Simplifications and renaming
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@ -121,9 +121,6 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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Univalent : Set (ℓa ⊔ ℓb)
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Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (idToIso A B)
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import Cat.Equivalence as E
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open E public using () renaming (Isomorphism to TypeIsomorphism)
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univalenceFromIsomorphism : {A B : Object}
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→ TypeIsomorphism (idToIso A B) → isEquiv (A ≡ B) (A ≅ B) (idToIso A B)
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univalenceFromIsomorphism = fromIso _ _
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@ -305,6 +302,9 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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iso-to-id : (A ≅ B) → (A ≡ B)
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iso-to-id = fst (toIso _ _ univalent)
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asTypeIso : TypeIsomorphism (idToIso A B)
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asTypeIso = toIso _ _ univalent
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-- | All projections are propositions.
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module Propositionality where
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-- | Terminal objects are propositional - a.k.a uniqueness of terminal
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@ -333,10 +333,8 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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right = Yprop _ _
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iso : X ≅ Y
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iso = X→Y , Y→X , left , right
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fromIso' : X ≅ Y → X ≡ Y
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fromIso' = fst (toIso (X ≡ Y) (X ≅ Y) univalent)
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p0 : X ≡ Y
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p0 = fromIso' iso
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p0 = iso-to-id iso
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p1 : (λ i → IsTerminal (p0 i)) [ Xit ≡ Yit ]
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p1 = lemPropF propIsTerminal p0
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res : Xt ≡ Yt
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@ -365,14 +363,8 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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right = Xprop _ _
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iso : X ≅ Y
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iso = Y→X , X→Y , right , left
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fromIso' : X ≅ Y → X ≡ Y
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fromIso' = fst (toIso (X ≡ Y) (X ≅ Y) univalent)
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p0 : X ≡ Y
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p0 = fromIso' iso
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p1 : (λ i → IsInitial (p0 i)) [ Xii ≡ Yii ]
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p1 = lemPropF propIsInitial p0
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res : Xi ≡ Yi
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res i = p0 i , p1 i
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res = lemSig propIsInitial _ _ (iso-to-id iso)
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module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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open RawCategory ℂ
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@ -495,72 +487,62 @@ module Opposite {ℓa ℓb : Level} where
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module _ {A B : ℂ.Object} where
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k : Equivalence.Isomorphism (ℂ.idToIso A B)
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k = toIso _ _ ℂ.univalent
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open Σ k renaming (fst to f ; snd to inv)
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open AreInverses inv
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open Σ k renaming (fst to η ; snd to inv-η)
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open AreInverses inv-η
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_⊙_ = Function._∘_
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infixr 9 _⊙_
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-- f : A ℂ.≅ B → A ≡ B
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flipDem : A ≅ B → A ℂ.≅ B
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flipDem (f , g , inv) = g , f , inv
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genericly : {ℓa ℓb ℓc : Level} {a : Set ℓa} {b : Set ℓb} {c : Set ℓc}
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→ a × b × c → b × a × c
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genericly (a , b , c) = (b , a , c)
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flopDem : A ℂ.≅ B → A ≅ B
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flopDem (f , g , inv) = g , f , inv
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shuffle : A ≅ B → A ℂ.≅ B
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shuffle (f , g , inv) = g , f , inv
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shuffle~ : A ℂ.≅ B → A ≅ B
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shuffle~ (f , g , inv) = g , f , inv
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-- Shouldn't be necessary to use `arrowsAreSets` here, but we have it,
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-- so why not?
