Portability | non-portable |
---|---|
Stability | experimental |
Maintainer | sjoerd@w3future.com |
Safe Haskell | Safe-Inferred |
- 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
- srcF :: Nat c d f g -> f
- tgtF :: Nat c d f g -> g
- data Nat where
- type Endo k = Nat k k
- compAssoc :: (Functor f, Functor g, Functor h, Dom f ~ Cod g, Dom g ~ Cod h) => f -> g -> h -> Nat (Dom h) (Cod f) ((f :.: g) :.: h) (f :.: (g :.: h))
- compAssocInv :: (Functor f, Functor g, Functor h, Dom f ~ Cod g, Dom g ~ Cod h) => f -> g -> h -> Nat (Dom h) (Cod f) (f :.: (g :.: h)) ((f :.: g) :.: h)
- idPrecomp :: Functor f => f -> Nat (Dom f) (Cod f) (f :.: Id (Dom f)) f
- idPrecompInv :: Functor f => f -> Nat (Dom f) (Cod f) f (f :.: Id (Dom f))
- idPostcomp :: Functor f => f -> Nat (Dom f) (Cod f) (Id (Cod f) :.: f) f
- idPostcompInv :: Functor f => f -> Nat (Dom f) (Cod f) f (Id (Cod f) :.: f)
- constPrecomp :: (Category c1, Functor f) => Const c1 (Dom f) x -> f -> Nat c1 (Cod f) (f :.: Const c1 (Dom f) x) (Const c1 (Cod f) (f :% x))
- constPrecompInv :: (Category c1, Functor f) => Const c1 (Dom f) x -> f -> Nat c1 (Cod f) (Const c1 (Cod f) (f :% x)) (f :.: Const c1 (Dom f) x)
- constPostcomp :: Functor f => Const (Cod f) c2 x -> f -> Nat (Dom f) c2 (Const (Cod f) c2 x :.: f) (Const (Dom f) c2 x)
- constPostcompInv :: Functor f => Const (Cod f) c2 x -> f -> Nat (Dom f) c2 (Const (Dom f) c2 x) (Const (Cod f) c2 x :.: f)
- data FunctorCompose k = FunctorCompose
- data Precompose where
- Precompose :: f -> Precompose f d
- data Postcompose where
- Postcompose :: f -> Postcompose f c
- data Wrap f h = Wrap f h
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 k => CartesianClosed (Presheaves k) | The category of presheaves on a category |
(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, HasBinaryCoproducts d) => HasBinaryCoproducts (Nat c d) | The functor coproduct |
(Category c, HasBinaryProducts d) => HasBinaryProducts (Nat c d) | The functor product |
(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, HasTerminalObject d) => HasTerminalObject (Nat c d) | The constant functor to the terminal object is itself the terminal object in its functor category. |
Functor isomorphisms
compAssoc :: (Functor f, Functor g, Functor h, Dom f ~ Cod g, Dom g ~ Cod h) => f -> g -> h -> Nat (Dom h) (Cod f) ((f :.: g) :.: h) (f :.: (g :.: h))Source
compAssocInv :: (Functor f, Functor g, Functor h, Dom f ~ Cod g, Dom g ~ Cod h) => f -> g -> h -> Nat (Dom h) (Cod f) (f :.: (g :.: h)) ((f :.: g) :.: h)Source
constPrecomp :: (Category c1, Functor f) => Const c1 (Dom f) x -> f -> Nat c1 (Cod f) (f :.: Const c1 (Dom f) x) (Const c1 (Cod f) (f :% x))Source
constPrecompInv :: (Category c1, Functor f) => Const c1 (Dom f) x -> f -> Nat c1 (Cod f) (Const c1 (Cod f) (f :% x)) (f :.: Const c1 (Dom f) x)Source
constPostcomp :: Functor f => Const (Cod f) c2 x -> f -> Nat (Dom f) c2 (Const (Cod f) c2 x :.: f) (Const (Dom f) c2 x)Source
constPostcompInv :: Functor f => Const (Cod f) c2 x -> f -> Nat (Dom f) c2 (Const (Dom f) c2 x) (Const (Cod f) c2 x :.: f)Source
Related functors
data FunctorCompose k Source
Category k => Functor (FunctorCompose k) | Composition of endofunctors is a functor. |
Category k => TensorProduct (FunctorCompose k) | Functor composition makes the category of endofunctors monoidal, with the identity functor as unit. |
data Precompose whereSource
Precompose :: f -> Precompose f d |
(Functor f, Category d) => Functor (Precompose f d) |
|
data Postcompose whereSource
Postcompose :: f -> Postcompose f c |
(Functor f, Category c) => Functor (Postcompose f c) |
|