susp
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3 changed files with 53 additions and 2 deletions
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@ -4,9 +4,11 @@ module CubicalHott.Chapter6 where
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open import CubicalHott.Prelude
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open import CubicalHott.Chapter3
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open import Cubical.Data.Equality.Conversion
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open import Cubical.HITs.Susp
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open import Cubical.HITs.S1
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open import Cubical.Data.Int
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open import Cubical.Data.Nat
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open import Cubical.Data.Bool
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private
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variable
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@ -91,7 +93,6 @@ module lemma632 where
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-- g : isSet Type
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-- g x y p q = J (λ y' p' → (q' : x ≡ y') → p' ≡ q') goal p q
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-- Corollary 6.10.13
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module corollary61013 where
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@ -76,4 +76,19 @@ module section81 where
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-- Lemma 8.1.8
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encode-decode : (x : S¹) → (c : code {l} x) → encode {l} x (decode x c) ≡ c
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encode-decode base c = {! !}
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encode-decode (loop i) c = {! !}
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encode-decode (loop i) c = {! !}
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-- Definition 8.4.1
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Type* = Σ Type (λ A → A)
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_→*_ : (A B : Type*) → Type
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_→*_ (A , a) (B , b) = Σ (A → B) (λ f → f a ≡ b)
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-- Definition 8.4.2
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Ω : Type* → Type*
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Ω (X , x) = (x ≡ x) , refl
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Ωf : {X Y : Type*} (f : X →* Y) → Type* → Type*
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Ωf (f , f₀) p = {! sym f₀ ∙ ? !}
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35
src/CubicalHott/Lemma651.agda
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35
src/CubicalHott/Lemma651.agda
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@ -0,0 +1,35 @@
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{-# OPTIONS --cubical #-}
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module CubicalHott.Lemma651 where
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open import CubicalHott.Prelude
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open import CubicalHott.Chapter3
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open import Cubical.HITs.Susp
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open import Cubical.HITs.S1
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open import Cubical.Data.Int
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open import Cubical.Data.Nat
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open import Cubical.Data.Bool
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-- Lemma 6.5.1
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module lemma651 where
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lemma : Susp Bool ≃ S¹
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lemma = isoToEquiv (iso f g f-g g-f) where
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f : Susp Bool → S¹
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f north = base
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f south = base
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f (merid true i) = refl {x = base} i
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f (merid false i) = loop i
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g : S¹ → Susp Bool
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g base = north
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g (loop i) = (merid false ∙ sym (merid true)) i
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f-g : section f g
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f-g base = refl
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f-g (loop i) = {! !}
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g-f : retract f g
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g-f north = refl
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g-f south = merid true
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g-f (merid true i) = {! !}
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g-f (merid false i) = {! !}
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