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{-# OPTIONS --type-in-type --without-K #-} | |
open import lib.Prelude | |
open Truncation | |
open Int | |
open LoopSpace | |
open Suspension | |
open import homotopy.Freudenthal | |
import homotopy.FreudenthalIteratedSuspension1 | |
open import homotopy.HStructure | |
open import homotopy.PiLessOfConnected | |
open import homotopy.Pi2HSusp | |
open import homotopy.KG1 | |
module homotopy.KGn where | |
-- KGn when G is π1(A) | |
module N+1 (A : Type) | |
(a0 : A) | |
(A-Connected : Connected (S (S -2)) A) | |
(A-level : NType (tl 1) A) | |
(H-A : H-Structure A a0) where | |
module FS = homotopy.FreudenthalIteratedSuspension1 A a0 A-Connected | |
KG : Positive → Type | |
KG n = Trunc (tlp n) (FS.Susp'^ n) | |
KG-Connected : ∀ (i : Nat) → Connected (tl i) (KG (i +1np)) | |
KG-Connected n = Connected.Trunc-connected _ _ _ (FS.Susp'^-Connected n) | |
KG-Connected'' : (n : Positive) → Connected (tlp n) (KG (n +1)) | |
KG-Connected'' n = coe (ap (NType -2) (ap2 Trunc (tl-pos2nat-tlp n) (ap KG (pos2nat-+1np n)))) (KG-Connected (pos2nat n)) | |
base^ : ∀ n → KG n | |
base^ n = [ FS.base'^ n ] | |
module Stable (k : Positive) | |
(n : Positive) | |
(indexing : Either (tlp k <tl tlp n) ((tlp k ≃ tlp n) × (tl 1 <tl tlp n))) where | |
stable : π k (KG n) (base^ n) ≃ π (k +1) (KG (n +1)) (base^ (n +1)) | |
stable = π k (KG n) (base^ n) ≃〈 π<=Trunc k n (lte indexing) (FS.base'^ n) 〉 | |
π k (FS.Susp'^ n) (FS.base'^ n) ≃〈 FS.Stable.stable k n (k<=n->k<=2n-2 k n indexing) 〉 | |
π (k +1) (FS.Susp'^ (n +1)) (FS.base'^ (n +1)) ≃〈 ! (π<=Trunc (k +1) (n +1) (<=SCong (lte indexing)) (FS.base'^ (n +1))) 〉 | |
π (k +1) (KG (n +1)) (base^ (n +1)) ∎ where | |
lte : (indexing : Either (tlp k <tl tlp n) ((tlp k ≃ tlp n) × (tl 1 <tl tlp n))) → tlp k <=tl tlp n | |
lte (Inl lt) = Inl lt | |
lte (Inr (eq , _)) = Inr eq | |
-- for talk | |
KG1 = A | |
stable2 : π (k +1) (KG (n +1)) (base^ (n +1)) ≃ π k (KG n) (base^ n) | |
stable2 = π (k +1) (KG (n +1)) (base^ (n +1)) ≃〈 (π<=Trunc (k +1) (n +1) (<=SCong (lte indexing)) (FS.base'^ (n +1))) 〉 | |
π (k +1) (Susp^ (S n -1pn) KG1) (FS.base'^ (n +1)) ≃〈 ! (FS.Stable.stable k n (k<=n->k<=2n-2 k n indexing)) 〉 | |
π k (Susp^ (n -1pn) KG1) (FS.base'^ n) ≃〈 ! (π<=Trunc k n (lte indexing) (FS.base'^ n)) 〉 | |
π k (KG n) (base^ n) ∎ | |
where | |
lte : (indexing : Either (tlp k <tl tlp n) ((tlp k ≃ tlp n) × (tl 1 <tl tlp n))) → tlp k <=tl tlp n | |
lte (Inl lt) = Inl lt | |
lte (Inr (eq , _)) = Inr eq | |
-- end for talk | |
module BelowDiagonal where | |
π1 : (n : Positive) → (π One (KG (n +1)) (base^ (n +1))) ≃ Unit | |
