Make sets a category according to HoTT
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@ -9,55 +9,95 @@ import Function
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open import Cat.Category
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open import Cat.Category.Functor
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open import Cat.Category.Product
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open Category
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module _ {ℓ : Level} where
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SetsRaw : RawCategory (lsuc ℓ) ℓ
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RawCategory.Object SetsRaw = Set ℓ
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RawCategory.Arrow SetsRaw = λ T U → T → U
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RawCategory.𝟙 SetsRaw = Function.id
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RawCategory._∘_ SetsRaw = Function._∘′_
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module _ (ℓ : Level) where
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private
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open RawCategory
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open IsCategory
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open import Cubical.Univalence
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open import Cubical.NType.Properties
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open import Cubical.Universe
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SetsRaw : RawCategory (lsuc ℓ) ℓ
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Object SetsRaw = Cubical.Universe.0-Set
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Arrow SetsRaw (T , _) (U , _) = T → U
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𝟙 SetsRaw = Function.id
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_∘_ SetsRaw = Function._∘′_
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setIsSet : (A : Set ℓ) → isSet A
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setIsSet A x y p q = {!ua!}
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SetsIsCategory : IsCategory SetsRaw
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assoc SetsIsCategory = refl
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proj₁ (ident SetsIsCategory) = funExt λ _ → refl
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proj₂ (ident SetsIsCategory) = funExt λ _ → refl
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arrowIsSet SetsIsCategory = {!!}
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arrowIsSet SetsIsCategory {B = (_ , s)} = setPi λ _ → s
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univalent SetsIsCategory = {!!}
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Sets : Category (lsuc ℓ) ℓ
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raw Sets = SetsRaw
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isCategory Sets = SetsIsCategory
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𝓢𝓮𝓽 Sets : Category (lsuc ℓ) ℓ
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Category.raw 𝓢𝓮𝓽 = SetsRaw
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Category.isCategory 𝓢𝓮𝓽 = SetsIsCategory
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Sets = 𝓢𝓮𝓽
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module _ {ℓ : Level} where
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private
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𝓢 = 𝓢𝓮𝓽 ℓ
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open Category 𝓢
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open import Cubical.Sigma
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module _ (0A 0B : Object) where
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private
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A : Set ℓ
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A = proj₁ 0A
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sA : isSet A
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sA = proj₂ 0A
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B : Set ℓ
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B = proj₁ 0B
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sB : isSet B
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sB = proj₂ 0B
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0A×0B : Object
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0A×0B = (A × B) , sigPresSet sA λ _ → sB
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module _ {X A B : Set ℓ} (f : X → A) (g : X → B) where
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_&&&_ : (X → A × B)
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_&&&_ x = f x , g x
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module _ {X A B : Set ℓ} (f : X → A) (g : X → B) where
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lem : Sets [ proj₁ ∘ (f &&& g)] ≡ f × Sets [ proj₂ ∘ (f &&& g)] ≡ g
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module _ {0X : Object} where
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X = proj₁ 0X
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module _ (f : X → A ) (g : X → B) where
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lem : proj₁ Function.∘′ (f &&& g) ≡ f × proj₂ Function.∘′ (f &&& g) ≡ g
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proj₁ lem = refl
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proj₂ lem = refl
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instance
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isProduct : {A B : Object Sets} → IsProduct Sets {A} {B} proj₁ proj₂
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isProduct f g = f &&& g , lem f g
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isProduct : IsProduct 𝓢 {0A} {0B} {0A×0B} proj₁ proj₂
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isProduct {X = X} f g = (f &&& g) , lem {0X = X} f g
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product : (A B : Object Sets) → Product {ℂ = Sets} A B
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product A B = record { obj = A × B ; proj₁ = proj₁ ; proj₂ = proj₂ ; isProduct = isProduct }
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product : Product {ℂ = 𝓢} 0A 0B
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product = record
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{ obj = 0A×0B
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; proj₁ = Data.Product.proj₁
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; proj₂ = Data.Product.proj₂
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; isProduct = λ { {X} → isProduct {X = X}}
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}
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instance
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SetsHasProducts : HasProducts Sets
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SetsHasProducts : HasProducts 𝓢
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SetsHasProducts = record { product = product }
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-- Covariant Presheaf
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Representable : {ℓ ℓ' : Level} → (ℂ : Category ℓ ℓ') → Set (ℓ ⊔ lsuc ℓ')
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Representable {ℓ' = ℓ'} ℂ = Functor ℂ (Sets {ℓ'})
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module _ {ℓa ℓb : Level} where
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module _ (ℂ : Category ℓa ℓb) where
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-- Covariant Presheaf
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Representable : Set (ℓa ⊔ lsuc ℓb)
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Representable = Functor ℂ (𝓢𝓮𝓽 ℓb)
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-- The "co-yoneda" embedding.
