Define the Traversing interface

Data/Profunctor/Traversing.idr

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+module Data.Profunctor.Traversing
+
+import Control.Monad.Identity
+import Data.Morphisms
+import Data.Profunctor.Types
+import Data.Profunctor.Functor
+import Data.Profunctor.Strong
+import Data.Profunctor.Closed
+
+%default total
+
+record Bazaar a b t where
+  constructor MkBazaar
+  getBazaar : forall f. Applicative f => (a -> f b) -> f t
+
+Functor (Bazaar a b) where
+  map f (MkBazaar g) = MkBazaar (map f . g)
+
+Applicative (Bazaar a b) where
+  pure a = MkBazaar $ \_ => pure a
+  mf <*> ma = MkBazaar $ \k => getBazaar mf k <*> getBazaar ma k
+
+sell : a -> Bazaar a b b
+sell a = MkBazaar ($ a)
+
+record Baz t b a where
+  constructor MkBaz
+  getBaz : forall f. Applicative f => (a -> f b) -> f t
+
+Functor (Baz t b) where
+  map f (MkBaz g) = MkBaz (g . (. f))
+
+
+sold : Baz t a a -> t
+sold m = runIdentity (getBaz m Id)
+
+Foldable (Baz t b) where
+  foldr f i bz = getBaz bz @{appEndo} f i
+    where
+      -- Equivalent to `Const (Endomorphism acc)`
+      appEndo : Applicative (\_ => acc -> acc)
+      appEndo = MkApplicative @{MkFunctor (const id)} (const id) (.)
+
+Traversable (Baz t b) where
+  traverse f bz = map (\m => MkBaz (getBazaar m)) $ getBaz bz @{Compose} $ \x => sell <$> f x
+
+
+public export
+interface (Strong p, Choice p) => Traversing p where
+  traverse' : Traversable f => p a b -> p (f a) (f b)
+  traverse' = wander traverse
+
+  wander : (forall f. Applicative f => (a -> f b) -> s -> f t) -> p a b -> p s t
+  wander f = dimap (\s => MkBaz $ \afb => f afb s) sold . traverse'