| 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 |
Data.Functor.Yoneda
Description
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.
Instances
| 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 Yoneda fliftYoneda and lowerYoneda
liftYoneda.lowerYoneda≡idlowerYoneda.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≡idranToYoneda.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≡idriftToYoneda.yonedaToRift≡id
riftToYoneda :: Rift Identity f a -> Yoneda f a Source