From 4e65fc2a32bb73ad3f6ccadbd79181d4786fb05b Mon Sep 17 00:00:00 2001 From: Michael Zhang <mail@mzhang.io> Date: Fri, 24 Jan 2025 05:46:08 -0600 Subject: [PATCH] wip --- src/ThesisWork/HomotopyGroupLES2/Step2.agda | 46 ++++++++++++++++----- src/ThesisWork/HomotopyGroupLES2/Step3.agda | 4 ++ 2 files changed, 39 insertions(+), 11 deletions(-) diff --git a/src/ThesisWork/HomotopyGroupLES2/Step2.agda b/src/ThesisWork/HomotopyGroupLES2/Step2.agda index a928907..f67269f 100644 --- a/src/ThesisWork/HomotopyGroupLES2/Step2.agda +++ b/src/ThesisWork/HomotopyGroupLES2/Step2.agda @@ -13,7 +13,7 @@ open import Cubical.Structures.Successor open import ThesisWork.Exactness open import ThesisWork.HomotopyGroupLES2.Util -open import ThesisWork.HomotopyGroupLES2.Step1 +open import ThesisWork.HomotopyGroupLES2.Step1 renaming (A to A' ; f^ to f^') open import ThesisWork.HomotopyGroupLES2.Lemma415 private @@ -23,6 +23,8 @@ private pattern 3+ n = suc (suc (suc n)) module _ {X∙ @ (X , x) : Pointed ℓ} {Y∙ @ (Y , y) : Pointed ℓ} (f∙ @ (f , f₀) : X∙ →∙ Y∙) where + A = A' f∙ + f^ = f^' f∙ F∙ = Fiber f∙ B : (n : ℕ) → Pointed ℓ @@ -37,28 +39,50 @@ module _ {X∙ @ (X , x) : Pointed ℓ} {Y∙ @ (Y , y) : Pointed ℓ} (f∙ @ ( g^ 2 = δ f∙ g^ (3+ n) = -Ω (g^ n) - _ : A f∙ 3 ≡ B 3 + _ : A 3 ≡ B 3 _ = ua∙ (eqv∙ f∙ .fst) (eqv∙ f∙ .snd) module Lemma416 where - η : (n : ℕ) → A f∙ n ≃∙ B n + η : (n : ℕ) → A n ≃∙ B n η 0 = idEquiv∙ Y∙ η 1 = idEquiv∙ X∙ η 2 = idEquiv∙ F∙ η (3+ n) = - A f∙ (3+ n) ≃∙⟨ eqv∙ (f^ f∙ n) ⟩ - Ω (A f∙ n) ≃∙⟨ cong-≃∙ Ω (η n) ⟩ + A (3+ n) ≃∙⟨ eqv∙ (f^ n) ⟩ + Ω (A n) ≃∙⟨ cong-≃∙ Ω (η n) ⟩ Ω (B n) ∎≃∙ - eqvfg : (n : ℕ) → (≃∙map (η n) ∘∙ f^ f∙ n) ∙∼ (g^ n ∘∙ ≃∙map (η (suc n))) - eqvfg 0 = {! !} - eqvfg 1 = {! !} - eqvfg 2 = {! !} - eqvfg (3+ n) = {! !} + A≡B : A ≡ B + A≡B = funExt (λ n → ua∙ (η n .fst) (η n .snd)) + postulate + -- TODO: Finish + f∙∼g : (n : ℕ) → (≃∙map (η n) ∘∙ f^ n) ∙∼P (g^ n ∘∙ ≃∙map (η (suc n))) + -- eqvfg 0 = {! !} + -- eqvfg 1 = {! !} + -- eqvfg 2 = {! !} + -- eqvfg (3+ n) = {! !} + + -- TODO: Is this actually easier? + -- Potential point of deviation from FVD thesis + f≡g' : (n : ℕ) → PathP (λ i → A≡B i (suc n) →∙ A≡B i n) (f^ n) (g^ n) + f≡g' 0 i = {! !} + -- at i = i0, A^ 1 →∙ A^ 0 + -- at i = i1, B^ 1 →∙ B^ 0 + f≡g' 1 i = {! !} + f≡g' 2 i = {! !} + f≡g' (3+ n) = {! !} + + f≡g : PathP (λ i → (n : ℕ) → A≡B i (suc n) →∙ A≡B i n) f^ g^ + f≡g = funExt f≡g' + + open Lemma416 using (f≡g) + + BⁿisExact : (n : ℕ) → isExactAt∙ (g^ (suc n)) (g^ n) + BⁿisExact n = transp (λ i → isExactAt∙ ((f≡g i) (suc n)) ((f≡g i) n)) i0 (AⁿisExact f∙ n) open ExactSequence∙ BSeq : ExactSequence∙ ℕ+ BSeq .seq = B BSeq .fun = g^ - BSeq .exactness = {! !} \ No newline at end of file + BSeq .exactness = BⁿisExact \ No newline at end of file diff --git a/src/ThesisWork/HomotopyGroupLES2/Step3.agda b/src/ThesisWork/HomotopyGroupLES2/Step3.agda index 8009ac6..6e2f433 100644 --- a/src/ThesisWork/HomotopyGroupLES2/Step3.agda +++ b/src/ThesisWork/HomotopyGroupLES2/Step3.agda @@ -1,3 +1,7 @@ {-# OPTIONS --cubical #-} module ThesisWork.HomotopyGroupLES2.Step3 where + +open import ThesisWork.HomotopyGroupLES2.Step2 renaming (B to B' ; g^ to g^') + +module _ {X∙ @ (X , x) : Pointed ℓ} {Y∙ @ (Y , y) : Pointed ℓ} (f∙ @ (f , f₀) : X∙ →∙ Y∙) where