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Frederik Hanghøj Iversen 2018-05-08 14:50:45 +02:00
parent 10c3c36305
commit 4e7506f06a
1 changed files with 26 additions and 30 deletions
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src/Cat/Category.agda
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@ -658,14 +658,9 @@ module Opposite {ℓa ℓb : Level} where
open IsPreCategory isPreCategory open IsPreCategory isPreCategory
module _ {A B : ℂ.Object} where module _ {A B : ℂ.Object} where
k : TypeIsomorphism (ℂ.idToIso A B) open Σ (toIso _ _ (ℂ.univalent {A} {B}))
k = toIso _ _ ℂ.univalent renaming (fst to idToIso* ; snd to inv*)
open Σ k renaming (fst to η ; snd to inv-η) open AreInverses {f = ℂ.idToIso A B} {idToIso*} inv*
open AreInverses {f = ℂ.idToIso A B} {η} inv-η
genericly : {ℓa ℓb ℓc : Level} {a : Set ℓa} {b : Set ℓb} {c : Set ℓc}
→ a × b × c → b × a × c
genericly (a , b , c) = (b , a , c)
shuffle : A ≊ B → A ℂ.≊ B shuffle : A ≊ B → A ℂ.≊ B
shuffle (f , g , inv) = g , f , inv shuffle (f , g , inv) = g , f , inv
@ -673,37 +668,38 @@ module Opposite {ℓa ℓb : Level} where
shuffle~ : A ℂ.≊ B → A ≊ B shuffle~ : A ℂ.≊ B → A ≊ B
shuffle~ (f , g , inv) = g , f , inv shuffle~ (f , g , inv) = g , f , inv
-- Shouldn't be necessary to use `arrowsAreSets` here, but we have it,
-- so why not?
lem : (p : A ≡ B) → idToIso A B p ≡ shuffle~ (ℂ.idToIso A B p) lem : (p : A ≡ B) → idToIso A B p ≡ shuffle~ (ℂ.idToIso A B p)
lem p = isoEq refl lem p = isoEq refl
ζ : A ≊ B → A ≡ B isoToId* : A ≊ B → A ≡ B
ζ = η ∘ shuffle isoToId* = idToIso* ∘ shuffle
-- inv : AreInverses (ℂ.idToIso A B) f inv : AreInverses (idToIso A B) isoToId*
inv-ζ : AreInverses (idToIso A B) ζ inv =
-- recto-verso : ℂ.idToIso A B <<< f ≡ idFun (A ℂ.≊ B) ( funExt (λ x → begin
inv-ζ = record (isoToId* ∘ idToIso A B) x
{ fst = funExt (λ x → begin ≡⟨⟩
(ζ ∘ idToIso A B) x ≡⟨⟩ (idToIso* ∘ shuffle ∘ idToIso A B) x
(η ∘ shuffle ∘ idToIso A B) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → η ∘ shuffle ∘ φ) (funExt lem)) ⟩ ≡⟨ cong (λ φ → φ x) (cong (λ φ → idToIso* ∘ shuffle ∘ φ) (funExt lem)) ⟩
(η ∘ shuffle ∘ shuffle~ ∘ ℂ.idToIso A B) x ≡⟨⟩ (idToIso* ∘ shuffle ∘ shuffle~ ∘ ℂ.idToIso A B) x
(η ∘ ℂ.idToIso A B) x ≡⟨ (λ i → verso-recto i x) ⟩ ≡⟨⟩
(idToIso* ∘ ℂ.idToIso A B) x
≡⟨ (λ i → verso-recto i x) ⟩
x ∎) x ∎)
; snd = funExt (λ x → begin , funExt (λ x → begin
(idToIso A B ∘ η ∘ shuffle) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ∘ η ∘ shuffle) (funExt lem)) ⟩ (idToIso A B ∘ idToIso* ∘ shuffle) x
(shuffle~ ∘ ℂ.idToIso A B ∘ η ∘ shuffle) x ≡⟨ cong (λ φ → φ x) (cong (λ φ → shuffle~ ∘ φ ∘ shuffle) recto-verso) ⟩ ≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ∘ idToIso* ∘ shuffle) (funExt lem)) ⟩
(shuffle~ ∘ shuffle) x ≡⟨⟩ (shuffle~ ∘ ℂ.idToIso A B ∘ idToIso* ∘ shuffle) x
≡⟨ cong (λ φ → φ x) (cong (λ φ → shuffle~ ∘ φ ∘ shuffle) recto-verso) ⟩
(shuffle~ ∘ shuffle) x
≡⟨⟩
x ∎) x ∎)
} )
h : TypeIsomorphism (idToIso A B)
h = ζ , inv-ζ
isCategory : IsCategory opRaw isCategory : IsCategory opRaw
IsCategory.isPreCategory isCategory = isPreCategory IsCategory.isPreCategory isCategory = isPreCategory
IsCategory.univalent isCategory = univalenceFromIsomorphism h IsCategory.univalent isCategory
= univalenceFromIsomorphism (isoToId* , inv)
opposite : Category ℓa ℓb opposite : Category ℓa ℓb
Category.raw opposite = opRaw Category.raw opposite = opRaw