Rename id-to-iso to idToIso

This commit is contained in:
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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@ -6,7 +6,7 @@ Prove postulates in `Cat.Wishlist`:
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)
Prove univalence for the category of

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@ -67,7 +67,7 @@ module _ {a b : Level} ( : Category a b) where
IsPreCategory.arrowsAreSets isPreCategory = arrowsAreSets
module _ {A B : .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 = {!!}
univalent : Univalent

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@ -35,7 +35,7 @@ module Fun {c c' d d' : Level} ( : Category c c') (𝔻 : C
module _ (F : Functor 𝔻) where
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)
@ -175,7 +175,7 @@ module Fun {c c' d d' : Level} ( : Category c c') (𝔻 : C
re-ve : (x : A B) reverse (obverse x) x
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!}
univalent : Univalent

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

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@ -322,14 +322,14 @@ module Try0 {a b : Level} { : Category a b}
f i = f0 i , {!f1 i!}
prp : isSet (.Object × .Arrow Y A × .Arrow Y B)
prp = setSig {sA = {!!}} {(λ _ setSig {sA = .arrowsAreSets} {λ _ .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 = {!.arrowsAreSets!} , .arrowsAreSets _ _ (let k = fst (snd (p i)) 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 = {!!}
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 }
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
isCat : IsCategory raw