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[WIP] Arrows are sets in special product category

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Frederik Hanghøj Iversen 2018-04-03 14:46:36 +02:00
parent 467c5d9c0c
commit 1e5fb7d50a
1 changed files with 19 additions and 15 deletions
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34
src/Cat/Category/Product.agda
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@ -160,27 +160,31 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
l = ℂ.rightIdentity l = ℂ.rightIdentity
arrowsAreSets : ArrowsAreSets arrowsAreSets : ArrowsAreSets
arrowsAreSets {X , x0 , x1} {Y , y0 , y1} (f , f0 , f1) (g , g0 , g1) p q = {!!} arrowsAreSets {X , x0 , x1} {Y , y0 , y1} (f , f0 , f1) (g , g0 , g1) p q = pq
where where
prop : ∀ {X Y} (x y : ℂ.Arrow X Y) → isProp (x ≡ y) -- prop : ∀ {X Y} (x y : ℂ.Arrow X Y) → isProp (x ≡ y)
prop = ℂ.arrowsAreSets -- prop = ℂ.arrowsAreSets
a0 a1 : f ≡ g a0 a1 : f ≡ g
a0 i = proj₁ (p i) a0 i = proj₁ (p i)
a1 i = proj₁ (q i) a1 i = proj₁ (q i)
a : a0 ≡ a1 a : a0 ≡ a1
a = ℂ.arrowsAreSets _ _ a0 a1 a = ℂ.arrowsAreSets _ _ a0 a1
res : p ≡ q module _ (i : I) where
res i j = a i j , {!b i j!} , {!!} r : f ≡ g
where r = a i
-- b0 b1 : (λ j → (ℂ [ y0 ∘ a i j ]) ≡ x0) [ f0 ≡ g0 ] module _ (j : I) where
-- b0 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) (a i) prop0 : isProp (ℂ [ y0 ∘ r j ] ≡ x0)
-- b1 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) (a i) prop0 = ℂ.arrowsAreSets _ _
b0 : (λ j → (ℂ [ y0 ∘ a0 j ]) ≡ x0) [ f0 ≡ g0 ] prop1 : isProp (ℂ [ y1 ∘ r j ] ≡ x1)
b0 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) a0 prop1 = ℂ.arrowsAreSets _ _
b1 : (λ j → (ℂ [ y0 ∘ a1 j ]) ≡ x0) [ f0 ≡ g0 ] prop : isProp (ℂ [ y0 ∘ r j ] ≡ x0 × ℂ [ y1 ∘ r j ] ≡ x1)
b1 = lemPropF (λ x → prop (ℂ [ y0 ∘ x ]) x0) a1 prop = propSig prop0 (λ _ → prop1)
-- b : b0 ≡ b1 helper : (b0 b1 : (ℂ [ y0 ∘ r j ]) ≡ x0 × (ℂ [ y1 ∘ r j ]) ≡ x1) → b0 ≡ b1
-- b = {!!} helper _ _ = lemPropF (λ _ → prop) p
b : (ℂ [ y0 ∘ r j ]) ≡ x0 × (ℂ [ y1 ∘ r j ]) ≡ x1
b = {!!}
pq : p ≡ q
pq i j = a i j , b i j
open Univalence isIdentity open Univalence isIdentity