Make the category an index of PreCategory

src/Cat/Categories/Rel.agda

@@ -163,6 +163,5 @@ IsPreCategory.isAssociative isPreCategory = funExt is-isAssociative
 IsPreCategory.isIdentity    isPreCategory = funExt ident-l , funExt ident-r
 IsPreCategory.arrowsAreSets isPreCategory = {!!}
 
-Rel : PreCategory _ _
-PreCategory.raw Rel = RawRel
+Rel : PreCategory RawRel
 PreCategory.isPreCategory Rel = isPreCategory

src/Cat/Category.agda

@@ -142,16 +142,8 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
     -- This is not so straight-forward so you can assume it
     postulate from[Contr] : Univalent[Contr] → Univalent
 
--- | The mere proposition of being a category.
---
--- Also defines a few lemmas:
---
---     iso-is-epi  : Isomorphism f → Epimorphism {X = X} f
---     iso-is-mono : Isomorphism f → Monomorphism {X = X} f
---
--- Sans `univalent` this would be what is referred to as a pre-category in
--- [HoTT].
-record IsPreCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
+module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
+  record IsPreCategory : Set (lsuc (ℓa ⊔ ℓb)) where
     open RawCategory ℂ public
     field
       isAssociative : IsAssociative
@@ -281,23 +273,22 @@ record IsPreCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (ls
       preorder≅ : Preorder _ _ _
       preorder≅ = record { Carrier = Object ; _≈_ = _≡_ ; _∼_ = _≅_ ; isPreorder = isPreorder }
 
-record PreCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
+  record PreCategory : Set (lsuc (ℓa ⊔ ℓb)) where
     field
-    raw            : RawCategory ℓa ℓb
-    isPreCategory  : IsPreCategory raw
+      isPreCategory  : IsPreCategory
     open IsPreCategory isPreCategory public
 
--- Definition 9.6.1 in [HoTT]
-record StrictCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
+  -- Definition 9.6.1 in [HoTT]
+  record StrictCategory : Set (lsuc (ℓa ⊔ ℓb)) where
     field
-    preCategory : PreCategory ℓa ℓb
+      preCategory : PreCategory
     open PreCategory preCategory
     field
       objectsAreSets : isSet Object
 
-record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
+  record IsCategory : Set (lsuc (ℓa ⊔ ℓb)) where
     field
-    isPreCategory : IsPreCategory ℂ
+      isPreCategory : IsPreCategory
     open IsPreCategory isPreCategory public
     field
       univalent : Univalent
@@ -387,7 +378,6 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
         res : Xi ≡ Yi
         res i = p0 i , p1 i
 
--- | Propositionality of being a category
 module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
   open RawCategory ℂ
   open Univalence
@@ -442,10 +432,6 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
       IsCategory.isPreCategory (done i)
         = propIsPreCategory X.isPreCategory Y.isPreCategory i
       IsCategory.univalent     (done i) = eqUni i
-      -- IsCategory.isAssociative (done i) = Prop.propIsAssociative X.isAssociative Y.isAssociative i
-      -- IsCategory.isIdentity    (done i) = isIdentity i
-      -- IsCategory.arrowsAreSets (done i) = Prop.propArrowIsSet X.arrowsAreSets Y.arrowsAreSets i
-      -- IsCategory.univalent     (done i) = eqUni i
 
   propIsCategory : isProp (IsCategory ℂ)
   propIsCategory = done