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Rename id-to-iso to idToIso

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Frederik Hanghøj Iversen 2018-04-05 15:21:54 +02:00
parent b5f89322ac
commit e69ace21a0
5 changed files with 27 additions and 27 deletions
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2
BACKLOG.md
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@ -6,7 +6,7 @@ Prove postulates in `Cat.Wishlist`:
Prove that these two formulations of univalence are equivalent: Prove that these two formulations of univalence are equivalent:
∀ A B → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B) ∀ A B → isEquiv (A ≡ B) (A ≅ B) (idToIso A B)
∀ A → isContr (Σ[ X ∈ Object ] A ≅ X) ∀ A → isContr (Σ[ X ∈ Object ] A ≅ X)
Prove univalence for the category of Prove univalence for the category of

2
src/Cat/Categories/Free.agda
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@ -67,7 +67,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
IsPreCategory.arrowsAreSets isPreCategory = arrowsAreSets IsPreCategory.arrowsAreSets isPreCategory = arrowsAreSets
module _ {A B : ℂ.Object} where module _ {A B : ℂ.Object} where
eqv : isEquiv (A ≡ B) (A ≅ B) (Univalence.id-to-iso isIdentity A B) eqv : isEquiv (A ≡ B) (A ≅ B) (Univalence.idToIso isIdentity A B)
eqv = {!!} eqv = {!!}
univalent : Univalent univalent : Univalent

4
src/Cat/Categories/Fun.agda
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@ -35,7 +35,7 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
module _ (F : Functor ℂ 𝔻) where module _ (F : Functor ℂ 𝔻) where
center : Σ[ G ∈ Object ] (F ≅ G) center : Σ[ G ∈ Object ] (F ≅ G)
center = F , id-to-iso F F refl center = F , idToIso F F refl
open Σ center renaming (snd to isoF) open Σ center renaming (snd to isoF)
@ -175,7 +175,7 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
re-ve : (x : A ≡ B) → reverse (obverse x) ≡ x re-ve : (x : A ≡ B) → reverse (obverse x) ≡ x
re-ve = {!!} re-ve = {!!}
done : isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B) done : isEquiv (A ≡ B) (A ≅ B) (idToIso A B)
done = {!gradLemma obverse reverse ve-re re-ve!} done = {!gradLemma obverse reverse ve-re re-ve!}
univalent : Univalent univalent : Univalent

38
src/Cat/Category.agda
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@ -114,11 +114,11 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
-- | Extract an isomorphism from an equality -- | Extract an isomorphism from an equality
-- --
-- [HoTT §9.1.4] -- [HoTT §9.1.4]
id-to-iso : (A B : Object) → A ≡ B → A ≅ B idToIso : (A B : Object) → A ≡ B → A ≅ B
id-to-iso A B eq = transp (\ i → A ≅ eq i) (idIso A) idToIso A B eq = transp (\ i → A ≅ eq i) (idIso A)
Univalent : Set (ℓa ⊔ ℓb) Univalent : Set (ℓa ⊔ ℓb)
Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B) Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (idToIso A B)
-- A perhaps more readable version of univalence: -- A perhaps more readable version of univalence:
Univalent≃ = {A B : Object} → (A ≡ B) ≃ (A ≅ B) Univalent≃ = {A B : Object} → (A ≡ B) ≃ (A ≅ B)
@ -149,7 +149,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
-- Some error with primComp -- Some error with primComp
isoAY : A ≅ Y isoAY : A ≅ Y
isoAY = {!id-to-iso A Y q!} isoAY = {!idToIso A Y q!}
lem : PathP (λ i → A ≅ q i) (idIso A) isoY lem : PathP (λ i → A ≅ q i) (idIso A) isoY
lem = d* isoAY lem = d* isoAY
@ -548,7 +548,7 @@ module Opposite {ℓa ℓb : Level} where
module _ {A B : ℂ.Object} where module _ {A B : ℂ.Object} where
open import Cat.Equivalence as Equivalence hiding (_≅_) open import Cat.Equivalence as Equivalence hiding (_≅_)
k : Equivalence.Isomorphism (ℂ.id-to-iso A B) k : Equivalence.Isomorphism (ℂ.idToIso A B)
k = Equiv≃.toIso _ _ ℂ.univalent k = Equiv≃.toIso _ _ ℂ.univalent
open Σ k renaming (fst to f ; snd to inv) open Σ k renaming (fst to f ; snd to inv)
open AreInverses inv open AreInverses inv
@ -568,11 +568,11 @@ module Opposite {ℓa ℓb : Level} where
-- Shouldn't be necessary to use `arrowsAreSets` here, but we have it, -- Shouldn't be necessary to use `arrowsAreSets` here, but we have it,
-- so why not? -- so why not?
lem : (p : A ≡ B) → id-to-iso A B p ≡ flopDem (ℂ.id-to-iso A B p) lem : (p : A ≡ B) → idToIso A B p ≡ flopDem (ℂ.idToIso A B p)
lem p i = l≡r i lem p i = l≡r i
where where
l = id-to-iso A B p l = idToIso A B p
r = flopDem (ℂ.id-to-iso A B p) r = flopDem (ℂ.idToIso A B p)
open Σ l renaming (fst to l-obv ; snd to l-areInv) open Σ l renaming (fst to l-obv ; snd to l-areInv)
open Σ l-areInv renaming (fst to l-invs ; snd to l-iso) open Σ l-areInv renaming (fst to l-invs ; snd to l-iso)
open Σ l-iso renaming (fst to l-l ; snd to l-r) open Σ l-iso renaming (fst to l-l ; snd to l-r)
@ -593,27 +593,27 @@ module Opposite {ℓa ℓb : Level} where
ff : A ≅ B → A ≡ B ff : A ≅ B → A ≡ B
ff = f ⊙ flipDem ff = f ⊙ flipDem
-- inv : AreInverses (ℂ.id-to-iso A B) f -- inv : AreInverses (ℂ.idToIso A B) f
invv : AreInverses (id-to-iso A B) ff invv : AreInverses (idToIso A B) ff
-- recto-verso : ℂ.id-to-iso A B ∘ f ≡ idFun (A ℂ.≅ B) -- recto-verso : ℂ.idToIso A B ∘ f ≡ idFun (A ℂ.≅ B)
invv = record invv = record
{ verso-recto = funExt (λ x → begin { verso-recto = funExt (λ x → begin
(ff ⊙ id-to-iso A B) x ≡⟨⟩ (ff ⊙ idToIso A B) x ≡⟨⟩
(f ⊙ flipDem ⊙ id-to-iso A B) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → f ⊙ flipDem ⊙ φ) (funExt lem)) ⟩ (f ⊙ flipDem ⊙ idToIso A B) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → f ⊙ flipDem ⊙ φ) (funExt lem)) ⟩
(f ⊙ flipDem ⊙ flopDem ⊙ ℂ.id-to-iso A B) x ≡⟨⟩ (f ⊙ flipDem ⊙ flopDem ⊙ ℂ.idToIso A B) x ≡⟨⟩
(f ⊙ ℂ.id-to-iso A B) x ≡⟨ (λ i → verso-recto i x) ⟩ (f ⊙ ℂ.idToIso A B) x ≡⟨ (λ i → verso-recto i x) ⟩
x ∎) x ∎)
; recto-verso = funExt (λ x → begin ; recto-verso = funExt (λ x → begin
(id-to-iso A B ⊙ f ⊙ flipDem) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ⊙ f ⊙ flipDem) (funExt lem)) ⟩ (idToIso A B ⊙ f ⊙ flipDem) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ⊙ f ⊙ flipDem) (funExt lem)) ⟩
(flopDem ⊙ ℂ.id-to-iso A B ⊙ f ⊙ flipDem) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → flopDem ⊙ φ ⊙ flipDem) recto-verso) ⟩ (flopDem ⊙ ℂ.idToIso A B ⊙ f ⊙ flipDem) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → flopDem ⊙ φ ⊙ flipDem) recto-verso) ⟩
(flopDem ⊙ flipDem) x ≡⟨⟩ (flopDem ⊙ flipDem) x ≡⟨⟩
x ∎) x ∎)
} }
h : Equivalence.Isomorphism (id-to-iso A B) h : Equivalence.Isomorphism (idToIso A B)
h = ff , invv h = ff , invv
univalent : isEquiv (A ≡ B) (A ≅ B) univalent : isEquiv (A ≡ B) (A ≅ B)
(Univalence.id-to-iso (swap ℂ.isIdentity) A B) (Univalence.idToIso (swap ℂ.isIdentity) A B)
univalent = Equiv≃.fromIso _ _ h univalent = Equiv≃.fromIso _ _ h
isCategory : IsCategory opRaw isCategory : IsCategory opRaw

