# Category theory/Natural transformation

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+ | === Commutative diagram === |
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+ | |||

+ | Let <math>\mathcal C</math>, <math>\mathcal D</math> denote categories. |
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+ | Let <math>\Phi, \Psi : \mathcal C \to \mathcal D</math> be functors. |
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+ | Let us define the <math>\eta : \Phi \to \Psi</math> natural transformation. |
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+ | ............ |
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=== Vertical arrows: sides of objects === |
=== Vertical arrows: sides of objects === |

## Revision as of 20:26, 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 Commutative diagram

Let , denote categories. Let be functors. Let us define the natural transformation.

............

### 1.2 Vertical arrows: sides of objects

… showing how the natural transformation works.

maybeToList :: Maybe a -> [a]

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

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

Nothing |
[] |

Just 0 |
[0] |

Just 1 |
[1] |

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

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

Nothing |
[] |

Just True |
[True] |

Just False |
[False] |

### 1.3 Horizontal arrows: sides of functors

even :: Int -> Bool

#### 1.3.1 Side of Φ functor

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

Nothing |
Nothing |

Just 0 |
Just True |

Just 1 |
Just False |

#### 1.3.2 Side of Ψ functor

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

[] |
[] |

[0] |
[True] |

[1] |
[False] |

### 1.4 Commutativity of the diagram

both paths span between

Maybe Int -> [Bool] | ||

map even . maybeToList |
maybeToList . map even | |

Nothing |
[] |
[] |

Just 0 |
[True] |
[True] |

Just 1 |
[False] |
[False] |

### 1.5 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.