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lem : (p : A ≡ B) → idToIso A B p ≡ flopDem (ℂ.idToIso A B p)
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lem p i = l≡r i
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lem : (p : A ≡ B) → idToIso A B p ≡ shuffle~ (ℂ.idToIso A B p)
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lem p = Σ≡ refl (Σ≡ refl (Σ≡ (ℂ.arrowsAreSets _ _ l-l r-l) (ℂ.arrowsAreSets _ _ l-r r-r)))
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where
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l = idToIso A B p
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r = flopDem (ℂ.idToIso A B p)
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r = shuffle~ (ℂ.idToIso A B p)
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open Σ l renaming (fst to l-obv ; snd to l-areInv)
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open Σ l-areInv renaming (fst to l-invs ; snd to l-iso)
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open Σ l-iso renaming (fst to l-l ; snd to l-r)
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open Σ r renaming (fst to r-obv ; snd to r-areInv)
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open Σ r-areInv renaming (fst to r-invs ; snd to r-iso)
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open Σ r-iso renaming (fst to r-l ; snd to r-r)
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l-obv≡r-obv : l-obv ≡ r-obv
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l-obv≡r-obv = refl
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l-invs≡r-invs : l-invs ≡ r-invs
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l-invs≡r-invs = refl
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l-l≡r-l : l-l ≡ r-l
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l-l≡r-l = ℂ.arrowsAreSets _ _ l-l r-l
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l-r≡r-r : l-r ≡ r-r
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l-r≡r-r = ℂ.arrowsAreSets _ _ l-r r-r
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l≡r : l ≡ r
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l≡r i = l-obv≡r-obv i , l-invs≡r-invs i , l-l≡r-l i , l-r≡r-r i
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ff : A ≅ B → A ≡ B
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ff = f ⊙ flipDem
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ζ : A ≅ B → A ≡ B
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ζ = η ⊙ shuffle
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-- inv : AreInverses (ℂ.idToIso A B) f
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invv : AreInverses (idToIso A B) ff
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inv-ζ : AreInverses (idToIso A B) ζ
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-- recto-verso : ℂ.idToIso A B ∘ f ≡ idFun (A ℂ.≅ B)
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invv = record
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inv-ζ = record
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{ verso-recto = funExt (λ x → begin
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(ff ⊙ idToIso A B) x ≡⟨⟩
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(f ⊙ flipDem ⊙ idToIso A B) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → f ⊙ flipDem ⊙ φ) (funExt lem)) ⟩
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(f ⊙ flipDem ⊙ flopDem ⊙ ℂ.idToIso A B) x ≡⟨⟩
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(f ⊙ ℂ.idToIso A B) x ≡⟨ (λ i → verso-recto i x) ⟩
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(ζ ⊙ idToIso A B) x ≡⟨⟩
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(η ⊙ shuffle ⊙ idToIso A B) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → η ⊙ shuffle ⊙ φ) (funExt lem)) ⟩
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(η ⊙ shuffle ⊙ shuffle~ ⊙ ℂ.idToIso A B) x ≡⟨⟩
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(η ⊙ ℂ.idToIso A B) x ≡⟨ (λ i → verso-recto i x) ⟩
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x ∎)
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; recto-verso = funExt (λ x → begin
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(idToIso A B ⊙ f ⊙ flipDem) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ⊙ f ⊙ flipDem) (funExt lem)) ⟩
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(flopDem ⊙ ℂ.idToIso A B ⊙ f ⊙ flipDem) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → flopDem ⊙ φ ⊙ flipDem) recto-verso) ⟩
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(flopDem ⊙ flipDem) x ≡⟨⟩
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(idToIso A B ⊙ η ⊙ shuffle) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ⊙ η ⊙ shuffle) (funExt lem)) ⟩
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(shuffle~ ⊙ ℂ.idToIso A B ⊙ η ⊙ shuffle) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → shuffle~ ⊙ φ ⊙ shuffle) recto-verso) ⟩
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(shuffle~ ⊙ shuffle) x ≡⟨⟩
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x ∎)
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}
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h : Equivalence.Isomorphism (idToIso A B)
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h = ff , invv
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univalent : isEquiv (A ≡ B) (A ≅ B)
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(Univalence.idToIso (swap ℂ.isIdentity) A B)
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univalent = fromIso _ _ h
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h = ζ , inv-ζ
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isCategory : IsCategory opRaw
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IsCategory.isPreCategory isCategory = isPreCategory
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IsCategory.univalent isCategory = univalent
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IsCategory.univalent isCategory = univalenceFromIsomorphism h
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opposite : Category ℓa ℓb
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Category.raw opposite = opRaw
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