[WIP] Propositionality for products
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@ -121,205 +121,215 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object
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propHasProducts = propHasProducts'
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module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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(let module ℂ = Category ℂ) {A B : ℂ.Object} (p : Product ℂ A B) where
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(let module ℂ = Category ℂ) {A B : ℂ.Object} where
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-- open Product p hiding (raw)
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open import Data.Product
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raw : RawCategory _ _
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raw = record
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{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X A × ℂ.Arrow X B
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; Arrow = λ{ (X , xa , xb) (Y , ya , yb)
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→ Σ[ xy ∈ ℂ.Arrow X Y ]
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( ℂ [ ya ∘ xy ] ≡ xa)
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× ℂ [ yb ∘ xy ] ≡ xb
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module _ where
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raw : RawCategory _ _
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raw = record
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{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X A × ℂ.Arrow X B
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; Arrow = λ{ (X , xa , xb) (Y , ya , yb)
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→ Σ[ xy ∈ ℂ.Arrow X Y ]
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( ℂ [ ya ∘ xy ] ≡ xa)
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× ℂ [ yb ∘ xy ] ≡ xb
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}
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; 𝟙 = λ{ {A , f , g} → ℂ.𝟙 {A} , ℂ.rightIdentity , ℂ.rightIdentity}
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; _∘_ = λ { {A , a0 , a1} {B , b0 , b1} {C , c0 , c1} (f , f0 , f1) (g , g0 , g1)
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→ (f ℂ.∘ g)
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, (begin
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ℂ [ c0 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
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ℂ [ ℂ [ c0 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f0 ⟩
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ℂ [ b0 ∘ g ] ≡⟨ g0 ⟩
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a0 ∎
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)
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, (begin
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ℂ [ c1 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
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ℂ [ ℂ [ c1 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f1 ⟩
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ℂ [ b1 ∘ g ] ≡⟨ g1 ⟩
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a1 ∎
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)
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}
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; 𝟙 = λ{ {A , f , g} → ℂ.𝟙 {A} , ℂ.rightIdentity , ℂ.rightIdentity}
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; _∘_ = λ { {A , a0 , a1} {B , b0 , b1} {C , c0 , c1} (f , f0 , f1) (g , g0 , g1)
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→ (f ℂ.∘ g)
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, (begin
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ℂ [ c0 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
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ℂ [ ℂ [ c0 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f0 ⟩
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ℂ [ b0 ∘ g ] ≡⟨ g0 ⟩
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a0 ∎
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)
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, (begin
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ℂ [ c1 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
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ℂ [ ℂ [ c1 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f1 ⟩
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ℂ [ b1 ∘ g ] ≡⟨ g1 ⟩
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a1 ∎
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)
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}
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}
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open RawCategory raw
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open RawCategory raw
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isAssocitaive : IsAssociative
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isAssocitaive {A , a0 , a1} {B , _} {C , c0 , c1} {D , d0 , d1} {ff@(f , f0 , f1)} {gg@(g , g0 , g1)} {hh@(h , h0 , h1)} i
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= s0 i , rl i , rr i
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where
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l = hh ∘ (gg ∘ ff)
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r = hh ∘ gg ∘ ff
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-- s0 : h ℂ.∘ (g ℂ.∘ f) ≡ h ℂ.∘ g ℂ.∘ f
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s0 : proj₁ l ≡ proj₁ r
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s0 = ℂ.isAssociative {f = f} {g} {h}
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prop0 : ∀ a → isProp (ℂ [ d0 ∘ a ] ≡ a0)
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prop0 a = ℂ.arrowsAreSets (ℂ [ d0 ∘ a ]) a0
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rl : (λ i → (ℂ [ d0 ∘ s0 i ]) ≡ a0) [ proj₁ (proj₂ l) ≡ proj₁ (proj₂ r) ]
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rl = lemPropF prop0 s0
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prop1 : ∀ a → isProp ((ℂ [ d1 ∘ a ]) ≡ a1)
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prop1 a = ℂ.arrowsAreSets _ _
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rr : (λ i → (ℂ [ d1 ∘ s0 i ]) ≡ a1) [ proj₂ (proj₂ l) ≡ proj₂ (proj₂ r) ]
