Rename ident to isIdentity

src/Cat/Categories/Cat.agda

@@ -91,13 +91,13 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
       issSet = setSig {sA = C.arrowIsSet} {sB = λ x → D.arrowIsSet}
       ident' : IsIdentity :𝟙:
       ident'
-        = Σ≡ (fst C.ident) (fst D.ident)
+        = Σ≡ (fst C.isIdentity) (fst D.isIdentity)
-        , Σ≡ (snd C.ident) (snd D.ident)
+        , Σ≡ (snd C.isIdentity) (snd D.isIdentity)
       postulate univalent : Univalence.Univalent :rawProduct: ident'
       instance
         :isCategory: : IsCategory :rawProduct:
         IsCategory.isAssociative :isCategory: = Σ≡ C.isAssociative D.isAssociative
-        IsCategory.ident :isCategory: = ident'
+        IsCategory.isIdentity :isCategory: = ident'
         IsCategory.arrowIsSet :isCategory: = issSet
         IsCategory.univalent :isCategory: = univalent
 
@@ -107,13 +107,13 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
       proj₁ : Catt [ :product: , ℂ ]
       proj₁ = record
         { raw = record { func* = fst ; func→ = fst }
-        ; isFunctor = record { ident = refl ; distrib = refl }
+        ; isFunctor = record { isIdentity = refl ; distrib = refl }
         }
 
       proj₂ : Catt [ :product: , 𝔻 ]
       proj₂ = record
         { raw = record { func* = snd ; func→ = snd }
-        ; isFunctor = record { ident = refl ; distrib = refl }
+        ; isFunctor = record { isIdentity = refl ; distrib = refl }
         }
 
       module _ {X : Object Catt} (x₁ : Catt [ X , ℂ ]) (x₂ : Catt [ X , 𝔻 ]) where
@@ -124,7 +124,7 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
             ; func→ = λ x → func→ x₁ x , func→ x₂ x
             }
           ; isFunctor = record
-            { ident   = Σ≡ x₁.ident x₂.ident
+            { isIdentity   = Σ≡ x₁.isIdentity x₂.isIdentity
             ; distrib = Σ≡ x₁.distrib x₂.distrib
             }
           }
@@ -230,7 +230,7 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
 
         -- NaturalTransformation F G × ℂ .Arrow A B
         -- :ident: : :func→: {c} {c} (identityNat F , ℂ .𝟙) ≡ 𝔻 .𝟙
-        -- :ident: = trans (proj₂ 𝔻.ident) (F .ident)
+        -- :ident: = trans (proj₂ 𝔻.isIdentity) (F .isIdentity)
         --   where
         --     open module 𝔻 = IsCategory (𝔻 .isCategory)
         -- Unfortunately the equational version has some ambigous arguments.
@@ -239,8 +239,8 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
           :func→: {c} {c} (𝟙 (Product.obj (:obj: ×p ℂ)) {c}) ≡⟨⟩
           :func→: {c} {c} (identityNat F , 𝟙 ℂ)             ≡⟨⟩
           𝔻 [ identityTrans F C ∘ func→ F (𝟙 ℂ)]           ≡⟨⟩
-          𝔻 [ 𝟙 𝔻 ∘ func→ F (𝟙 ℂ)]                        ≡⟨ proj₂ 𝔻.ident ⟩
+          𝔻 [ 𝟙 𝔻 ∘ func→ F (𝟙 ℂ)]                        ≡⟨ proj₂ 𝔻.isIdentity ⟩
-          func→ F (𝟙 ℂ)                                    ≡⟨ F.ident ⟩
+          func→ F (𝟙 ℂ)                                    ≡⟨ F.isIdentity ⟩
           𝟙 𝔻                                               ∎
           where
             open module 𝔻 = Category 𝔻
@@ -313,7 +313,7 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
           ; func→ = λ {dom} {cod} → :func→: {dom} {cod}
           }
         ; isFunctor = record
-          { ident = λ {o} → :ident: {o}
+          { isIdentity = λ {o} → :ident: {o}
           ; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y}
           }
         }

src/Cat/Categories/Fam.agda

@@ -29,8 +29,8 @@ module _ (ℓa ℓb : Level) where
       isAssociative = Σ≡ refl refl
 
     module _ {A B : Obj'} {f : Arr A B} where
-      ident : B ⟨ f ∘ one ⟩ ≡ f × B ⟨ one {B} ∘ f ⟩ ≡ f
+      isIdentity : B ⟨ f ∘ one ⟩ ≡ f × B ⟨ one {B} ∘ f ⟩ ≡ f
-      ident = (Σ≡ refl refl) , Σ≡ refl refl
+      isIdentity = (Σ≡ refl refl) , Σ≡ refl refl
 
