Some stuff about CwF's

src/Cat/Cubical.agda

@@ -7,14 +7,62 @@ open import Data.Product
 open import Data.Sum
 open import Data.Unit
 open import Data.Empty
+open import Data.Product
 
 open import Cat.Category
+open import Cat.Functor
 
 -- See chapter 1 for a discussion on how presheaf categories are CwF's.
 
 -- See section 6.8 in Huber's thesis for details on how to implement the
 -- categorical version of CTT
 
+module CwF {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
+  open Category
+  open Functor
+  open import Function
+  open import Cubical
+
+  module _ {ℓa ℓb : Level} where
+    private
+      Obj = Σ[ A ∈ Set ℓa ] (A → Set ℓb)
+      Arr : Obj → Obj → Set (ℓa ⊔ ℓb)
+      Arr (A , B) (A' , B') = Σ[ f ∈ (A → A') ] ({x : A} → B x → B' (f x))
+      one : {o : Obj} → Arr o o
+      proj₁ one = λ x → x
+      proj₂ one = λ b → b
+      _:⊕:_ : {a b c : Obj} → Arr b c → Arr a b → Arr a c
+      (g , g') :⊕: (f , f') = g ∘ f , g' ∘ f'
+
+      module _ {A B C D : Obj} {f : Arr A B} {g : Arr B C} {h : Arr C D} where
+        :assoc: : (_:⊕:_ {A} {C} {D} h (_:⊕:_ {A} {B} {C} g f)) ≡ (_:⊕:_ {A} {B} {D} (_:⊕:_ {B} {C} {D} h g) f)
+        :assoc: = {!!}
+
+      module _ {A B : Obj} {f : Arr A B} where
+        :ident: : (_:⊕:_ {A} {A} {B} f one) ≡ f × (_:⊕:_ {A} {B} {B} one f) ≡ f
+        :ident: = {!!}
+
+      instance
+        :isCategory: : IsCategory Obj Arr one (λ {a b c} → _:⊕:_ {a} {b} {c})
+        :isCategory: = record
+          { assoc = λ {A} {B} {C} {D} {f} {g} {h} → :assoc: {A} {B} {C} {D} {f} {g} {h}
+          ; ident = {!!}
+          }
+    Fam : Category (lsuc (ℓa ⊔ ℓb)) (ℓa ⊔ ℓb)
+    Fam = record
+      { Object = Obj
+      ; Arrow = Arr
+      ; 𝟙 = one
+      ; _⊕_ = λ {a b c} → _:⊕:_ {a} {b} {c}
+      }
+
+  Contexts = ℂ .Object
+  Substitutions = ℂ .Arrow
+
+  record CwF : Set {!ℓa ⊔ ℓb!} where
+    field
+      Terms : Functor (Opposite ℂ) Fam
+
 module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
   -- Ns is the "namespace"
   ℓo = (lsuc lzero ⊔ ℓ)
@@ -49,5 +97,5 @@ module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
     ; Arrow = Mor
     ; 𝟙 = {!!}
     ; _⊕_ = {!!}
-    ; isCategory = ?
+    ; isCategory = {!!}
     }