Portability | non-portable |
---|---|

Stability | experimental |

Maintainer | sjoerd@w3future.com |

- type :~> f g = (c ~ Dom f, c ~ Dom g, d ~ Cod f, d ~ Cod g) => Nat c d f g
- type Component f g z = Cod f (f :% z) (g :% z)
- newtype Com f g z = Com {}
- (!) :: (Category c, Category d) => Nat c d f g -> c a b -> d (f :% a) (g :% b)
- o :: (Category c, Category d, Category e) => Nat d e j k -> Nat c d f g -> Nat c e (j :.: f) (k :.: g)
- natId :: Functor f => f -> Nat (Dom f) (Cod f) f f
- data Nat where
- type Endo (~>) = Nat ~> ~>
- data FunctorCompose (~>) = FunctorCompose
- data Precompose where
- Precompose :: f -> Precompose f d

- data Postcompose where
- Postcompose :: f -> Postcompose f c

- data Wrap f h = Wrap f h
- type Presheaves (~>) = Nat (Op ~>) (->)
- class Functor f => Representable f where
- type RepresentingObject f :: *
- represent :: Dom f ~ Op c => f -> (c :-*: RepresentingObject f) :~> f
- unrepresent :: Dom f ~ Op c => f -> f :~> (c :-*: RepresentingObject f)

- data YonedaEmbedding where
- YonedaEmbedding :: Category ~> => YonedaEmbedding ~>

- data Yoneda f = Yoneda
- fromYoneda :: (Functor f, Cod f ~ (->)) => f -> Nat (Dom f) (->) (Yoneda f) f
- toYoneda :: (Functor f, Cod f ~ (->)) => f -> Nat (Dom f) (->) f (Yoneda f)

# Natural transformations

type :~> f g = (c ~ Dom f, c ~ Dom g, d ~ Cod f, d ~ Cod g) => Nat c d f gSource

`f :~> g`

is a natural transformation from functor f to functor g.

type Component f g z = Cod f (f :% z) (g :% z)Source

A component for an object `z`

is an arrow from `F z`

to `G z`

.

A newtype wrapper for components, which can be useful for helper functions dealing with components.

(!) :: (Category c, Category d) => Nat c d f g -> c a b -> d (f :% a) (g :% b)Source

'n ! a' returns the component for the object `a`

of a natural transformation `n`

.
This can be generalized to any arrow (instead of just identity arrows).

o :: (Category c, Category d, Category e) => Nat d e j k -> Nat c d f g -> Nat c e (j :.: f) (k :.: g)Source

Horizontal composition of natural transformations.

natId :: Functor f => f -> Nat (Dom f) (Cod f) f fSource

The identity natural transformation of a functor.

# Functor category

Natural transformations are built up of components,
one for each object `z`

in the domain category of `f`

and `g`

.

Nat :: (Functor f, Functor g, c ~ Dom f, c ~ Dom g, d ~ Cod f, d ~ Cod g) => f -> g -> (forall z. Obj c z -> Component f g z) -> Nat c d f g |

(Category c, Category d) => Category (Nat c d) | Functor category D^C. Objects of D^C are functors from C to D. Arrows of D^C are natural transformations. |

(Category c, HasTerminalObject d) => HasTerminalObject (Nat c d) | The constant functor to the terminal object is itself the terminal object in its functor category. |

(Category c, HasInitialObject d) => HasInitialObject (Nat c d) | The constant functor to the initial object is itself the initial object in its functor category. |

(Category c, HasBinaryProducts d) => HasBinaryProducts (Nat c d) | |

(Category c, HasBinaryCoproducts d) => HasBinaryCoproducts (Nat c d) |

# Related functors

data FunctorCompose (~>) Source

Composition of endofunctors is a functor.

Category ~> => Functor (FunctorCompose ~>) | |

Category ~> => HasUnit (FunctorCompose ~>) | |

Category ~> => TensorProduct (FunctorCompose ~>) |

data Precompose whereSource

`Precompose f d`

is the functor such that `Precompose f d :% g = g :.: f`

,
for functors `g`

that compose with `f`

and with codomain `d`

.

Precompose :: f -> Precompose f d |

(Functor f, Category d) => Functor (Precompose f d) |

data Postcompose whereSource

`Postcompose f c`

is the functor such that `Postcompose f c :% g = f :.: g`

,
for functors `g`

that compose with `f`

and with domain `c`

.

Postcompose :: f -> Postcompose f c |

(Functor f, Category c) => Functor (Postcompose f c) |

`Wrap f h`

is the functor such that `Wrap f h :% g = f :.: g :.: h`

,
for functors `g`

that compose with `f`

and `h`

.

Wrap f h |

## Presheaves

type Presheaves (~>) = Nat (Op ~>) (->)Source

class Functor f => Representable f whereSource

A functor F: Op(C) -> Set is representable if it is naturally isomorphic to the contravariant hom-functor.

type RepresentingObject f :: *Source

represent :: Dom f ~ Op c => f -> (c :-*: RepresentingObject f) :~> fSource

unrepresent :: Dom f ~ Op c => f -> f :~> (c :-*: RepresentingObject f)Source

Category ~> => Representable (:-*: ~> x) |

## Yoneda

data YonedaEmbedding whereSource

The Yoneda embedding functor.

YonedaEmbedding :: Category ~> => YonedaEmbedding ~> |

Category ~> => Functor (YonedaEmbedding ~>) |