Stuff about the free category

src/Cat/Categories/Free.agda

@@ -2,41 +2,61 @@
 module Cat.Categories.Free where
 
 open import Agda.Primitive
-open import Cubical hiding (Path)
+open import Cubical hiding (Path ; isSet ; empty)
 open import Data.Product
 
-open import Cat.Category as C
+open import Cat.Category
+
+open IsCategory
+open Category
+
+-- data Path {ℓ : Level} {A : Set ℓ} : (a b : A) → Set ℓ where
+--   emptyPath : {a : A} → Path a a
+--   concatenate : {a b c : A} → Path a b → Path b c → Path a b
 
 module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
-  private
-    open module ℂ = Category ℂ
+  module ℂ = Category ℂ
 
-  postulate
-    Path : (a b : ℂ.Object) → Set ℓ'
-    emptyPath : (o : ℂ.Object) → Path o o
-    concatenate : {a b c : ℂ.Object} → Path b c → Path a b → Path a c
+  -- import Data.List
+  -- P : (a b : Object ℂ) → Set (ℓ ⊔ ℓ')
+  -- P = {!Data.List.List ?!}
+  -- Generalized paths:
+  -- data P {ℓ : Level} {A : Set ℓ} (R : A → A → Set ℓ) : (a b : A) → Set ℓ where
+  --   e : {a : A} → P R a a
+  --   c : {a b c : A} → R a b → P R b c → P R a c
+
+  -- Path's are like lists with directions.
+  -- This implementation is specialized to categories.
+  data Path : (a b : Object ℂ) → Set (ℓ ⊔ ℓ') where
+    empty : {A : Object ℂ} → Path A A
+    cons : ∀ {A B C} → ℂ [ B , C ] → Path A B → Path A C
+
+  concatenate : ∀ {A B C : Object ℂ}  → Path B C → Path A B → Path A C
+  concatenate empty p = p
+  concatenate (cons x q) p = cons x (concatenate q p)
 
   private
-    module _ {A B C D : ℂ.Object} {r : Path A B} {q : Path B C} {p : Path C D} where
-      postulate
-        p-assoc : concatenate {A} {C} {D} p (concatenate {A} {B} {C} q r)
-          ≡ concatenate {A} {B} {D} (concatenate {B} {C} {D} p q) r
-    module _ {A B : ℂ.Object} {p : Path A B} where
-      postulate
-        ident-r : concatenate {A} {A} {B} p (emptyPath A) ≡ p
-        ident-l : concatenate {A} {B} {B} (emptyPath B) p ≡ p
-
-  RawFree : RawCategory ℓ ℓ'
+    module _ {A B C D : Object ℂ} where
+      p-assoc : {r : Path A B} {q : Path B C} {p : Path C D} → concatenate p (concatenate q r) ≡ concatenate (concatenate p q) r
+      p-assoc {r} {q} {p} = {!!}
+    module _ {A B : Object ℂ} {p : Path A B} where
+      -- postulate
+      --   ident-r : concatenate {A} {A} {B} p (lift 𝟙) ≡ p
+      --   ident-l : concatenate {A} {B} {B} (lift 𝟙) p ≡ p
+    module _ {A B : Object ℂ} where
+      isSet : IsSet (Path A B)
+      isSet = {!!}
+  RawFree : RawCategory ℓ (ℓ ⊔ ℓ')
   RawFree = record
-    { Object = ℂ.Object
+    { Object = Object ℂ
     ; Arrow = Path
-    ; 𝟙 = λ {o} → emptyPath o
-    ; _∘_ = λ {a b c} → concatenate {a} {b} {c}
+    ; 𝟙 = λ {o} → {!lift 𝟙!}
+    ; _∘_ = λ {a b c} → {!concatenate {a} {b} {c}!}
     }
   RawIsCategoryFree : IsCategory RawFree
   RawIsCategoryFree = record
-    { assoc = p-assoc
-    ; ident = ident-r , ident-l
+    { assoc = {!p-assoc!}
+    ; ident = {!ident-r , ident-l!}
     ; arrowIsSet = {!!}
     ; univalent = {!!}
     }