Rename assoc to isAssociative

src/Cat/Categories/Cat.agda

@@ -43,8 +43,8 @@ module _ (ℓ ℓ' : Level) where
       }
   private
     open RawCategory RawCat
-    assoc : IsAssociative
+    isAssociative : IsAssociative
-    assoc {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
+    isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
     -- TODO: Rename `ident'` to `ident` after changing how names are exposed in Functor.
     ident' : IsIdentity identity
     ident' = ident-r , ident-l
@@ -96,7 +96,7 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
       postulate univalent : Univalence.Univalent :rawProduct: ident'
       instance
         :isCategory: : IsCategory :rawProduct:
-        IsCategory.assoc :isCategory: = Σ≡ C.assoc D.assoc
+        IsCategory.isAssociative :isCategory: = Σ≡ C.isAssociative D.isAssociative
         IsCategory.ident :isCategory: = ident'
         IsCategory.arrowIsSet :isCategory: = issSet
         IsCategory.univalent :isCategory: = univalent
@@ -288,15 +288,15 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
             𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
               ≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩
             𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
-              ≡⟨ sym assoc ⟩
+              ≡⟨ sym isAssociative ⟩
             𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
-              ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩
+              ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) isAssociative ⟩
             𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
               ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
             𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
-              ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩
+              ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym isAssociative) ⟩
             𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
-              ≡⟨ assoc ⟩
+              ≡⟨ isAssociative ⟩
             𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
               ≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
             𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]

src/Cat/Categories/Fam.agda

@@ -25,8 +25,8 @@ module _ (ℓa ℓb : Level) where
     c ⟨ g ∘ f ⟩ = _∘_ {c = c} g f
 
     module _ {A B C D : Obj'} {f : Arr A B} {g : Arr B C} {h : Arr C D} where
-      assoc : (D ⟨ h ∘ C ⟨ g ∘ f ⟩ ⟩) ≡ D ⟨ D ⟨ h ∘ g ⟩ ∘ f ⟩
+      isAssociative : (D ⟨ h ∘ C ⟨ g ∘ f ⟩ ⟩) ≡ D ⟨ D ⟨ h ∘ g ⟩ ∘ f ⟩
-      assoc = Σ≡ refl refl
+      isAssociative = Σ≡ refl refl
 
     module _ {A B : Obj'} {f : Arr A B} where
       ident : B ⟨ f ∘ one ⟩ ≡ f × B ⟨ one {B} ∘ f ⟩ ≡ f
@@ -44,7 +44,7 @@ module _ (ℓa ℓb : Level) where
     instance
       isCategory : IsCategory RawFam
       isCategory = record
-        { assoc = λ {A} {B} {C} {D} {f} {g} {h} → assoc {D = D} {f} {g} {h}
+        { 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}
         ; arrowIsSet = {!!}
         ; univalent = {!!}

src/Cat/Categories/Free.agda

@@ -26,16 +26,16 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
   open Category ℂ
 
   private
-    p-assoc : {A B C D : Object} {r : Path Arrow A B} {q : Path Arrow B C} {p : Path Arrow C D}
+    p-isAssociative : {A B C D : Object} {r : Path Arrow A B} {q : Path Arrow B C} {p : Path Arrow C D}
       → p ++ (q ++ r) ≡ (p ++ q) ++ r
-    p-assoc {r = r} {q} {empty} = refl
+    p-isAssociative {r = r} {q} {empty} = refl
-    p-assoc {A} {B} {C} {D} {r = r} {q} {cons x p} = begin
+    p-isAssociative {A} {B} {C} {D} {r = r} {q} {cons x p} = begin
       cons x p ++ (q ++ r)   ≡⟨ cong (cons x) lem ⟩
       cons x ((p ++ q) ++ r) ≡⟨⟩
       (cons x p ++ q) ++ r ∎
       where
         lem : p ++ (q ++ r) ≡ ((p ++ q) ++ r)
-        lem = p-assoc {r = r} {q} {p}
+        lem = p-isAssociative {r = r} {q} {p}
 