 
     RawFam : RawCategory (lsuc (ℓa ⊔ ℓb)) (ℓa ⊔ ℓb)
@@ -45,7 +45,7 @@ module _ (ℓa ℓb : Level) where
       isCategory : IsCategory RawFam
       isCategory = record
         { isAssociative = λ {A} {B} {C} {D} {f} {g} {h} → isAssociative {D = D} {f} {g} {h}
-        ; ident = λ {A} {B} {f} → ident {A} {B} {f = f}
+        ; isIdentity = λ {A} {B} {f} → isIdentity {A} {B} {f = f}
         ; arrowIsSet = {!!}
         ; univalent = {!!}
         }

src/Cat/Categories/Free.agda

@@ -58,7 +58,7 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
   RawIsCategoryFree : IsCategory RawFree
   RawIsCategoryFree = record
     { isAssociative = λ { {f = f} {g} {h} → p-isAssociative {r = f} {g} {h}}
-    ; ident = ident-r , ident-l
+    ; isIdentity = ident-r , ident-l
     ; arrowIsSet = {!!}
     ; univalent = {!!}
     }

src/Cat/Categories/Fun.agda

@@ -60,8 +60,8 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
   identityNatural : (F : Functor ℂ 𝔻) → Natural F F (identityTrans F)
   identityNatural F {A = A} {B = B} f = begin
     𝔻 [ identityTrans F B ∘ F→ f ]  ≡⟨⟩
-    𝔻 [ 𝟙 𝔻 ∘  F→ f ]              ≡⟨ proj₂ 𝔻.ident ⟩
+    𝔻 [ 𝟙 𝔻 ∘  F→ f ]              ≡⟨ proj₂ 𝔻.isIdentity ⟩
-    F→ f                            ≡⟨ sym (proj₁ 𝔻.ident) ⟩
+    F→ f                            ≡⟨ sym (proj₁ 𝔻.isIdentity) ⟩
     𝔻 [ F→ f ∘ 𝟙 𝔻 ]               ≡⟨⟩
     𝔻 [ F→ f ∘ identityTrans F A ]  ∎
     where
@@ -143,10 +143,10 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       eq-r : ∀ C → (𝔻 [ f' C ∘ identityTrans A C ]) ≡ f' C
       eq-r C = begin
         𝔻 [ f' C ∘ identityTrans A C ] ≡⟨⟩
-        𝔻 [ f' C ∘ 𝔻.𝟙 ]  ≡⟨ proj₁ 𝔻.ident ⟩
+        𝔻 [ f' C ∘ 𝔻.𝟙 ]  ≡⟨ proj₁ 𝔻.isIdentity ⟩
         f' C ∎
       eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
-      eq-l C = proj₂ 𝔻.ident
+      eq-l C = proj₂ 𝔻.isIdentity
       ident-r : (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f
       ident-r = lemSig allNatural _ _ (funExt eq-r)
       ident-l : (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f
@@ -169,7 +169,7 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
     :isCategory: : IsCategory RawFun
     :isCategory: = record
       { isAssociative = λ {A B C D} → :isAssociative: {A} {B} {C} {D}
-      ; ident = λ {A B} → :ident: {A} {B}
+      ; isIdentity = λ {A B} → :ident: {A} {B}
       ; arrowIsSet = λ {F} {G} → naturalTransformationIsSets {F} {G}
       ; univalent = {!!}
       }

src/Cat/Categories/Rel.agda

@@ -165,7 +165,7 @@ RawRel = record
 RawIsCategoryRel : IsCategory RawRel
 RawIsCategoryRel = record
   { isAssociative = funExt is-isAssociative
-  ; ident = funExt ident-l , funExt ident-r
+  ; isIdentity = funExt ident-l , funExt ident-r
   ; arrowIsSet = {!!}
   ; univalent = {!!}
   }

src/Cat/Categories/Sets.agda

@@ -26,8 +26,8 @@ module _ (ℓ : Level) where
 
     SetsIsCategory : IsCategory SetsRaw
     isAssociative SetsIsCategory = refl
-    proj₁ (ident SetsIsCategory) = funExt λ _ → refl
+    proj₁ (isIdentity SetsIsCategory) = funExt λ _ → refl
-    proj₂ (ident SetsIsCategory) = funExt λ _ → refl
+    proj₂ (isIdentity SetsIsCategory) = funExt λ _ → refl
     arrowIsSet SetsIsCategory {B = (_ , s)} = setPi λ _ → s
     univalent SetsIsCategory = {!!}
 
