Fix typo, rename implicit variables, implement presheaf

src/Cat/Categories/Fun.agda

@@ -24,8 +24,8 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       → (f : ℂ .Arrow A B)
       → 𝔻 ._⊕_ (θ B) (F .func→ f) ≡ 𝔻 ._⊕_ (G .func→ f) (θ A)
 
-    NaturalTranformation : Set (ℓc ⊔ ℓc' ⊔ ℓd')
+    NaturalTransformation : Set (ℓc ⊔ ℓc' ⊔ ℓd')
-    NaturalTranformation = Σ Transformation Natural
+    NaturalTransformation = Σ Transformation Natural
 
     -- NaturalTranformation : Set (ℓc ⊔ (ℓc' ⊔ ℓd'))
     -- NaturalTranformation = ∀ (θ : Transformation) {A B : ℂ .Object} → (f : ℂ .Arrow A B) → 𝔻 ._⊕_ (θ B) (F .func→ f) ≡ 𝔻 ._⊕_ (G .func→ f) (θ A)
@@ -45,37 +45,38 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       F→ = F .func→
       open module 𝔻 = IsCategory (𝔻 .isCategory)
 
-  identityNat : (F : Functor ℂ 𝔻) → NaturalTranformation F F
+  identityNat : (F : Functor ℂ 𝔻) → NaturalTransformation F F
   identityNat F = identityTrans F , identityNatural F
 
-  module _ {a b c : Functor ℂ 𝔻} where
+  module _ {F G H : Functor ℂ 𝔻} where
     private
       _𝔻⊕_ = 𝔻 ._⊕_
-      _∘nt_ : Transformation b c → Transformation a b → Transformation a c
+      _∘nt_ : Transformation G H → Transformation F G → Transformation F H
       (θ ∘nt η) C = θ C 𝔻⊕ η C
 
-    _:⊕:_ : NaturalTranformation b c → NaturalTranformation a b → NaturalTranformation a c
+    NatComp _:⊕:_ : NaturalTransformation G H → NaturalTransformation F G → NaturalTransformation F H
     proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η
     proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin
-      ((θ ∘nt η) B) 𝔻⊕ (a .func→ f)    ≡⟨⟩
+      ((θ ∘nt η) B) 𝔻⊕ (F .func→ f)    ≡⟨⟩
-      (θ B 𝔻⊕ η B) 𝔻⊕ (a .func→ f)     ≡⟨ sym assoc ⟩
+      (θ B 𝔻⊕ η B) 𝔻⊕ (F .func→ f)     ≡⟨ sym assoc ⟩
-      θ B 𝔻⊕ (η B 𝔻⊕ (a .func→ f))     ≡⟨ cong (λ φ → θ B 𝔻⊕ φ) (ηNat f) ⟩
+      θ B 𝔻⊕ (η B 𝔻⊕ (F .func→ f))     ≡⟨ cong (λ φ → θ B 𝔻⊕ φ) (ηNat f) ⟩
-      θ B 𝔻⊕ ((b .func→ f) 𝔻⊕ η A)     ≡⟨ assoc ⟩
+      θ B 𝔻⊕ ((G .func→ f) 𝔻⊕ η A)     ≡⟨ assoc ⟩
-      (θ B 𝔻⊕ (b .func→ f)) 𝔻⊕ η A     ≡⟨ cong (λ φ → φ 𝔻⊕ η A) (θNat f) ⟩
+      (θ B 𝔻⊕ (G .func→ f)) 𝔻⊕ η A     ≡⟨ cong (λ φ → φ 𝔻⊕ η A) (θNat f) ⟩
-      (((c .func→ f) 𝔻⊕ θ A) 𝔻⊕ η A)   ≡⟨ sym assoc ⟩
+      (((H .func→ f) 𝔻⊕ θ A) 𝔻⊕ η A)   ≡⟨ sym assoc ⟩
-      ((c .func→ f) 𝔻⊕ (θ A 𝔻⊕ η A))   ≡⟨⟩
+      ((H .func→ f) 𝔻⊕ (θ A 𝔻⊕ η A))   ≡⟨⟩
-      ((c .func→ f)  𝔻⊕ ((θ ∘nt η) A)) ∎
+      ((H .func→ f)  𝔻⊕ ((θ ∘nt η) A)) ∎
       where
         open IsCategory (𝔻 .isCategory)
+    NatComp = _:⊕:_
 
   private
-    module _ {A B C D : Functor ℂ 𝔻} {f : NaturalTranformation A B}
+    module _ {A B C D : Functor ℂ 𝔻} {f : NaturalTransformation A B}
-      {g : NaturalTranformation B C} {h : NaturalTranformation C D} where
+      {g : NaturalTransformation B C} {h : NaturalTransformation C D} where
       _g⊕f_ = _:⊕:_ {A} {B} {C}
       _h⊕g_ = _:⊕:_ {B} {C} {D}
       :assoc: : (_:⊕:_ {A} {C} {D} h (_:⊕:_ {A} {B} {C} g f)) ≡ (_:⊕:_ {A} {B} {D} (_:⊕:_ {B} {C} {D} h g) f)
       :assoc: = {!!}
-    module _ {A B : Functor ℂ 𝔻} {f : NaturalTranformation A B} where
+    module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where
       ident-r : (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f
       ident-r = {!!}
       ident-l : (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f
@@ -86,7 +87,7 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       :ident: = ident-r , ident-l
 
   instance
-    :isCategory: : IsCategory (Functor ℂ 𝔻) NaturalTranformation
+    :isCategory: : IsCategory (Functor ℂ 𝔻) NaturalTransformation
       (λ {F} → identityNat F) (λ {a} {b} {c} → _:⊕:_ {a} {b} {c})
     :isCategory: = record
       { assoc = λ {A B C D} → :assoc: {A} {B} {C} {D}
@@ -94,10 +95,22 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       }
 
   -- Functor categories. Objects are functors, arrows are natural transformations.
-  Fun : Category (ℓc ⊔ (ℓc' ⊔ (ℓd ⊔ ℓd'))) (ℓc ⊔ (ℓc' ⊔ ℓd'))
+  Fun : Category (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
   Fun = record
     { Object = Functor ℂ 𝔻
-    ; Arrow = NaturalTranformation
+    ; Arrow = NaturalTransformation
     ; 𝟙 = λ {F} → identityNat F
-    ; _⊕_ = λ {a b c} → _:⊕:_ {a} {b} {c}
+    ; _⊕_ = λ {F G H} → _:⊕:_ {F} {G} {H}
+    }
+
+module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
+  open import Cat.Categories.Sets
+
+  -- Restrict the functors to Presheafs.
+  Presh : Category (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
+  Presh = record
+    { Object = Presheaf ℂ
+    ; Arrow = NaturalTransformation
+    ; 𝟙 = λ {F} → identityNat F
+    ; _⊕_ = λ {F G H} → NatComp {F = F} {G = G} {H = H}
     }