     ident-r : ∀ {A} {B} {p : Path Arrow A B} → concatenate p empty ≡ p
     ident-r {p = empty} = refl
@@ -57,7 +57,7 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
     }
   RawIsCategoryFree : IsCategory RawFree
   RawIsCategoryFree = record
-    { assoc = λ { {f = f} {g} {h} → p-assoc {r = f} {g} {h}}
+    { isAssociative = λ { {f = f} {g} {h} → p-isAssociative {r = f} {g} {h}}
     ; ident = ident-r , ident-l
     ; arrowIsSet = {!!}
     ; univalent = {!!}

src/Cat/Categories/Fun.agda

@@ -84,11 +84,11 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
     proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η
     proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin
       𝔻 [ (θ ∘nt η) B ∘ F.func→ f ]     ≡⟨⟩
-      𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym assoc ⟩
+      𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym isAssociative ⟩
       𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.func→ f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩
-      𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ assoc ⟩
+      𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ isAssociative ⟩
       𝔻 [ 𝔻 [ θ B ∘ G.func→ f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩
-      𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym assoc ⟩
+      𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym isAssociative ⟩
       𝔻 [ H.func→ f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩
       𝔻 [ H.func→ f ∘ (θ ∘nt η) A ]     ∎
       where
@@ -130,9 +130,9 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       R = (_:⊕:_ {A} {B} {D} (_:⊕:_ {B} {C} {D} ζ' η') θ')
     _g⊕f_ = _:⊕:_ {A} {B} {C}
     _h⊕g_ = _:⊕:_ {B} {C} {D}
-    :assoc: : L ≡ R
+    :isAssociative: : L ≡ R
-    :assoc: = lemSig (naturalIsProp {F = A} {D})
+    :isAssociative: = lemSig (naturalIsProp {F = A} {D})
-      L R (funExt (λ x → assoc))
+      L R (funExt (λ x → isAssociative))
       where
         open Category 𝔻
 
@@ -168,7 +168,7 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
   instance
     :isCategory: : IsCategory RawFun
     :isCategory: = record
-      { assoc = λ {A B C D} → :assoc: {A} {B} {C} {D}
+      { isAssociative = λ {A B C D} → :isAssociative: {A} {B} {C} {D}
       ; ident = λ {A B} → :ident: {A} {B}
       ; arrowIsSet = λ {F} {G} → naturalTransformationIsSets {F} {G}
       ; univalent = {!!}

src/Cat/Categories/Rel.agda

@@ -149,10 +149,10 @@ module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset
       ≃ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
     equi = fwd Cubical.FromStdLib., isequiv
 
-    -- assocc : Q + (R + S) ≡ (Q + R) + S
+    -- isAssociativec : Q + (R + S) ≡ (Q + R) + S
-  is-assoc : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
+  is-isAssociative : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
          ≡ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
-  is-assoc = equivToPath equi
+  is-isAssociative = equivToPath equi
 
 RawRel : RawCategory (lsuc lzero) (lsuc lzero)
 RawRel = record
@@ -164,7 +164,7 @@ RawRel = record
 
 RawIsCategoryRel : IsCategory RawRel
 RawIsCategoryRel = record
-  { assoc = funExt is-assoc
+  { isAssociative = funExt is-isAssociative
   ; ident = funExt ident-l , funExt ident-r
   ; arrowIsSet = {!!}
   ; univalent = {!!}

src/Cat/Categories/Sets.agda

@@ -25,7 +25,7 @@ module _ (ℓ : Level) where
     _∘_ SetsRaw = Function._∘′_
 
     SetsIsCategory : IsCategory SetsRaw
-    assoc SetsIsCategory = refl
+    isAssociative SetsIsCategory = refl
     proj₁ (ident SetsIsCategory) = funExt λ _ → refl
     proj₂ (ident SetsIsCategory) = funExt λ _ → refl
     arrowIsSet SetsIsCategory {B = (_ , s)} = setPi λ _ → s
@@ -99,7 +99,7 @@ module _ {ℓa ℓb : Level} where
       }
     ; isFunctor = record
       { ident = funExt λ _ → proj₂ ident
-      ; distrib = funExt λ x → sym assoc
+      ; distrib = funExt λ x → sym isAssociative
       }
     }
     where
@@ -114,7 +114,7 @@ module _ {ℓa ℓb : Level} where
     }
     ; isFunctor = record
       { ident = funExt λ x → proj₁ ident
-      ; distrib = funExt λ x → assoc
+      ; distrib = funExt λ x → isAssociative
       }
     }
     where

