more wip on ch 6
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@ -259,7 +259,56 @@ module S² where
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S² : Set
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S² : Set
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base : S²
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base : S²
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surf : refl {x = base} ≡ refl
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surf : refl {x = base} ≡ refl
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-- TODO: Define recursion principle
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open S² hiding (base)
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open S² hiding (base)
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```
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```
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## 6.5 Suspensions
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## 6.5 Suspensions
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```
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module Suspension where
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private
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variable
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A : Set l
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postulate
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Susp : Set → Set
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N : Susp A
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S : Susp A
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merid : A → (N {A} ≡ S {A})
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rec-Susp : {B : Set l} → (n s : B) → (m : A → n ≡ s) → Susp A → B
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rec-Susp-N : {B : Set l} → (n s : B) → (m : A → n ≡ s) → rec-Susp n s m N ≡ n
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rec-Susp-S : {B : Set l} → (n s : B) → (m : A → n ≡ s) → rec-Susp n s m S ≡ s
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{-# REWRITE rec-Susp-N #-}
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{-# REWRITE rec-Susp-S #-}
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rec-Susp-merid : {B : Set l} (n s : B) → (m : A → n ≡ s) → (a : A) → ap (rec-Susp n s m) (merid a) ≡ m a
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open Suspension public
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```
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### Lemma 6.5.1
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```
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lemma6∙5∙1 : Susp 𝟚 ≃ S¹
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lemma6∙5∙1 = f , qinv-to-isequiv (mkQinv g forward backward)
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where
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open S¹
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m : 𝟚 → base ≡ base
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m true = refl
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m false = loop
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f : Susp 𝟚 → S¹
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f s = rec-Susp base base m s
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g : S¹ → Susp 𝟚
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g s = rec-S¹ N (merid false ∙ sym (merid true)) s
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forward : (f ∘ g) ∼ id
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forward x = rec-S¹ {P = λ x → f (g x) ≡ x} refl {! refl !} x
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backward : (g ∘ f) ∼ id
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backward x = {! !}
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```
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3
src/HottBook/Chapter6Exercises.lagda.md
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3
src/HottBook/Chapter6Exercises.lagda.md
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@ -0,0 +1,3 @@
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```
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module HottBook.Chapter6Exercises where
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```
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