# Category theory/Natural transformation

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

− | |||

| | | | ||

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

| <hask>map even:: Maybe Int -> Maybe Bool</hask> | | <hask>map even:: Maybe Int -> Maybe Bool</hask> | ||

|- | |- | ||

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

− | |+ <math>\Psi(f) \cdot \eta_X = \eta_Y \cdot \Phi(f)</math> | + | |+ <math>\Psi(f) \cdot \eta_X = \eta_Y \cdot \Phi(f)</math>, both paths span between <math>\Phi(X) \to \Psi(Y)</math> |

− | | | + | | RowSpan=2| |

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

+ | |- | ||

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

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

## Revision as of 20:02, 2 October 2006

## Contents |

## 1 Example: maybeToList

maybeToList

map even $ maybeToList $ Just 5

yields the same as

maybeToList $ map even $ Just 5

yields: both yield

[False]

### 1.1 Vertical arrows: sides of objects

… showing the operation of the natural transformation.

maybeToList :: Maybe a -> [a]

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

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

Nothing |
[] |

Just 0 |
[0] |

Just 1 |
[1] |

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

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

Nothing |
[] |

Just True |
[True] |

Just False |
[False] |

### 1.2 Horizontal arrows: sides of functors

even :: Int -> Bool

#### 1.2.1 Side of Φ functor

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

Nothing |
Nothing |

Just 0 |
Just True |

Just 1 |
Just False |

#### 1.2.2 Side of Ψ functor

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

[] |
[] |

[0] |
[T]rue |

[1] |
[F]alse |

### 1.3 Commutativity of diagram

Maybe Int -> [Bool] | ||

map even . maybeToList |
maybeToList . map even | |

Nothing |
[] |
[] |

Just 0 |
[True] |
[True] |

Just 1 |
[False] |
[False] |

### 1.4 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.