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rr = lemPropF prop1 s0
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isIdentity : IsIdentity 𝟙
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isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = leftIdentity , rightIdentity
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where
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leftIdentity : 𝟙 ∘ (f , f0 , f1) ≡ (f , f0 , f1)
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leftIdentity i = l i , rl i , rr i
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isAssocitaive : IsAssociative
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isAssocitaive
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{A , a0 , a1}
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{B , b0 , b1}
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{C , c0 , c1}
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{D , d0 , d1}
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{ff@(f , f0 , f1)}
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{gg@(g , g0 , g1)}
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{hh@(h , h0 , h1)}
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i
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= s0 i , rl i , rr i
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where
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L = 𝟙 ∘ (f , f0 , f1)
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R : Arrow AA BB
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R = f , f0 , f1
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l : proj₁ L ≡ proj₁ R
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l = ℂ.leftIdentity
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prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
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prop0 a = ℂ.arrowsAreSets _ _
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rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
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rl = lemPropF prop0 l
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prop1 : ∀ a → isProp (ℂ [ b1 ∘ a ] ≡ a1)
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prop1 _ = ℂ.arrowsAreSets _ _
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rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
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rr = lemPropF prop1 l
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rightIdentity : (f , f0 , f1) ∘ 𝟙 ≡ (f , f0 , f1)
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rightIdentity i = l i , rl i , {!!}
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l = hh ∘ (gg ∘ ff)
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r = hh ∘ gg ∘ ff
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-- s0 : h ℂ.∘ (g ℂ.∘ f) ≡ h ℂ.∘ g ℂ.∘ f
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s0 : proj₁ l ≡ proj₁ r
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s0 = ℂ.isAssociative {f = f} {g} {h}
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prop0 : ∀ a → isProp (ℂ [ d0 ∘ a ] ≡ a0)
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prop0 a = ℂ.arrowsAreSets (ℂ [ d0 ∘ a ]) a0
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rl : (λ i → (ℂ [ d0 ∘ s0 i ]) ≡ a0) [ proj₁ (proj₂ l) ≡ proj₁ (proj₂ r) ]
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rl = lemPropF prop0 s0
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prop1 : ∀ a → isProp ((ℂ [ d1 ∘ a ]) ≡ a1)
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prop1 a = ℂ.arrowsAreSets _ _
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rr : (λ i → (ℂ [ d1 ∘ s0 i ]) ≡ a1) [ proj₂ (proj₂ l) ≡ proj₂ (proj₂ r) ]
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rr = lemPropF prop1 s0
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isIdentity : IsIdentity 𝟙
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isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = leftIdentity , rightIdentity
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where
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L = (f , f0 , f1) ∘ 𝟙
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R : Arrow AA BB
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R = (f , f0 , f1)
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l : ℂ [ f ∘ ℂ.𝟙 ] ≡ f
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l = ℂ.rightIdentity
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prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
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prop0 _ = ℂ.arrowsAreSets _ _
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rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
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rl = lemPropF prop0 l
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prop1 : ∀ a → isProp ((ℂ [ b1 ∘ a ]) ≡ a1)
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prop1 _ = ℂ.arrowsAreSets _ _
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rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
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rr = lemPropF prop1 l
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leftIdentity : 𝟙 ∘ (f , f0 , f1) ≡ (f , f0 , f1)
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leftIdentity i = l i , rl i , rr i
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where
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L = 𝟙 ∘ (f , f0 , f1)
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R : Arrow AA BB
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R = f , f0 , f1
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l : proj₁ L ≡ proj₁ R
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l = ℂ.leftIdentity
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prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
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prop0 a = ℂ.arrowsAreSets _ _
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rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
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rl = lemPropF prop0 l
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prop1 : ∀ a → isProp (ℂ [ b1 ∘ a ] ≡ a1)
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prop1 _ = ℂ.arrowsAreSets _ _
