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Make progress with univalence in product-category

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Frederik Hanghøj Iversen 2018-04-12 13:16:25 +02:00
parent 7fcd8e631a
commit 5bbb40b664
1 changed files with 25 additions and 2 deletions
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27
src/Cat/Category/Product.agda
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@ -250,8 +250,31 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
≈ ((X , xa , xb) ≅ (Y , ya , yb))
step2
= ( λ{ ((f , f~ , inv-f) , p , q)
→ ( f , {!ℂ.9-1-9'!} , {!!})
, ( f~ , {!!} , {!!})
→
let
r : X ≡ Y
r = ℂ.isoToId (f , f~ , inv-f)
r-arrow : (ℂ.Arrow X A) ≡ (ℂ.Arrow Y A)
r-arrow i = ℂ.Arrow (r i) A
t : coe r-arrow xa ≡ ℂ.identity ℂ.<<< xa ℂ.<<< f~
t = {!? ≡ ?!}
lem : ∀ {x} → ℂ.idToIso _ _ (ℂ.isoToId x) ≡ x
lem {x} i = snd ℂ.inverse-from-to-iso' i x
h : ya ≡ xa ℂ.<<< f~
h = begin
ya ≡⟨ {!!} ⟩
coe r-arrow xa
≡⟨ ℂ.9-1-9 r refl xa ⟩
fst (ℂ.idToIso _ _ refl) ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))
≡⟨ cong (λ φ → φ ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))) (subst-neutral {P = λ x → ℂ.Arrow x x}) ⟩
ℂ.identity ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))
≡⟨ cong (λ φ → ℂ.identity ℂ.<<< xa ℂ.<<< φ) (cong (λ x → (fst (snd x))) lem) ⟩
ℂ.identity ℂ.<<< xa ℂ.<<< f~
≡⟨ cong (ℂ._<<< f~) ℂ.leftIdentity ⟩
xa ℂ.<<< f~ ∎
in
( f , {!h!} , {!!})
, ( f~ , sym h , {!!})
, lemA (fst inv-f)
, lemA (snd inv-f)
}