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Progress on univalence

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Frederik Hanghøj Iversen 2018-04-13 09:40:09 +02:00
parent 5bbb40b664
commit 0ced448fa6
2 changed files with 64 additions and 27 deletions
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2
src/Cat/Category.agda
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@ -356,7 +356,7 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
fst (idToIso _ _ refl) <<< f <<< (fst (snd (idToIso _ _ refl))) ∎
where
lem : ∀ {x} → PathP (λ _ → Arrow x x) identity (fst (idToIso x x refl))
lem = sym (subst-neutral {P = λ x → Arrow x x})
lem = sym subst-neutral
D : ∀ a'' → a ≡ a'' → Set _
D a'' p' = coe (λ i → Arrow (p' i) (q i)) f ≡ fst (idToIso b b' q) <<< f <<< (fst (snd (idToIso _ _ p')))

55
src/Cat/Category/Product.agda
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@ -258,22 +258,45 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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
lem : ∀ {A B} {x : A ℂ.≅ B} → ℂ.idToIso A B (ℂ.isoToId x) ≡ x
lem = λ{ {x = x} i → snd ℂ.inverse-from-to-iso' i x}
lem' : ∀ {A B} {x : A ≡ B} → ℂ.isoToId (ℂ.idToIso _ _ x) ≡ x
lem' = λ{ {x = x} i → fst ℂ.inverse-from-to-iso' i x}
h : ya ≡ xa ℂ.<<< f~
h = begin
ya ≡⟨ {!!} ⟩
ya ≡⟨ sym (coe-lem p) ⟩
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}) ⟩
≡⟨ cong (λ φ → φ ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))) subst-neutral ⟩
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~ ∎
complicated : Y ≡ X
complicated = (λ z → sym (ℂ.isoToId (f , f~ , inv-f)) z)
in
( f , {!h!} , {!!})
( f , (begin
ℂ [ ya ∘ f ] ≡⟨⟩
ya ℂ.<<< f ≡⟨ sym ℂ.leftIdentity ⟩
ℂ.identity ℂ.<<< (ya ℂ.<<< f)
≡⟨ ℂ.isAssociative ⟩
ℂ.identity ℂ.<<< ya ℂ.<<< f
≡⟨ cong (λ φ → φ ℂ.<<< ya ℂ.<<< f) (sym subst-neutral) ⟩
fst (ℂ.idToIso _ _ refl) ℂ.<<< ya ℂ.<<< f
≡⟨ cong (λ φ → fst (ℂ.idToIso _ _ refl) ℂ.<<< ya ℂ.<<< φ)
(begin
f ≡⟨ {!cong (λ x → (fst (snd x))) ((lem {x = f~ , f , swap inv-f}))!} ⟩
-- f ≡⟨ cong fst (sym (lem {f , f~ , inv-f})) ⟩
-- fst ( ℂ.idToIso X Y ( ℂ.isoToId (f , f~ , inv-f))) ≡⟨ {!fst inv-f!} ⟩
fst (snd (ℂ.idToIso Y X (λ z → sym (ℂ.isoToId (f , f~ , inv-f)) z))) ∎)
⟩
_ ℂ.<<< ya ℂ.<<< fst (snd (ℂ.idToIso _ _ _))
≡⟨ sym (ℂ.9-1-9 (sym r) refl ya) ⟩
coe (sym r-arrow) ya
≡⟨ coe-lem (λ i → p (~ i)) ⟩
xa ∎) , {!!})
, ( f~ , sym h , {!!})
, lemA (fst inv-f)
, lemA (snd inv-f)
@ -283,14 +306,28 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
let
iso : X ℂ.≅ Y
iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~
helper : PathP (λ i → ℂ.Arrow (ℂ.isoToId {!!} i) A) xa ya
helper = {!!}
in iso , helper , {!!}})
p : X ≡ Y
p = ℂ.isoToId iso
lem : xa ≡ _
lem = begin
xa ≡⟨ sym (fst (snd f)) ⟩
ya ℂ.<<< fst f ≡⟨ sym ℂ.leftIdentity ⟩
ℂ.identity ℂ.<<< (ya ℂ.<<< fst f) ≡⟨ ℂ.isAssociative ⟩
ℂ.identity ℂ.<<< ya ℂ.<<< fst f ≡⟨ {!sym (ℂ.9-1-9 ? refl ya)!} ⟩
_ ℂ.<<< ya ℂ.<<< _ ≡⟨ sym (ℂ.9-1-9 (sym p) refl ya) ⟩
coe (λ i → ℂ.Arrow (p (~ i)) A) ya ∎
lem0 = begin
xa ≡⟨ {!!} ⟩
coe _ xa ≡⟨ {!!} ⟩
ya ℂ.<<< fst f ∎
helper : PathP (λ i → ℂ.Arrow (p i) A) xa ya
helper = {!snd f!}
in iso , ? , {!!}})
, record
{ fst = funExt (λ x → lemSig
(λ x → propSig prop0 (λ _ → prop1))
_ _
(Σ≡ {!!} (ℂ.propIsomorphism _ _ _)))
(Σ≡ {!ℂ!} (ℂ.propIsomorphism _ _ _)))
; snd = funExt (λ{ (f , _) → lemSig propIsomorphism _ _ {!refl!}})
}
where