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rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
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rr = lemPropF prop1 l
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rightIdentity : (f , f0 , f1) ∘ 𝟙 ≡ (f , f0 , f1)
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rightIdentity i = l i , rl i , rr i
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where
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L = (f , f0 , f1) ∘ 𝟙
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R : Arrow AA BB
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R = (f , f0 , f1)
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l : ℂ [ f ∘ ℂ.𝟙 ] ≡ f
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l = ℂ.rightIdentity
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prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
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prop0 _ = ℂ.arrowsAreSets _ _
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rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
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rl = lemPropF prop0 l
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prop1 : ∀ a → isProp ((ℂ [ b1 ∘ a ]) ≡ a1)
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prop1 _ = ℂ.arrowsAreSets _ _
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rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
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rr = lemPropF prop1 l
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cat : IsCategory raw
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cat = record
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{ isAssociative = isAssocitaive
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; isIdentity = isIdentity
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; arrowsAreSets = {!!}
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; univalent = {!!}
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}
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arrowsAreSets : ArrowsAreSets
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arrowsAreSets {X , x0 , x1} {Y , y0 , y1} (f , f0 , f1) (g , g0 , g1) p q = {!!}
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where
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prop : ∀ {X Y} (x y : ℂ.Arrow X Y) → isProp (x ≡ y)
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prop = ℂ.arrowsAreSets
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a0 a1 : f ≡ g
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a0 i = proj₁ (p i)
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a1 i = proj₁ (q i)
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a : a0 ≡ a1
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a = ℂ.arrowsAreSets _ _ a0 a1
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res : p ≡ q
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res i j = a i j , {!b i j!} , {!!}
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where
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-- b0 b1 : (λ j → (ℂ [ y0 ∘ a i j ]) ≡ x0) [ f0 ≡ g0 ]
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-- b0 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) (a i)
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-- b1 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) (a i)
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b0 : (λ j → (ℂ [ y0 ∘ a0 j ]) ≡ x0) [ f0 ≡ g0 ]
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b0 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) a0
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b1 : (λ j → (ℂ [ y0 ∘ a1 j ]) ≡ x0) [ f0 ≡ g0 ]
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b1 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) a1
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-- b : b0 ≡ b1
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-- b = {!!}
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module cat = IsCategory cat
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isCat : IsCategory raw
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isCat = record
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{ isAssociative = isAssocitaive
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; isIdentity = isIdentity
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; arrowsAreSets = arrowsAreSets
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; univalent = {!!}
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}
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lemma : Terminal ≃ Product ℂ A B
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lemma = {!!}
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cat : Category _ _
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cat = record
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{ raw = raw
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; isCategory = isCat
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}
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open Category cat
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open import Cat.Equivalence
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lemma : Terminal ≃ Product ℂ A B
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lemma = Equiv≃.fromIsomorphism Terminal (Product ℂ A B) (f , g , inv)
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where
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f : Terminal → Product ℂ A B
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f ((X , x0 , x1) , uniq) = p
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where
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rawP : RawProduct ℂ A B
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rawP = record
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{ object = X
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; proj₁ = x0
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; proj₂ = x1
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}
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-- open RawProduct rawP renaming (proj₁ to x0 ; proj₂ to x1)
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module _ {Y : ℂ.Object} (p0 : ℂ [ Y , A ]) (p1 : ℂ [ Y , B ]) where
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uy : isContr (Arrow (Y , p0 , p1) (X , x0 , x1))