@@ -98,7 +98,7 @@ module _ {ℓa ℓb : Level} where
       ; func→ = ℂ [_∘_]
       }
     ; isFunctor = record
-      { ident = funExt λ _ → proj₂ ident
+      { isIdentity = funExt λ _ → proj₂ isIdentity
       ; distrib = funExt λ x → sym isAssociative
       }
     }
@@ -113,7 +113,7 @@ module _ {ℓa ℓb : Level} where
       ; func→ = λ f g → ℂ [ g ∘ f ]
     }
     ; isFunctor = record
-      { ident = funExt λ x → proj₁ ident
+      { isIdentity = funExt λ x → proj₁ isIdentity
       ; distrib = funExt λ x → isAssociative
       }
     }

src/Cat/Category.agda

@@ -83,9 +83,9 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
 -- Univalence is indexed by a raw category as well as an identity proof.
 module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
   open RawCategory ℂ
-  module _ (ident : IsIdentity 𝟙) where
+  module _ (isIdentity : IsIdentity 𝟙) where
     idIso : (A : Object) → A ≅ A
-    idIso A = 𝟙 , (𝟙 , ident)
+    idIso A = 𝟙 , (𝟙 , isIdentity)
 
     -- Lemma 9.1.4 in [HoTT]
     id-to-iso : (A B : Object) → A ≡ B → A ≅ B
@@ -102,9 +102,9 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
   open Univalence ℂ public
   field
     isAssociative : IsAssociative
-    ident : IsIdentity 𝟙
+    isIdentity : IsIdentity 𝟙
     arrowIsSet : ArrowsAreSets
-    univalent : Univalent ident
+    univalent : Univalent isIdentity
 
 -- `IsCategory` is a mere proposition.
 module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
@@ -142,14 +142,14 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
             open Cubical.NType.Properties
             geq : g ≡ g'
             geq = begin
-              g            ≡⟨ sym (fst ident) ⟩
+              g            ≡⟨ sym (fst isIdentity) ⟩
               g ∘ 𝟙        ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
               g ∘ (f ∘ g') ≡⟨ isAssociative ⟩
               (g ∘ f) ∘ g' ≡⟨ cong (λ φ → φ ∘ g') η ⟩
-              𝟙 ∘ g'       ≡⟨ snd ident ⟩
+              𝟙 ∘ g'       ≡⟨ snd isIdentity ⟩
               g'           ∎
 
-    propUnivalent : isProp (Univalent ident)
+    propUnivalent : isProp (Univalent isIdentity)
     propUnivalent a b i = propPi (λ iso → propHasLevel ⟨-2⟩) a b i
 
   private
@@ -162,27 +162,27 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
       -- projections of `IsCategory` - I've arbitrarily chosed to use this
       -- result from `x : IsCategory C`. I don't know which (if any) possibly
       -- adverse effects this may have.
-      ident : (λ _ → IsIdentity 𝟙) [ X.ident ≡ Y.ident ]
+      isIdentity : (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ Y.isIdentity ]
-      ident = propIsIdentity x X.ident Y.ident
+      isIdentity = propIsIdentity x X.isIdentity Y.isIdentity
       done : x ≡ y
       U : ∀ {a : IsIdentity 𝟙}
-        → (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ]
+        → (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ a ]
         → (b : Univalent a)
         → Set _
       U eqwal bbb =
         (λ i → Univalent (eqwal i))
         [ X.univalent ≡ bbb ]
       P : (y : IsIdentity 𝟙)
-        → (λ _ → IsIdentity 𝟙) [ X.ident ≡ y ] → Set _
+        → (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ y ] → Set _
       P y eq = ∀ (b' : Univalent y) → U eq b'
-      helper : ∀ (b' : Univalent X.ident)
+      helper : ∀ (b' : Univalent X.isIdentity)
-        → (λ _ → Univalent X.ident) [ X.univalent ≡ b' ]
+        → (λ _ → Univalent X.isIdentity) [ X.univalent ≡ b' ]
       helper univ = propUnivalent x X.univalent univ
-      foo = pathJ P helper Y.ident ident
+      foo = pathJ P helper Y.isIdentity isIdentity
-      eqUni : U ident Y.univalent
+      eqUni : U isIdentity Y.univalent
       eqUni = foo Y.univalent
       IC.isAssociative      (done i) = propIsAssociative x X.isAssociative Y.isAssociative i
-      IC.ident      (done i) = ident i
+      IC.isIdentity      (done i) = isIdentity i
       IC.arrowIsSet (done i) = propArrowIsSet x X.arrowIsSet Y.arrowIsSet i
       IC.univalent  (done i) = eqUni i
 
@@ -217,7 +217,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
 
     OpIsCategory : IsCategory OpRaw
     IsCategory.isAssociative OpIsCategory = sym isAssociative
-    IsCategory.ident OpIsCategory = swap ident
+    IsCategory.isIdentity OpIsCategory = swap isIdentity
     IsCategory.arrowIsSet OpIsCategory = arrowIsSet
     IsCategory.univalent OpIsCategory = {!!}
 
