Move various type-synonyms to RawCategory
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@ -25,19 +25,46 @@ open import Cat.Wishlist
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syntax ∃!-syntax (λ x → B) = ∃![ x ] B
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record RawCategory (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
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record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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no-eta-equality
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field
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Object : Set ℓ
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Arrow : Object → Object → Set ℓ'
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𝟙 : {o : Object} → Arrow o o
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Object : Set ℓa
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Arrow : Object → Object → Set ℓb
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𝟙 : {A : Object} → Arrow A A
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_∘_ : {A B C : Object} → Arrow B C → Arrow A B → Arrow A C
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infixl 10 _∘_
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domain : { a b : Object } → Arrow a b → Object
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domain {a = a} _ = a
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codomain : { a b : Object } → Arrow a b → Object
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codomain {b = b} _ = b
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IsAssociative : Set (ℓa ⊔ ℓb)
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IsAssociative = ∀ {A B C D} {f : Arrow A B} {g : Arrow B C} {h : Arrow C D}
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→ h ∘ (g ∘ f) ≡ (h ∘ g) ∘ f
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IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
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IsIdentity id = {A B : Object} {f : Arrow A B}
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→ f ∘ id ≡ f × id ∘ f ≡ f
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IsInverseOf : ∀ {A B} → (Arrow A B) → (Arrow B A) → Set ℓb
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IsInverseOf = λ f g → g ∘ f ≡ 𝟙 × f ∘ g ≡ 𝟙
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Isomorphism : ∀ {A B} → (f : Arrow A B) → Set ℓb
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Isomorphism {A} {B} f = Σ[ g ∈ Arrow B A ] IsInverseOf f g
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_≅_ : (A B : Object) → Set ℓb
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_≅_ A B = Σ[ f ∈ Arrow A B ] (Isomorphism f)
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module _ {A B : Object} where
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Epimorphism : {X : Object } → (f : Arrow A B) → Set ℓb
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Epimorphism {X} f = ( g₀ g₁ : Arrow B X ) → g₀ ∘ f ≡ g₁ ∘ f → g₀ ≡ g₁
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Monomorphism : {X : Object} → (f : Arrow A B) → Set ℓb
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Monomorphism {X} f = ( g₀ g₁ : Arrow X A ) → f ∘ g₀ ≡ f ∘ g₁ → g₀ ≡ g₁
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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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@ -47,15 +74,6 @@ record RawCategory (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
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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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module Raw = RawCategory ℂ
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IsAssociative : Set (ℓa ⊔ ℓb)
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IsAssociative = ∀ {A B C D} {f : Arrow A B} {g : Arrow B C} {h : Arrow C D}
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→ h ∘ (g ∘ f) ≡ (h ∘ g) ∘ f
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IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
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IsIdentity id = {A B : Object} {f : Arrow A B}
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→ f ∘ id ≡ f × id ∘ f ≡ f
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field
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assoc : IsAssociative
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ident : IsIdentity 𝟙
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@ -72,9 +90,6 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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propArrowIsSet : isProp (∀ {A B} → isSet (Arrow A B))
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propArrowIsSet a b i = isSetIsProp a b i
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IsInverseOf : ∀ {A B} → (Arrow A B) → (Arrow B A) → Set ℓb
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IsInverseOf = λ f g → g ∘ f ≡ 𝟙 × f ∘ g ≡ 𝟙
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propIsInverseOf : ∀ {A B f g} → isProp (IsInverseOf {A} {B} f g)
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propIsInverseOf x y = λ i →
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let
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@ -84,15 +99,6 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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hh = arrowIsSet _ _ (snd x) (snd y)
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in h i , hh i
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Isomorphism : ∀ {A B} → (f : Arrow A B) → Set ℓb
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Isomorphism {A} {B} f = Σ[ g ∈ Arrow B A ] IsInverseOf f g
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inverse : ∀ {A B} {f : Arrow A B} → Isomorphism f → Arrow B A
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inverse iso = fst iso
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_≅_ : (A B : Object) → Set ℓb
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_≅_ A B = Σ[ f ∈ Arrow A B ] (Isomorphism f)
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idIso : (A : Object) → A ≅ A
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idIso A = 𝟙 , (𝟙 , ident)
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@ -107,13 +113,6 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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field
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univalent : Univalent
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module _ {A B : Object} where
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Epimorphism : {X : Object } → (f : Arrow A B) → Set ℓb
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Epimorphism {X} f = ( g₀ g₁ : Arrow B X ) → g₀ ∘ f ≡ g₁ ∘ f → g₀ ≡ g₁
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Monomorphism : {X : Object} → (f : Arrow A B) → Set ℓb
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Monomorphism {X} f = ( g₀ g₁ : Arrow X A ) → f ∘ g₀ ≡ f ∘ g₁ → g₀ ≡ g₁
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module _ {A B : Object} {f : Arrow A B} where
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isoIsProp : isProp (Isomorphism f)
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isoIsProp a@(g , η , ε) a'@(g' , η' , ε') =
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@ -161,7 +160,7 @@ module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
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open RawCategory ℂ
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Pp : (x.ident ≡ y.ident) → I → Set (ℓa ⊔ ℓb)
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Pp eqIdent i = {A B : Object} →
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isEquiv (A ≡ B) (A x.≅ B)
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isEquiv (A ≡ B) (A ≅ B)
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(λ A≡B →
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transp
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(λ j →
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@ -191,9 +190,12 @@ record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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raw : RawCategory ℓa ℓb
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{{isCategory}} : IsCategory raw
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-- What happens if we just open this up to the public?
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private
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module ℂ = RawCategory raw
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open RawCategory raw public using ( Isomorphism ; Epimorphism ; Monomorphism )
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Object : Set ℓa
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Object = ℂ.Object
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