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uy = uniq {Y , p0 , p1}
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open Σ uy renaming (proj₁ to Y→X ; proj₂ to contractible)
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open Σ Y→X renaming (proj₁ to p0×p1 ; proj₂ to cond)
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ump : ∃![ f×g ] (ℂ [ x0 ∘ f×g ] ≡ p0 P.× ℂ [ x1 ∘ f×g ] ≡ p1)
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ump = p0×p1 , cond , λ {y} x → let k = contractible (y , x) in λ i → proj₁ (k i)
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isP : IsProduct ℂ A B rawP
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isP = record { ump = ump }
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p : Product ℂ A B
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p = record
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{ raw = rawP
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; isProduct = isP
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}
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g : Product ℂ A B → Terminal
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g p = o , t
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where
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module p = Product p
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module isp = IsProduct p.isProduct
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o : Object
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o = p.object , p.proj₁ , p.proj₂
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module _ {Xx : Object} where
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open Σ Xx renaming (proj₁ to X ; proj₂ to x)
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ℂXo : ℂ [ X , isp.object ]
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ℂXo = isp._P[_×_] (proj₁ x) (proj₂ x)
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ump = p.ump (proj₁ x) (proj₂ x)
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Xoo = proj₁ (proj₂ ump)
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Xo : Arrow Xx o
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Xo = ℂXo , Xoo
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contractible : ∀ y → Xo ≡ y
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contractible (y , yy) = res
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where
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k : ℂXo ≡ y
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k = proj₂ (proj₂ ump) (yy)
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prp : ∀ a → isProp
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( (ℂ [ p.proj₁ ∘ a ] ≡ proj₁ x)
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× (ℂ [ p.proj₂ ∘ a ] ≡ proj₂ x)
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)
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prp ab ac ad i
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= ℂ.arrowsAreSets _ _ (proj₁ ac) (proj₁ ad) i
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, ℂ.arrowsAreSets _ _ (proj₂ ac) (proj₂ ad) i
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h :
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( λ i
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→ ℂ [ p.proj₁ ∘ k i ] ≡ proj₁ x
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× ℂ [ p.proj₂ ∘ k i ] ≡ proj₂ x
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) [ Xoo ≡ yy ]
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h = lemPropF prp k
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res : (ℂXo , Xoo) ≡ (y , yy)
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res i = k i , h i
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t : IsTerminal o
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t {Xx} = Xo , contractible
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ve-re : ∀ x → g (f x) ≡ x
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ve-re x = Propositionality.propTerminal _ _
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re-ve : ∀ x → f (g x) ≡ x
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re-ve x = {!!}
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inv : AreInverses f g
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inv = record
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{ verso-recto = funExt ve-re
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; recto-verso = funExt re-ve
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}
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thm : isProp (Product ℂ A B)
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thm = equivPreservesNType {n = ⟨-1⟩} lemma cat.Propositionality.propTerminal
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-- open Univalence ℂ.isIdentity
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-- open import Cat.Equivalence hiding (_≅_)
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-- k : {A B : ℂ.Object} → isEquiv (A ≡ B) (A ℂ.≅ B) (ℂ.id-to-iso A B)
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-- k = ℂ.univalent
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-- module _ {X' Y' : Σ[ X ∈ ℂ.Object ] (ℂ [ X , A ] × ℂ [ X , B ])} where
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-- open Σ X' renaming (proj₁ to X) using ()
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-- open Σ (proj₂ X') renaming (proj₁ to Xxa ; proj₂ to Xxb)
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-- open Σ Y' renaming (proj₁ to Y) using ()
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-- open Σ (proj₂ Y') renaming (proj₁ to Yxa ; proj₂ to Yxb)
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-- module _ (p : X ≡ Y) where
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-- D : ∀ y → X ≡ y → Set _
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-- D y q = ∀ b → (λ i → ℂ [ q i , A ]) [ Xxa ≡ b ]
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-- -- Not sure this is actually provable - but if it were it might involve
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-- -- something like the ump of the product -- in which case perhaps the
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-- -- objects of the category I'm constructing should not merely be the
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-- -- data-part of the product but also the laws.
