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@ -11,6 +11,7 @@ open import HottBook.Chapter1
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open import HottBook.Chapter1Util
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open import HottBook.Chapter2Lemma221 public
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open import HottBook.Chapter2Lemma231 public
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open import HottBook.Chapter2Definition217 public
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private
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variable
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@ -84,36 +85,34 @@ module lemma2∙1∙4 {l : Level} {A : Set l} where
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iv {x} {y} {z} {w} refl refl refl = refl
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```
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### Definition 2.1.7 (pointed type)
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This comes first, since it is needed to define Theorem 2.1.6.
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```
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pointed : (l : Level) → Set (lsuc l)
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pointed l = Σ (Set l) (λ A → A)
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```
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### Definition 2.1.8 (loop space)
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```
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Ω : {l : Level} → pointed l → Set l
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Ω (A , a) = a ≡ a
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```
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### Theorem 2.1.6 (Eckmann-Hilton)
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```
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module theorem2∙1∙6 where
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Ω² : {l : Level} → pointed l → Set l
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Ω² p = Ω (Ω p , refl)
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Ω-type : (A : Set) → (a : A) → Set
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Ω-type A a = refl {x = a} ≡ refl
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compose : {l : Level} {p : pointed l} → (Ω² p) × (Ω² p) → Ω² p
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compose (a , b) = a ∙ b
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compose1 : {A : Set} {a : A} → fst (Ω (A , a)) → fst (Ω (A , a)) → fst (Ω (A , a))
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compose1 x y = x ∙ y
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-- commute : {l : Level} {p : pointed l} → (α β : Ω² p) → α ∙ β ≡ β ∙ α
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-- commute {l} {p @ (A , a₀)} α β = {! !}
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Ω² : {l : Level} → Set* l → Set* l
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Ω² {l} = Ω[_] {l = l} 2
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compose2 : {l : Level} {A : Set l} {a : A} → fst (Ω² (A , a)) → fst (Ω² (A , a)) → fst (Ω² (A , a))
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compose2 x y = x ∙ y
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compose2-commutative : {l : Level} {A : Set l} {a : A} (x y : fst (Ω² (A , a))) → compose2 x y ≡ compose2 y x
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compose2-commutative {l} {A} {a} x y = {! !}
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```
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### Definition 2.1.7 (pointed type)
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{{#include HottBook.Chapter2Definition217.md:pointedtype}}
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### Definition 2.1.8 (loop space)
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{{#include HottBook.Chapter2Definition217.md:loopspace}}
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## 2.2 Functions are functors
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### Lemma 2.2.1
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30
src/HottBook/Chapter2Definition217.lagda.md
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30
src/HottBook/Chapter2Definition217.lagda.md
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@ -0,0 +1,30 @@
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```
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{-# OPTIONS --rewriting #-}
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module HottBook.Chapter2Definition217 where
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open import Agda.Primitive
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open import HottBook.Chapter1
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```
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[//]: <> (ANCHOR: pointedtype)
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```
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Set* : (l : Level) → Set (lsuc l)
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Set* l = Σ (Set l) (λ A → A)
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```
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[//]: <> (ANCHOR_END: pointedtype)
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[//]: <> (ANCHOR: loopspace)
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```
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Ω : {l : Level} → Set* l → Set* l
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Ω (A , a) = (a ≡ a) , refl
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Ω[_] : {l : Level} → ℕ → Set* l → Set* l
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Ω[ zero ] (A , a) = (A , a)
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Ω[ suc n ] (A , a) = Ω[ n ] (Ω (A , a))
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```
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[//]: <> (ANCHOR_END: loopspace)
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@ -300,10 +300,10 @@ module Suspension where
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A : Set l
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postulate
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Susp : Set → Set
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Susp : Set l → Set l
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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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merid : A → (N {A = A} ≡ S {A = 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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@ -371,7 +371,7 @@ module definition6∙5∙2 where
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### Lemma 6.5.3
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```
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Map* : {l : Level} → (A B : pointed l) → Set l
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Map* : {l : Level} → (A B : Set* l) → Set l
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Map* (A , a₀) (B , b₀) = Σ (A → B) (λ f → f a₀ ≡ b₀)
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```
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@ -379,13 +379,13 @@ Adjoining a disjoint basepoint
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```
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module lol where
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_₊ : {l : Level} → Set l → pointed l
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_₊ : {l : Level} → Set l → Set* l
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A ₊ = A + Lift 𝟙 , inr (lift tt)
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```
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```
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open lol
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lemma6∙5∙3 : {l : Level} (A : Set l) → (B* @ (B , b₀) : pointed l)
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lemma6∙5∙3 : {l : Level} (A : Set l) → (B* @ (B , b₀) : Set* l)
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→ (Map* (A ₊) B*) ≃ (A → B)
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lemma6∙5∙3 A B* @ (B , b₀) = f , qinv-to-isequiv (mkQinv g forward {! !})
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where
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@ -409,6 +409,24 @@ lemma6∙5∙3 A B* @ (B , b₀) = f , qinv-to-isequiv (mkQinv g forward {! !}
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-- g (f' x) gives us b₀
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```
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### Lemma 6.5.4
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```
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Susp* : {l : Level} → Set* l → Set* l
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Susp* (A , a₀) = Susp A , N
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lemma6∙5∙4 : {l : Level}
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→ (A* @ (A , a₀) : Set* l)
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→ (B* @ (B , b₀) : Set* l)
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→ Map* {l = l} (Susp* A*) B* ≃ Map* A* (Ω B*)
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lemma6∙5∙4 A* B* = f , {! !}
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where
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f : Map* (Susp* A*) B* → Map* A* (Ω B*)
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f (f' , p) = {! f'' , ? !}
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where
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f'' : A* → Ω B*
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```
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## 6.6 Cell complexes
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```
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