module Data.Category.Presheaf where
import Prelude (($))
import Data.Category
import Data.Category.Functor
import Data.Category.NaturalTransformation
import Data.Category.Limit
import Data.Category.CartesianClosed
import Data.Category.Yoneda
type Presheaves (~>) = Nat (Op (~>)) (->)
type PShExponential (~>) y z = (Presheaves (~>) :-*: z) :.: Opposite
( ProductFunctor (Presheaves (~>))
:.: Tuple2 (Presheaves (~>)) (Presheaves (~>)) y
:.: YonedaEmbedding (~>)
)
pshExponential :: Category (~>) => Obj (Presheaves (~>)) y -> Obj (Presheaves (~>)) z -> PShExponential (~>) y z
pshExponential y z = hom_X z :.: Opposite (ProductFunctor :.: Tuple2 y :.: yonedaEmbedding)
type instance Exponential (Presheaves (~>)) y z = PShExponential (~>) y z
instance Category (~>) => CartesianClosed (Presheaves (~>)) where
apply yn@(Nat y _ _) zn@(Nat z _ _) = Nat (pshExponential yn zn :*: y) z $ \(Op i) (n, yi) -> (n ! Op i) (i, yi)
tuple yn zn@(Nat z _ _) = Nat z (pshExponential yn (zn *** yn)) $ \(Op i) zi -> (Nat (hom_X i) z $ \_ j2i -> (z % Op j2i) zi) *** yn
zn ^^^ yn = Nat (pshExponential (tgt yn) (src zn)) (pshExponential (src yn) (tgt zn)) $ \(Op i) n -> zn . n . (natId (hom_X i) *** yn)