|- | |- | ||

Line 74: | Line 117: | ||

|- | |- | ||

| <hask>[0]</hask> | | <hask>[0]</hask> | ||

− | | <hask>[ | + | | <hask>[True]</hask> |

|- | |- | ||

| <hask>[1]</hask> | | <hask>[1]</hask> | ||

− | | <hask>[ | + | | <hask>[False]</hask> |

|} | |} | ||

− | === Commutativity of diagram === | + | === Commutativity of the diagram === |

+ | |||

+ | :<math>\Psi(f) \cdot \eta_X = \eta_Y \cdot \Phi(f)</math> | ||

+ | both paths span between | ||

+ | :<math>\Phi(X) \to \Psi(Y)</math> | ||

{| Border=2 CellPadding=2 CellSpacing=2 | {| Border=2 CellPadding=2 CellSpacing=2 | ||

− | | | + | | RowSpan=2| |

− | | | + | | ColSpan=2|<hask>Maybe Int -> [Bool]</hask> |

+ | |- | ||

| <hask>map even . maybeToList</hask> | | <hask>map even . maybeToList</hask> | ||

− | | <hask>maybeToList . | + | | <hask>maybeToList . fmap even</hask> |

|- | |- | ||

| <hask>Nothing</hask> | | <hask>Nothing</hask> | ||

Line 100: | Line 148: | ||

| <hask>[False]</hask> | | <hask>[False]</hask> | ||

|} | |} | ||

+ | |||

+ | === Remarks === | ||

+ | |||

+ | * <hask>even</hask> has a more general type (<hask>Integral a => a -> Bool</hask>) 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 modify the [[#Definition|commutative diagram]], see its [[Media:Natural_transformation.tex|source code]] (in LaTeX using <code>amscd</code>). | ||

+ | |||

+ | == Operations == | ||

+ | |||

+ | === Mixed === | ||

+ | |||

+ | The “mixed” operations described below will be important also in understanding the definition of “monad” concept in category theory. | ||

+ | |||

+ | ==== 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 enumeration of items (e.g. names of acquired languages) | ||

+ | |||

+ | <haskell> | ||

+ | spouse :: Parser (Maybe String) | ||

+ | languages :: Parser [String] | ||

+ | </haskell> | ||

+ | |||

+ | 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 to list format and exclude any Maybe's). (Perhaps, all this example is impractical and exaggerated, because in real life we should solve the whole thing in other ways.) | ||

+ | |||

+ | Thus, we want to build a parser combinator (we could notate it graphically with something like <math>\mathrm?\!\!\to\!\!\mathrm*</math>) which converts a “zero-ore-one-occurrence” like parser to a “zero-or-one-or-many-occurrences” like parser. | ||

+ | |||

+ | We can convert <hask>Maybe</hask> to list with <hask>maybeToList</hask> | ||

+ | But if we want to do something similar with a ''parser'' on Maybe's to achieve a ''parser'' on list, then <hask>maybeToList</hask> is not enough alone, we must <hask>fmap</hask> it. | ||

+ | E.g. if we want to convert a parser like <hask>spouse</hask> to be of the same type as <hask>languages</hask>: | ||

+ | <haskell> | ||

+ | fmap maybeToList spouse | ||

+ | </haskell> | ||

+ | Let us see the types: | ||

+ | We start with | ||

+ | <haskell> | ||

+ | spouse :: Parser (Maybe String) | ||

+ | </haskell> | ||

+ | :<math>\Lambda(\Phi(X))</math> | ||

+ | or using notion of composing functors | ||

+ | :<math>(\Lambda \Phi)(X)</math> | ||

+ | We want to achieve | ||

+ | <haskell> | ||

+ | fmap maybeToList spouse :: Parser [String] | ||

+ | </haskell> | ||

+ | :<math>\Lambda(\Psi(X))</math> | ||

+ | :<math>(\Lambda \Psi)(X)</math> | ||

+ | thus we can infer | ||

+ | <haskell> | ||

+ | fmap maybeToList :: Parser (Maybe [String]) -> Parser [String] | ||

+ | </haskell> | ||

+ | :<math>(\Lambda\eta)_X \in \mathrm{Hom}_{\mathcal D}((\Lambda\Phi)(X),\;(\Lambda\Psi)(X))</math> | ||

+ | 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 <math>\Lambda\eta</math>: | ||

+ | :To each <math>X \in \mathbf{Ob}(\mathcal C)</math> we associate <math>(\Lambda\eta)_X \in \mathrm{Hom}_{\mathcal D}((\Lambda\Phi)(X),\;(\Lambda\Psi)(X))</math> | ||

+ | :<math>\Lambda\eta : \Lambda\Phi \to \Lambda\Psi</math> | ||

+ | :<math>(\Lambda\eta)_X = \Lambda(\eta_X)</math> | ||

+ | |||

+ | Summary: | ||

+ | |||

+ | :Let <math>\mathcal C, \mathcal D, \mathcal E</math> be categories | ||

+ | :<math>\Phi, \Psi : \mathcal C \to \mathcal D</math> functors | ||

+ | :<math>\Lambda : \mathcal D \to \mathcal E</math> functor | ||

+ | :<math>\eta : \Phi \to \Psi</math> natural transformation | ||

+ | Then let us define a new natural transformation: | ||

+ | :<math>\Lambda\eta : \Lambda\Phi \to \Lambda\Psi</math> | ||

+ | :<math>(\Lambda\eta)_X = \Lambda(\eta_X)</math> | ||

+ | |||

+ | ==== Natural transformation and functor ==== | ||

+ | |||

+ | :Let <math>\mathcal C, \mathcal D, \mathcal E</math> be categories | ||

+ | :<math>\Delta : \mathcal C \to \mathcal D</math> functor | ||

+ | :<math>\Phi, \Psi : \mathcal D \to \mathcal E</math> functors | ||

+ | :<math>\eta : \Phi \to \Psi</math> natural transformation | ||

+ | Then let us define a new natural transformation: | ||

+ | :<math>\eta\Delta : \Phi\Delta \to \Psi\Delta</math> | ||

+ | :<math>(\eta\Delta)_X = \eta_{\Delta(X)}</math> | ||

+ | |||

+ | It can be illustrated by Haskell examples, too. Understanding it is made harder (easier?) by the fact that Haskell's type inference “(dis)solves” the main point, thus there is no “materialized” manifestation of it. | ||

