More guarded cat ideas

formalizations/guarded-cubical/GuardedCat.agda

@@ -8,6 +8,11 @@ open import Cubical.Foundations.Prelude
 open import Cubical.Categories.Category
 open import Cubical.Categories.Functor
 open import Cubical.Categories.Adjoint
+open import Cubical.Categories.Functor
+open import Cubical.Categories.Instances.Functors
+open import Cubical.Categories.Constructions.BinProduct
+
+open import Cubical.Categories.Instances.Sets
 
 variable
   ℓ ℓ' : Level
@@ -24,7 +29,21 @@ variable
          ; isSetHom = λ x y → λ p:x≡y q:x≡y i j t → Category.isSetHom (C t) (x t) (y t) (λ k → p:x≡y k t) ((λ k → q:x≡y k t)) i j
          }
 
-next-c : ∀ (C : Category ℓ ℓ') → Functor C (▸c (next C))
+
+▹c : ∀ (C : Category ℓ ℓ') → Category ℓ ℓ'
+▹c C = ▸c (next C)
+
+▹F : ∀ {C D : Category ℓ ℓ'} → Functor C D → Functor (▹c C) (▹c D)
+▹F F = record
+  { F-ob = λ x t → F-ob (x t)
+  ; F-hom = λ x t → F-hom (x t)
+  ; F-id = λ i t → F-id i
+  ; F-seq = λ f g i t → F-seq (f t) (g t) i
+  }
+  where module F = Functor F
+        open F
+
+next-c : ∀ (C : Category ℓ ℓ') → Functor C (▹c C)
 next-c C = record
   { F-ob = λ x t → x
   ; F-hom = λ z t → z
@@ -41,3 +60,42 @@ record GuardedCat : Type (ℓ-suc (ℓ-max ℓ ℓ')) where
     θ-UMP : isRightAdjoint (next-c C)
 
 -- Conjecture: Every GuardedCat has an initial object ⊥ = fix θ₀
+
+
+PROF : (ℓ' : Level) → (C : Category ℓ ℓ') (D : Category ℓ ℓ') → Category (ℓ-max ℓ (ℓ-suc ℓ')) (ℓ-max ℓ ℓ')
+PROF ℓ' C D = FUNCTOR ((C ^op) × D) (SET ℓ')
+
+Profunctor : (C : Category ℓ ℓ') (D : Category ℓ ℓ') → Type (ℓ-max ℓ (ℓ-suc ℓ'))
+Profunctor C D = Category.ob (PROF _ C D)
+
+-- e : A → B
+-- p : B → ▹ A
+
+-- e a ~> b
+-- iff (next a) ~> p b
+-- defines e as a relative left adjoint
+-- doesn't uniquely determine p
+
+-- a▹ ~> p b
+-- iff ▹ (λ t → e (a▹ t) ~> b)
+-- iff (▹ e) a▹ ~> (next b)
+-- defines p as a relative right adjoint
+-- doesn't uniquely determine e
+
+-- e seems more fundamental than p tbh...but which of these holds period?
+
+
+-- e : A → B
+-- p : B → Option (▹ A)
+
+-- e a ~> b
+-- iff some (next a) ~> p b
+-- defines e as a relative left adjoint
+-- doesn't uniquely determine p
+
+-- a?▹ ~> p b
+-- iff Option (▹ e) a▹ ~> (next b)
+-- defines p as a relative right adjoint
+-- doesn't uniquely determine e
+
+-- ▹p : Profunctor C D → (Profunctor (▹ C) (▹ D))