π1 n = π1Connected≃Unit (tlp n) _ (base^ (n +1)) (KG-Connected'' n) (1<=pos n) | |
πk : (k n : Positive) → (tlp k <tl tlp n) → π k (KG n) (base^ n) ≃ Unit | |
πk One One (ltSR (ltSR (ltSR ()))) | |
πk One (S n) lt = π1 n | |
πk (S k) One lt = Sums.abort (pos-not-<=0 k (Inl (lt-unS lt))) | |
πk (S k) (S n) lt = π (k +1) (KG (n +1)) (base^ (n +1)) ≃〈 ! (Stable.stable k n (Inl (lt-unS lt))) 〉 | |
π k (KG n) (base^ n) ≃〈 πk k n (lt-unS lt) 〉 | |
Unit ∎ | |
module OnDiagonal where | |
π1 : π One (KG One) (base^ One) ≃ π One A a0 | |
π1 = τ₀ (Path {Trunc (tl 1) A} [ a0 ] [ a0 ]) ≃〈 ap τ₀ (ap-Loop≃ One (UnTrunc.path _ _ A-level) (ap≃ (type≃β (UnTrunc.eqv _ _ A-level)))) 〉 | |
τ₀ (Path {A} a0 a0) ∎ | |
Two : Positive | |
Two = S One | |
π2 : π Two (KG Two) (base^ Two) ≃ π One A a0 | |
π2 = π Two (KG Two) (base^ Two) ≃〈 id 〉 | |
Trunc (tl 0) (Loop Two (Trunc (tl 2) (Susp A)) [ No ]) ≃〈 ap (Trunc (tl 0)) (Loop-Trunc0 Two) 〉 | |
Trunc (tl 0) (Trunc (tl 0) (Loop Two (Susp A) No)) ≃〈 FuseTrunc.path (tl 0) (tl 0) _ 〉 | |
Trunc (tl 0) (Loop Two (Susp A) No) ≃〈 π2Susp A a0 A-level A-Connected H-A 〉 | |
Trunc (tl 0) (Loop One A a0) ≃〈 id 〉 | |
π One A a0 ∎ | |
πn : (n : Positive) → π n (KG n) (base^ n) ≃ π One A a0 | |
πn One = π1 | |
πn (S One) = π2 | |
πn (S (S n)) = πn (S n) ∘ ! (Stable.stable (S n) (S n) (Inr (id , >pos->1 n (S n) ltS))) | |
module AboveDiagonal where | |
πabove : (k n : Positive) → tlp n <tl tlp k → π k (KG n) (base^ n) ≃ Unit | |
πabove k n lt = Contractible≃Unit (use-level { -2} (Trunc-level-better (Loop-level-> (tlp n) k Trunc-level lt))) | |
module Explicit (G : AbelianGroup) where | |
module KG1 = K1 (fst G) | |
module KGn = N+1 (KG1.KG1) KG1.base KG1.Pi0.KG1-Connected KG1.level (H-on-KG1.H-KG1 G) | |
KG : Positive -> Type | |
KG One = KG1.KG1 | |
KG (S n) = KGn.KG (S n) | |
KGbase : ∀ n → KG n | |
KGbase One = KG1.base | |
KGbase (S n) = KGn.base^ (S n) | |
πn-KGn-is-G : ∀ n → π n (KG n) (KGbase n) ≃ (Group.El (fst G)) | |
πn-KGn-is-G One = KG1.Pi1.π1[KGn]-is-G | |
πn-KGn-is-G (S n) = KG1.Pi1.π1[KGn]-is-G ∘ KGn.OnDiagonal.πn (S n) | |
πk-KGn-trivial : ∀ k n → Either (tlp k <tl tlp n) (tlp n <tl tlp k) | |
→ π k (KG n) (KGbase n) ≃ Unit | |
πk-KGn-trivial k One (Inl k<n) with pos-not-<=0 k (lt-unS-right k<n) | |
... | () | |
πk-KGn-trivial k (S n) (Inl k<n) = KGn.BelowDiagonal.πk k (S n) k<n | |
πk-KGn-trivial k One (Inr n<k) = Contractible≃Unit (use-level { -2} (Trunc-level-better (Loop-level-> (tlp One) k KG1.level n<k))) | |
πk-KGn-trivial k (S n) (Inr n<k) = KGn.AboveDiagonal.πabove k (S n) n<k | |
-- todo: | |
-- spectrum: | |
-- Path (KG n+1) No No ≃ KG n | |
-- set k = n, and cancel redundant truncations | |