Factor univalence out to a seperate module
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@ -65,20 +65,10 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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Monomorphism : {X : Object} → (f : Arrow A B) → Set ℓb
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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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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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-- Univalence is indexed by a raw category as well as an identity proof.
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module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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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 RawCategory ℂ
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module Raw = RawCategory ℂ
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module _ (ident : IsIdentity 𝟙) where
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field
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assoc : IsAssociative
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ident : IsIdentity 𝟙
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arrowIsSet : ∀ {A B : Object} → isSet (Arrow A B)
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idIso : (A : Object) → A ≅ A
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idIso : (A : Object) → A ≅ A
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idIso A = 𝟙 , (𝟙 , ident)
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idIso A = 𝟙 , (𝟙 , ident)
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@ -90,8 +80,21 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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-- equivalent formulations in the book.
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-- equivalent formulations in the book.
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Univalent : Set (ℓa ⊔ ℓb)
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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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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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field
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field
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univalent : Univalent
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assoc : IsAssociative
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ident : IsIdentity 𝟙
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arrowIsSet : ∀ {A B : Object} → isSet (Arrow A B)
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univalent : Univalent ident
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-- `IsCategory` is a mere proposition.
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-- `IsCategory` is a mere proposition.
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module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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@ -136,7 +139,7 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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𝟙 ∘ g' ≡⟨ snd ident ⟩
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𝟙 ∘ g' ≡⟨ snd ident ⟩
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g' ∎
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g' ∎
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propUnivalent : isProp Univalent
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propUnivalent : isProp (Univalent ident)
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propUnivalent a b i = propPi (λ iso → propHasLevel ⟨-2⟩) a b i
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propUnivalent a b i = propPi (λ iso → propHasLevel ⟨-2⟩) a b i
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private
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private
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@ -144,27 +147,21 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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module IC = IsCategory
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module IC = IsCategory
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module X = IsCategory x
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module X = IsCategory x
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module Y = IsCategory y
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module Y = IsCategory y
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open Univalence C
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-- In a few places I use the result of propositionality of the various
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-- In a few places I use the result of propositionality of the various
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-- projections of `IsCategory` - I've arbitrarily chosed to use this
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-- projections of `IsCategory` - I've arbitrarily chosed to use this
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-- result from `x : IsCategory C`. I don't know which (if any) possibly
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-- result from `x : IsCategory C`. I don't know which (if any) possibly
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-- adverse effects this may have.
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-- adverse effects this may have.
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ident : (λ _ → IsIdentity 𝟙) [ X.ident ≡ Y.ident ]
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ident : (λ _ → IsIdentity 𝟙) [ X.ident ≡ Y.ident ]
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ident = propIsIdentity x X.ident Y.ident
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ident = propIsIdentity x X.ident Y.ident
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-- A version of univalence indexed by the identity proof.
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-- Note of course that since it's defined where `RawCategory ℂ` has been opened
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-- this is specialized to that category.
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Univ : IsIdentity 𝟙 → Set _
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Univ idnt = {A B : Y.Raw.Object} →
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isEquiv (A ≡ B) (A ≅ B)
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(λ eq → transp (λ j → A ≅ eq j) (𝟙 , 𝟙 , idnt))
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done : x ≡ y
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done : x ≡ y
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U : ∀ {a : IsIdentity 𝟙} → (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ] → (b : Univ a) → Set _
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U : ∀ {a : IsIdentity 𝟙} → (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ] → (b : Univalent a) → Set _
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U eqwal bbb = (λ i → Univ (eqwal i)) [ X.univalent ≡ bbb ]
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U eqwal bbb = (λ i → Univalent (eqwal i)) [ X.univalent ≡ bbb ]
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P : (y : IsIdentity 𝟙)
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P : (y : IsIdentity 𝟙)
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→ (λ _ → IsIdentity 𝟙) [ X.ident ≡ y ] → Set _
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→ (λ _ → IsIdentity 𝟙) [ X.ident ≡ y ] → Set _
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P y eq = ∀ (b' : Univ y) → U eq b'
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P y eq = ∀ (b' : Univalent y) → U eq b'
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helper : ∀ (b' : Univ X.ident)
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helper : ∀ (b' : Univalent X.ident)
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→ (λ _ → Univ X.ident) [ X.univalent ≡ b' ]
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→ (λ _ → Univalent X.ident) [ X.univalent ≡ b' ]
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helper univ = propUnivalent x X.univalent univ
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helper univ = propUnivalent x X.univalent univ
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foo = pathJ P helper Y.ident ident
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foo = pathJ P helper Y.ident ident
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eqUni : U ident Y.univalent
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eqUni : U ident Y.univalent
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