@@ -243,7 +243,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
     module IsCat = IsCategory (ℂ .isCategory)
     rawIsCat : (i : I) → IsCategory (rawOp i)
     isAssociative (rawIsCat i) = IsCat.isAssociative
-    ident (rawIsCat i) = IsCat.ident
+    isIdentity (rawIsCat i) = IsCat.isIdentity
     arrowIsSet (rawIsCat i) = IsCat.arrowIsSet
     univalent (rawIsCat i) = IsCat.univalent
 

src/Cat/Category/Functor.agda

@@ -33,7 +33,7 @@ module _ {ℓc ℓc' ℓd ℓd'}
   record IsFunctor (F : RawFunctor) : 𝓤 where
     open RawFunctor F public
     field
-      ident   : IsIdentity
+      isIdentity   : IsIdentity
       distrib : IsDistributive
 
   record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
@@ -55,7 +55,7 @@ module _
 
   propIsFunctor : isProp (IsFunctor _ _ F)
   propIsFunctor isF0 isF1 i = record
-    { ident = 𝔻.arrowIsSet _ _ isF0.ident isF1.ident i
+    { isIdentity = 𝔻.arrowIsSet _ _ isF0.isIdentity isF1.isIdentity i
     ; distrib = 𝔻.arrowIsSet _ _ isF0.distrib isF1.distrib i
     }
     where
@@ -116,10 +116,10 @@ module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : F
     instance
       isFunctor' : IsFunctor A C _∘fr_
       isFunctor' = record
-        { ident = begin
+        { isIdentity = begin
           (F→ ∘ G→) (𝟙 A) ≡⟨ refl ⟩
-          F→ (G→ (𝟙 A))   ≡⟨ cong F→ (ident G)⟩
+          F→ (G→ (𝟙 A))   ≡⟨ cong F→ (isIdentity G)⟩
-          F→ (𝟙 B)        ≡⟨ ident F ⟩
+          F→ (𝟙 B)        ≡⟨ isIdentity F ⟩
           𝟙 C             ∎
         ; distrib = dist
         }
@@ -135,7 +135,7 @@ identity = record
     ; func→ = λ x → x
     }
   ; isFunctor = record
-    { ident = refl
+    { isIdentity = refl
     ; distrib = refl
     }
   }

src/Cat/Category/Properties.agda

@@ -17,25 +17,25 @@ module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : Category.Object
 
   iso-is-epi : Isomorphism f → Epimorphism {X = X} f
   iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
-    g₀              ≡⟨ sym (proj₁ ident) ⟩
+    g₀              ≡⟨ sym (proj₁ isIdentity) ⟩
     g₀ ∘ 𝟙          ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
     g₀ ∘ (f ∘ f-)   ≡⟨ isAssociative ⟩
     (g₀ ∘ f) ∘ f-   ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
     (g₁ ∘ f) ∘ f-   ≡⟨ sym isAssociative ⟩
     g₁ ∘ (f ∘ f-)   ≡⟨ cong (_∘_ g₁) right-inv ⟩
-    g₁ ∘ 𝟙          ≡⟨ proj₁ ident ⟩
+    g₁ ∘ 𝟙          ≡⟨ proj₁ isIdentity ⟩
     g₁              ∎
 
   iso-is-mono : Isomorphism f → Monomorphism {X = X} f
   iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
     begin
-    g₀            ≡⟨ sym (proj₂ ident) ⟩
+    g₀            ≡⟨ sym (proj₂ isIdentity) ⟩
     𝟙 ∘ g₀        ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
     (f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
     f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
     f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
     (f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
-    𝟙 ∘ g₁        ≡⟨ proj₂ ident ⟩
+    𝟙 ∘ g₁        ≡⟨ proj₂ isIdentity ⟩
     g₁            ∎
 
   iso-is-epi-mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
@@ -70,7 +70,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
     module _ {c : Category.Object ℂ} where
       eqTrans : (λ _ → Transformation (prshf c) (prshf c))
         [ (λ _ x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c) ]
-      eqTrans = funExt λ x → funExt λ x → ℂ.ident .proj₂
+      eqTrans = funExt λ x → funExt λ x → ℂ.isIdentity .proj₂
 
       open import Cubical.NType.Properties
       open import Cat.Categories.Fun
@@ -78,7 +78,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
       :ident: = lemSig (naturalIsProp {F = prshf c} {prshf c}) _ _ eq
         where
           eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c)
-          eq = funExt λ A → funExt λ B → proj₂ ℂ.ident
+          eq = funExt λ A → funExt λ B → proj₂ ℂ.isIdentity
 
   yoneda : Functor ℂ (Fun {ℂ = Opposite ℂ} {𝔻 = 𝓢})
   yoneda = record
@@ -87,7 +87,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
       ; func→ = :func→:
       }
     ; isFunctor = record
-      { ident = :ident:
+      { isIdentity = :ident:
       ; distrib = {!!}
       }
     }