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-- -- d : D X refl
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-- d : ∀ b → (λ i → ℂ [ X , A ]) [ Xxa ≡ b ]
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-- d b = {!!}
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-- kk : D Y p
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-- kk = pathJ D d Y p
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-- a : (λ i → ℂ [ p i , A ]) [ Xxa ≡ Yxa ]
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-- a = kk Yxa
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-- b : (λ i → ℂ [ p i , B ]) [ Xxb ≡ Yxb ]
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-- b = {!!}
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-- f : X' ≡ Y'
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-- f i = p i , a i , b i
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-- module _ (p : X' ≡ Y') where
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-- g : X ≡ Y
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-- g i = proj₁ (p i)
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-- step0 : (X' ≡ Y') ≃ (X ≡ Y)
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-- step0 = Equiv≃.fromIsomorphism _ _ (g , f , record { verso-recto = {!refl!} ; recto-verso = refl})
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-- step1 : (X ≡ Y) ≃ X ℂ.≅ Y
|
||||
-- step1 = ℂ.univalent≃
|
||||
|
||||
-- -- Just a reminder
|
||||
-- step1-5 : (X' ≅ Y') ≡ (X ℂ.≅ Y)
|
||||
-- step1-5 = refl
|
||||
|
||||
-- step2 : (X' ≡ Y') ≃ (X ℂ.≅ Y)
|
||||
-- step2 = Equivalence.compose step0 step1
|
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|
||||
-- univalent : isEquiv (X' ≡ Y') (X ℂ.≅ Y) (id-to-iso X' Y')
|
||||
-- univalent = proj₂ step2
|
||||
|
||||
-- isCategory : IsCategory raw
|
||||
-- isCategory = record
|
||||
-- { isAssociative = ℂ.isAssociative
|
||||
-- ; isIdentity = ℂ.isIdentity
|
||||
-- ; arrowsAreSets = ℂ.arrowsAreSets
|
||||
-- ; univalent = univalent
|
||||
-- }
|
||||
|
||||
-- category : Category _ _
|
||||
-- category = record
|
||||
-- { raw = raw
|
||||
-- ; isCategory = isCategory
|
||||
-- }
|
||||
|
||||
-- open Category category hiding (IsTerminal ; Object)
|
||||
|
||||
-- -- Essential turns `p : Product ℂ A B` into a triple
|
||||
-- productObject : Object
|
||||
-- productObject = Product.object p , Product.proj₁ p , Product.proj₂ p
|
||||
|
||||
-- productObjectIsTerminal : IsTerminal productObject
|
||||
-- productObjectIsTerminal = {!!}
|
||||
|
||||
-- proppp : isProp (IsTerminal productObject)
|
||||
-- proppp = Propositionality.propIsTerminal productObject
|
||||
|
||||
-- module Try1 {ℓa ℓb : Level} (A B : Set) where
|
||||
-- open import Data.Product
|
||||
-- raw : RawCategory _ _
|
||||
-- raw = record
|
||||
-- { Object = Σ[ X ∈ Set ] (X → A) × (X → B)
|
||||
-- ; Arrow = λ{ (X0 , f0 , g0) (X1 , f1 , g1) → X0 → X1}
|
||||
-- ; 𝟙 = λ x → x
|
||||
-- ; _∘_ = λ x x₁ x₂ → x (x₁ x₂)
|
||||
-- }
|
||||
|
||||
-- open RawCategory raw
|
||||
|
||||
-- isCategory : IsCategory raw
|
||||
-- isCategory = record
|
||||
-- { isAssociative = refl
|
||||
-- ; isIdentity = refl , refl
|
||||
-- ; arrowsAreSets = {!!}
|
||||
-- ; univalent = {!!}
|
||||
-- }
|
||||
|
||||
-- t : IsTerminal ((A × B) , proj₁ , proj₂)
|
||||
-- t = {!!}
|
||||
thm = equivPreservesNType {n = ⟨-1⟩} lemma Propositionality.propTerminal
|
||||
|
|
|
|||
Loading…
Reference in a new issue