Portability | non-portable (rank-2 polymorphism) |
---|---|
Stability | experimental |
Maintainer | Edward Kmett <ekmett@gmail.com> |
Left and right Kan extensions, expressed as higher order functors
See http://comonad.com/reader/2008/kan-extensions/ and http://comonad.com/reader/2008/kan-extensions-ii/ for motivation.
NB: Yoneda
, CoYoneda
, Density
, Codensity
have been factored
out into separate modules.
- newtype Ran g h a = Ran {
- runRan :: forall b. (a -> g b) -> h b
- toRan :: (Composition o, Functor k) => ((k `o` g) :~> h) -> k :~> Ran g h
- fromRan :: Composition o => (k :~> Ran g h) -> (k `o` g) :~> h
- adjointToRan :: Adjunction f g => f :~> Ran g Identity
- ranToAdjoint :: Adjunction f g => Ran g Identity :~> f
- ranToComposedAdjoint :: (Composition o, Adjunction f g) => Ran g h :~> (h `o` f)
- composedAdjointToRan :: (Functor h, Composition o, Adjunction f g) => (h `o` f) :~> Ran g h
- composeRan :: Composition o => Ran f (Ran g h) :~> Ran (f `o` g) h
- decomposeRan :: (Functor f, Composition o) => Ran (f `o` g) h :~> Ran f (Ran g h)
- data Lan g h a = forall b . Lan (g b -> a) (h b)
- toLan :: (Composition o, Functor f) => (h :~> (f `o` g)) -> Lan g h :~> f
- fromLan :: Composition o => (Lan g h :~> f) -> h :~> (f `o` g)
- adjointToLan :: Adjunction f g => g :~> Lan f Identity
- lanToAdjoint :: Adjunction f g => Lan f Identity :~> g
- composeLan :: (Functor f, Composition o) => Lan f (Lan g h) :~> Lan (f `o` g) h
- decomposeLan :: Composition o => Lan (f `o` g) h :~> Lan f (Lan g h)
- lanToComposedAdjoint :: (Functor h, Composition o, Adjunction f g) => Lan f h :~> (h `o` g)
- composedAdjointToLan :: (Composition o, Adjunction f g) => (h `o` g) :~> Lan f h
Right Kan Extensions
The right Kan Extension of h along g. An alternative definition in terms of Ends.
newtype RanT g h a b b' { (a -> g b) -> h b' }
type Ran g h a = End (RanT g h a)
toRan :: (Composition o, Functor k) => ((k `o` g) :~> h) -> k :~> Ran g hSource
Nat(k o
g, h) is isomorphic to Nat(k, Ran g h) (forwards)
fromRan :: Composition o => (k :~> Ran g h) -> (k `o` g) :~> hSource
Nat(k o
g, h) is isomorphic to Nat(k, Ran g h) (backwards)
adjointToRan :: Adjunction f g => f :~> Ran g IdentitySource
f -| g
iff Ran g Identity
exists (forward)
ranToAdjoint :: Adjunction f g => Ran g Identity :~> fSource
f -| g
iff Ran g Identity
exists (backwards)
ranToComposedAdjoint :: (Composition o, Adjunction f g) => Ran g h :~> (h `o` f)Source
composedAdjointToRan :: (Functor h, Composition o, Adjunction f g) => (h `o` f) :~> Ran g hSource
composeRan :: Composition o => Ran f (Ran g h) :~> Ran (f `o` g) hSource
The natural isomorphism from Ran f (Ran g h)
to Ran (f
(forwards)
o
g) h
decomposeRan :: (Functor f, Composition o) => Ran (f `o` g) h :~> Ran f (Ran g h)Source
The natural isomorphism from Ran f (Ran g h)
to Ran (f
(backwards)
o
g) h
Left Kan Extensions
Left Kan Extension
newtype LanT g h a b b' { (g b -> a, h b') }
type Lan g h a = Coend (LanT g h a)
forall b . Lan (g b -> a) (h b) |
toLan :: (Composition o, Functor f) => (h :~> (f `o` g)) -> Lan g h :~> fSource
Nat(h, f.g)
is isomorphic to Nat (Lan g h, f)
(forwards)
fromLan :: Composition o => (Lan g h :~> f) -> h :~> (f `o` g)Source
Nat(h, f.g)
is isomorphic to Nat (Lan g h, f)
(backwards)
adjointToLan :: Adjunction f g => g :~> Lan f IdentitySource
f -| g iff Lan f Identity is inhabited (forwards)
lanToAdjoint :: Adjunction f g => Lan f Identity :~> gSource
f -| g iff Lan f Identity is inhabited (backwards)
composeLan :: (Functor f, Composition o) => Lan f (Lan g h) :~> Lan (f `o` g) hSource
the natural isomorphism from Lan f (Lan g h)
to Lan (f
(forwards)
o
g) h
decomposeLan :: Composition o => Lan (f `o` g) h :~> Lan f (Lan g h)Source
the natural isomorphism from Lan f (Lan g h)
to Lan (f
(backwards)
o
g) h
lanToComposedAdjoint :: (Functor h, Composition o, Adjunction f g) => Lan f h :~> (h `o` g)Source
composedAdjointToLan :: (Composition o, Adjunction f g) => (h `o` g) :~> Lan f hSource