Copyright | (C) 2011-2013 Edward Kmett |
---|---|

License | BSD-style (see the file LICENSE) |

Maintainer | Edward Kmett <ekmett@gmail.com> |

Stability | provisional |

Portability | MPTCs, fundeps |

Safe Haskell | Trustworthy |

Language | Haskell98 |

The covariant form of the Yoneda lemma states that `f`

is naturally
isomorphic to `Yoneda f`

.

This is described in a rather intuitive fashion by Dan Piponi in

- newtype Yoneda f a = Yoneda {
- runYoneda :: forall b. (a -> b) -> f b

- liftYoneda :: Functor f => f a -> Yoneda f a
- lowerYoneda :: Yoneda f a -> f a
- maxF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a
- minF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a
- maxM :: (Monad m, Ord (m a)) => Yoneda m a -> Yoneda m a -> Yoneda m a
- minM :: (Monad m, Ord (m a)) => Yoneda m a -> Yoneda m a -> Yoneda m a
- yonedaToRan :: Yoneda f a -> Ran Identity f a
- ranToYoneda :: Ran Identity f a -> Yoneda f a
- yonedaToRift :: Yoneda f a -> Rift Identity f a
- riftToYoneda :: Rift Identity f a -> Yoneda f a

# Documentation

`Yoneda f a`

can be viewed as the partial application of `fmap`

to its second argument.

ComonadTrans Yoneda | |

MonadTrans Yoneda | |

(Functor f, MonadFree f m) => MonadFree f (Yoneda m) | |

Alternative f => Alternative (Yoneda f) | |

Monad m => Monad (Yoneda m) | |

Functor (Yoneda f) | |

MonadFix m => MonadFix (Yoneda m) | |

MonadPlus m => MonadPlus (Yoneda m) | |

Applicative f => Applicative (Yoneda f) | |

Foldable f => Foldable (Yoneda f) | |

Traversable f => Traversable (Yoneda f) | |

Distributive f => Distributive (Yoneda f) | |

Representable g => Representable (Yoneda g) | |

Comonad w => Comonad (Yoneda w) | |

Plus f => Plus (Yoneda f) | |

Traversable1 f => Traversable1 (Yoneda f) | |

Alt f => Alt (Yoneda f) | |

Apply f => Apply (Yoneda f) | |

Bind m => Bind (Yoneda m) | |

Extend w => Extend (Yoneda w) | |

Foldable1 f => Foldable1 (Yoneda f) | |

Adjunction f g => Adjunction (Yoneda f) (Yoneda g) | |

Eq (f a) => Eq (Yoneda f a) | |

Ord (f a) => Ord (Yoneda f a) | |

(Functor f, Read (f a)) => Read (Yoneda f a) | |

Show (f a) => Show (Yoneda f a) | |

type Rep (Yoneda g) = Rep g |

liftYoneda :: Functor f => f a -> Yoneda f a Source

The natural isomorphism between `f`

and

given by the Yoneda lemma
is witnessed by `Yoneda`

f`liftYoneda`

and `lowerYoneda`

`liftYoneda`

.`lowerYoneda`

≡`id`

`lowerYoneda`

.`liftYoneda`

≡`id`

lowerYoneda (liftYoneda fa) = -- definition lowerYoneda (Yoneda (f -> fmap f a)) -- definition (f -> fmap f fa) id -- beta reduction fmap id fa -- functor law fa

`lift`

=`liftYoneda`

lowerYoneda :: Yoneda f a -> f a Source

# as a right Kan extension

yonedaToRan :: Yoneda f a -> Ran Identity f a Source

`Yoneda f`

can be viewed as the right Kan extension of `f`

along the `Identity`

functor.

`yonedaToRan`

.`ranToYoneda`

≡`id`

`ranToYoneda`

.`yonedaToRan`

≡`id`

ranToYoneda :: Ran Identity f a -> Yoneda f a Source

# as a right Kan lift

yonedaToRift :: Yoneda f a -> Rift Identity f a Source

`Yoneda f`

can be viewed as the right Kan lift of `f`

along the `Identity`

functor.

`yonedaToRift`

.`riftToYoneda`

≡`id`

`riftToYoneda`

.`yonedaToRift`

≡`id`

riftToYoneda :: Rift Identity f a -> Yoneda f a Source