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representable : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} → Category.Object ℂ → Representable ℂ
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representable {ℂ = ℂ} A = record
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-- Contravariant Presheaf
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Presheaf : Set (ℓa ⊔ lsuc ℓb)
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Presheaf = Functor (Opposite ℂ) (𝓢𝓮𝓽 ℓb)
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-- The "co-yoneda" embedding.
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representable : {ℂ : Category ℓa ℓb} → Category.Object ℂ → Representable ℂ
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representable {ℂ = ℂ} A = record
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{ raw = record
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{ func* = λ B → ℂ [ A , B ]
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{ func* = λ B → ℂ [ A , B ] , arrowIsSet
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; func→ = ℂ [_∘_]
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}
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; isFunctor = record
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@ -66,17 +106,13 @@ representable {ℂ = ℂ} A = record
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}
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}
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where
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open IsCategory (isCategory ℂ)
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open Category ℂ
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-- Contravariant Presheaf
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Presheaf : ∀ {ℓ ℓ'} (ℂ : Category ℓ ℓ') → Set (ℓ ⊔ lsuc ℓ')
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Presheaf {ℓ' = ℓ'} ℂ = Functor (Opposite ℂ) (Sets {ℓ'})
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-- Alternate name: `yoneda`
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presheaf : {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} → Category.Object (Opposite ℂ) → Presheaf ℂ
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presheaf {ℂ = ℂ} B = record
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-- Alternate name: `yoneda`
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presheaf : {ℂ : Category ℓa ℓb} → Category.Object (Opposite ℂ) → Presheaf ℂ
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presheaf {ℂ = ℂ} B = record
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{ raw = record
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{ func* = λ A → ℂ [ A , B ]
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{ func* = λ A → ℂ [ A , B ] , arrowIsSet
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; func→ = λ f g → ℂ [ g ∘ f ]
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}
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; isFunctor = record
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@ -85,4 +121,4 @@ presheaf {ℂ = ℂ} B = record
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}
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}
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where
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open IsCategory (isCategory ℂ)
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open Category ℂ
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@ -84,6 +84,7 @@ module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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idIso : (A : Object) → A ≅ A
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idIso A = 𝟙 , (𝟙 , ident)
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-- Lemma 9.1.4 in [HoTT]
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id-to-iso : (A B : Object) → A ≡ B → A ≅ B
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id-to-iso A B eq = transp (\ i → A ≅ eq i) (idIso A)
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@ -93,12 +94,6 @@ module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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Univalent : Set (ℓa ⊔ ℓb)
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Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
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-- Thierry: All projections must be `isProp`'s
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-- According to definitions 9.1.1 and 9.1.6 in the HoTT book the
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-- arrows of a category form a set (arrow-is-set), and there is an
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-- equivalence between the equality of objects and isomorphisms
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-- (univalent).
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record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
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open RawCategory ℂ
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open Univalence ℂ public
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@ -192,14 +187,16 @@ record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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{{isCategory}} : IsCategory raw
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open RawCategory raw public
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open IsCategory isCategory public
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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open Category ℂ
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_[_,_] : (A : Object) → (B : Object) → Set ℓb
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_[_,_] = Arrow
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_[_∘_] : {A B C : Object} → (g : Arrow B C) → (f : Arrow A B) → Arrow A C
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_[_∘_] = _∘_
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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open Category ℂ
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@ -210,8 +207,6 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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RawCategory.𝟙 OpRaw = 𝟙
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RawCategory._∘_ OpRaw = Function.flip _∘_
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open IsCategory isCategory
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OpIsCategory : IsCategory OpRaw
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IsCategory.assoc OpIsCategory = sym assoc
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IsCategory.ident OpIsCategory = swap ident
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