8
src/Cat/Category/Product.agda
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@ -322,14 +322,14 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
f i = f0 i , {!f1 i!} f i = f0 i , {!f1 i!}
prp : isSet (ℂ.Object × ℂ.Arrow Y A × ℂ.Arrow Y B) prp : isSet (ℂ.Object × ℂ.Arrow Y A × ℂ.Arrow Y B)
prp = setSig {sA = {!!}} {(λ _ → setSig {sA = ℂ.arrowsAreSets} {λ _ → ℂ.arrowsAreSets})} prp = setSig {sA = {!!}} {(λ _ → setSig {sA = ℂ.arrowsAreSets} {λ _ → ℂ.arrowsAreSets})}
ve-re : (p : (X , x) ≡ (Y , y)) → f (id-to-iso _ _ p) ≡ p ve-re : (p : (X , x) ≡ (Y , y)) → f (idToIso _ _ p) ≡ p
-- ve-re p i j = {!ℂ.arrowsAreSets!} , ℂ.arrowsAreSets _ _ (let k = fst (snd (p i)) in {!!}) {!!} {!!} {!!} , {!!} -- ve-re p i j = {!ℂ.arrowsAreSets!} , ℂ.arrowsAreSets _ _ (let k = fst (snd (p i)) in {!!}) {!!} {!!} {!!} , {!!}
ve-re p = let k = prp {!!} {!!} {!!} {!p!} in {!!} ve-re p = let k = prp {!!} {!!} {!!} {!p!} in {!!}
re-ve : (iso : (X , x) ≅ (Y , y)) → id-to-iso _ _ (f iso) ≡ iso re-ve : (iso : (X , x) ≅ (Y , y)) → idToIso _ _ (f iso) ≡ iso
re-ve = {!!} re-ve = {!!}
iso : E.Isomorphism (id-to-iso (X , x) (Y , y)) iso : E.Isomorphism (idToIso (X , x) (Y , y))
iso = f , record { verso-recto = funExt ve-re ; recto-verso = funExt re-ve } iso = f , record { verso-recto = funExt ve-re ; recto-verso = funExt re-ve }
res : isEquiv ((X , x) ≡ (Y , y)) ((X , x) ≅ (Y , y)) (id-to-iso (X , x) (Y , y)) res : isEquiv ((X , x) ≡ (Y , y)) ((X , x) ≅ (Y , y)) (idToIso (X , x) (Y , y))
res = Equiv≃.fromIso _ _ iso res = Equiv≃.fromIso _ _ iso
isCat : IsCategory raw isCat : IsCategory raw