src/Cat/Category.agda

@@ -101,7 +101,7 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
   open RawCategory ℂ
   open Univalence ℂ public
   field
-    assoc : IsAssociative
+    isAssociative : IsAssociative
     ident : IsIdentity 𝟙
     arrowIsSet : ArrowsAreSets
     univalent : Univalent ident
@@ -144,7 +144,7 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
             geq = begin
               g            ≡⟨ sym (fst ident) ⟩
               g ∘ 𝟙        ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
-              g ∘ (f ∘ g') ≡⟨ assoc ⟩
+              g ∘ (f ∘ g') ≡⟨ isAssociative ⟩
               (g ∘ f) ∘ g' ≡⟨ cong (λ φ → φ ∘ g') η ⟩
               𝟙 ∘ g'       ≡⟨ snd ident ⟩
               g'           ∎
@@ -181,7 +181,7 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
       foo = pathJ P helper Y.ident ident
       eqUni : U ident Y.univalent
       eqUni = foo Y.univalent
-      IC.assoc      (done i) = propIsAssociative x X.assoc Y.assoc i
+      IC.isAssociative      (done i) = propIsAssociative x X.isAssociative Y.isAssociative i
       IC.ident      (done i) = ident i
       IC.arrowIsSet (done i) = propArrowIsSet x X.arrowIsSet Y.arrowIsSet i
       IC.univalent  (done i) = eqUni i
@@ -216,7 +216,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
     RawCategory._∘_ OpRaw = Function.flip _∘_
 
     OpIsCategory : IsCategory OpRaw
-    IsCategory.assoc OpIsCategory = sym assoc
+    IsCategory.isAssociative OpIsCategory = sym isAssociative
     IsCategory.ident OpIsCategory = swap ident
     IsCategory.arrowIsSet OpIsCategory = arrowIsSet
     IsCategory.univalent OpIsCategory = {!!}
@@ -242,7 +242,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
     open IsCategory
     module IsCat = IsCategory (ℂ .isCategory)
     rawIsCat : (i : I) → IsCategory (rawOp i)
-    assoc (rawIsCat i) = IsCat.assoc
+    isAssociative (rawIsCat i) = IsCat.isAssociative
     ident (rawIsCat i) = IsCat.ident
     arrowIsSet (rawIsCat i) = IsCat.arrowIsSet
     univalent (rawIsCat i) = IsCat.univalent

src/Cat/Category/Properties.agda

@@ -19,9 +19,9 @@ module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : Category.Object
   iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
     g₀              ≡⟨ sym (proj₁ ident) ⟩
     g₀ ∘ 𝟙          ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
-    g₀ ∘ (f ∘ f-)   ≡⟨ assoc ⟩
+    g₀ ∘ (f ∘ f-)   ≡⟨ isAssociative ⟩
     (g₀ ∘ f) ∘ f-   ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
-    (g₁ ∘ f) ∘ f-   ≡⟨ sym assoc ⟩
+    (g₁ ∘ f) ∘ f-   ≡⟨ sym isAssociative ⟩
     g₁ ∘ (f ∘ f-)   ≡⟨ cong (_∘_ g₁) right-inv ⟩
     g₁ ∘ 𝟙          ≡⟨ proj₁ ident ⟩
     g₁              ∎
@@ -31,9 +31,9 @@ module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : Category.Object
     begin
     g₀            ≡⟨ sym (proj₂ ident) ⟩
     𝟙 ∘ g₀        ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
-    (f- ∘ f) ∘ g₀ ≡⟨ sym assoc ⟩
+    (f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
     f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
-    f- ∘ (f ∘ g₁) ≡⟨ assoc ⟩
+    f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
     (f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
     𝟙 ∘ g₁        ≡⟨ proj₂ ident ⟩
     g₁            ∎
@@ -65,7 +65,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
 
     module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
       :func→: : NaturalTransformation (prshf A) (prshf B)
-      :func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.assoc
+      :func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.isAssociative
 
     module _ {c : Category.Object ℂ} where
       eqTrans : (λ _ → Transformation (prshf c) (prshf c))