+ | |||

+ | <haskell> | ||

+ | convert :: Maybe (Term a) -> [Term a] | ||

+ | </haskell> | ||

+ | |||

+ | Unlike [[#Functor and natural transformation|seen at <math>\mathrm?\!\!\to\!\!\mathrm*</math>]], the definition of this converter will not show much novelty: | ||

+ | |||

+ | <haskell> | ||

+ | convert = maybeToList | ||

+ | </haskell> | ||

+ | |||

+ | the most interesting thing is done automatically by type inference. | ||

+ | |||

+ | == 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. | ||

+ | * Wikipedia's [http://en.wikipedia.org/wiki/Natural_transformation Natural transformation] article | ||

+ | * [http://www.case.edu/artsci/math/wells/pub/ttt.html Toposes, Triples and Theories] written by Michael Barr and Charles Wells. | ||

+ | |||

+ | [[Category:Theoretical foundations]] |

## Revision as of 15:30, 5 October 2006

map even $ maybeToList $ Just 5

yields the same as

maybeToList $ fmap even $ Just 5

yields: both yield

[False]

In the followings, this example will be used to illustrate the notion of natural transformation. If the examples are exaggerated and/or the definitions are incomprehensible, try #External links.

## Contents |

## 1 Definition

- Let , denote categories.
- Let be functors.
- Let . Let .

Let us define the natural transformation. It associates to each object of a morphism of in the following way (usually, not sets are discussed here, but proper classes, so I do not use term “function” for this mapping):

- . We call η
_{A}the component of η at*A*.

Thus, the following diagram commutes (in ):

## 2 Example: maybeToList

As already mentioned

map even $ maybeToList $ Just 5

yields the same as

maybeToList $ fmap even $ Just 5

yields: both yield

[False]

This example will be shown in the light of the above definition in the followings.

### 2.1 Vertical arrows: sides of objects

… showing how the natural transformation works.

maybeToList :: Maybe a -> [a]

#### 2.1.1 Left: side of *X* object

maybeToList :: Maybe Int -> [Int] | |

Nothing |
[] |

Just 0 |
[0] |

Just 1 |
[1] |

#### 2.1.2 Right: side of *Y* object

maybeToList :: Maybe Bool -> [Bool] | |

Nothing |
[] |

Just True |
[True] |

Just False |
[False] |

### 2.2 Horizontal arrows: sides of functors

even :: Int -> Bool

#### 2.2.1 Side of Φ functor

fmap even:: Maybe Int -> Maybe Bool | |

Nothing |
Nothing |

Just 0 |
Just True |

Just 1 |
Just False |

#### 2.2.2 Side of Ψ functor

map even:: [Int] -> [Bool] | |

[] |
[] |

[0] |
[True] |

[1] |
[False] |

### 2.3 Commutativity of the diagram

both paths span between

Maybe Int -> [Bool] | ||

map even . maybeToList |
maybeToList . fmap even | |

Nothing |
[] |
[] |

Just 0 |
[True] |
[True] |

Just 1 |
[False] |
[False] |

### 2.4 Remarks

- has a more general type (even) than described hereIntegral a => a -> Bool
- 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 modify the commutative diagram, see its source code (in LaTeX using
`amscd`

).

## 3 Operations

### 3.1 Mixed

The “mixed” operations described below will be important also in understanding the definition of “monad” concept in category theory.

#### 3.1.1 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 enumeration 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 to list format and exclude any Maybe's). (Perhaps, all this example is impractical and exaggerated, because in real life we should solve the whole thing in other ways.)

Thus, we want to build a parser combinator (we could notate it graphically with something like ) which converts a “zero-ore-one-occurrence” like parser to a “zero-or-one-or-many-occurrences” like parser.

We can convert*parser*on Maybe's to achieve a

*parser*on list, then

` 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]

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 we associate
- (Λη)
_{X}= Λ(η_{X})

Summary:

- Let be categories
- functors
- functor
- natural transformation

Then let us define a new natural transformation:

- (Λη)
_{X}= Λ(η_{X})

#### 3.1.2 Natural transformation and functor

- Let be categories
- functor
- functors
- natural transformation

Then let us define a new natural transformation:

- (ηΔ)
_{X}= η_{Δ(X)}

It can be illustrated by Haskell examples, too. Understanding it is made harder (easier?) by the fact that Haskell's type inference “(dis)solves” the main point, thus there is no “materialized” manifestation of it.

convert :: Maybe (Term a) -> [Term a]

Unlike seen at , the definition of this converter will not show much novelty:

`convert = maybeToList`

the most interesting thing is done automatically by type inference.

## 4 External links

- The corresponding HaWiki article is not migrated here yet, so You can see it for more information.
- Wikipedia's Natural transformation article
- Toposes, Triples and Theories written by Michael Barr and Charles Wells.