feat(EM): Work on uniqueness of K(G,n)'s
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Floris van Doorn
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dc2a26745e
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78492bbe09
1 changed files with 82 additions and 10 deletions
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@ -87,7 +87,7 @@ namespace EM
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-- general case
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definition EMadd1_map [unfold 8] {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃ G)
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(r : Π(p q : Ω[succ n] X), e (@concat (Ω[n] X) pt pt pt p q) = e p * e q)
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(r : Π(p q : Ω (Ω[n] X)), e (p ⬝ q) = e p * e q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n → X :=
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begin
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revert X e r H1 H2, induction n with n f: intro X e r H1 H2,
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@ -96,14 +96,14 @@ namespace EM
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induction x with x, induction x with x,
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{ exact pt},
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{ exact pt},
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{ change carrier (Ω X), refine f _ _ _ _ _ (tr x),
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{ refine _⁻¹ᵉ ⬝e e, apply equiv_of_pequiv, apply pequiv_of_eq, apply loop_space_succ_eq_in},
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exact abstract begin
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intro p q, refine _ ⬝ !r, apply ap e, esimp,
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apply inv_tr_eq_of_eq_tr, symmetry,
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rewrite [- + ap_inv, - + tr_compose],
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refine !loop_space_succ_eq_in_concat ⬝ _, exact !tr_inv_tr ◾ !tr_inv_tr
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end end}
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change carrier (Ω X), refine f _ _ _ _ _ (tr x),
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{ refine _⁻¹ᵉ ⬝e e, apply equiv_of_pequiv, apply pequiv_of_eq, apply loop_space_succ_eq_in},
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exact abstract begin
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intro p q, refine _ ⬝ !r, apply ap e, esimp,
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apply inv_tr_eq_of_eq_tr, symmetry,
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rewrite [- + ap_inv, - + tr_compose],
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refine !loop_space_succ_eq_in_concat ⬝ _, exact !tr_inv_tr ◾ !tr_inv_tr
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end end
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end
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definition pEMadd1_pmap [constructor] {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃ G)
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@ -121,7 +121,7 @@ namespace EM
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revert X e r H1 H2, induction n with n IH: intro X e r H1 H2,
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{ refine !idp_con ⬝ _, refine !ap_compose'⁻¹ ⬝ _, esimp, apply elim_pth},
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{ replace (succ (succ n)) with ((succ n) + 1), rewrite [apn_succ],
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unfold [ap1], exact sorry}
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exact sorry}
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--exact !idp_con ⬝ !elim_pth
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end
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@ -157,9 +157,81 @@ namespace EM
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intro p q, esimp, exact to_respect_mul e (tr p) (tr q)
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end
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definition EM_pequiv_succ {G : CommGroup} {X : Type*} {n : ℕ} (e : πg[n+1] X ≃g G)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EM G (succ n) ≃* X :=
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pEMadd1_pequiv e
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definition EM_pequiv_zero {G : CommGroup} {X : Type*} (e : X ≃* pType_of_Group G) : EM G 0 ≃* X :=
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proof e⁻¹ᵉ* qed
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definition EM_spectrum /-[constructor]-/ (G : CommGroup) : spectrum :=
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spectrum.Mk (K G) (λn, (loop_EM G n)⁻¹ᵉ*)
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/- uniqueness of K(G,n), method 2: -/
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-- definition freudenthal_homotopy_group_pequiv (A : Type*) {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
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-- : π*[k + 1] (psusp A) ≃* π*[k] A :=
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-- calc
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-- π*[k + 1] (psusp A) ≃* π*[k] (Ω (psusp A)) : pequiv_of_eq (phomotopy_group_succ_in (psusp A) k)
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-- ... ≃* Ω[k] (ptrunc k (Ω (psusp A))) : phomotopy_group_pequiv_loop_ptrunc k (Ω (psusp A))
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-- ... ≃* Ω[k] (ptrunc k A) : loopn_pequiv_loopn k (freudenthal_pequiv A H)
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-- ... ≃* π*[k] A : (phomotopy_group_pequiv_loop_ptrunc k A)⁻¹ᵉ*
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definition iterate_psusp_succ_pequiv (n : ℕ) (A : Type*) :
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iterate_psusp (succ n) A ≃* iterate_psusp n (psusp A) :=
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begin
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induction n with n IH,
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{ reflexivity},
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{ exact psusp_equiv IH}
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end
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definition is_conn_psusp [instance] (n : trunc_index) (A : Type*)
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[H : is_conn n A] : is_conn (n .+1) (psusp A) :=
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is_conn_susp n A
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definition iterated_freudenthal_pequiv (A : Type*) {n k m : ℕ} [HA : is_conn n A] (H : k ≤ 2 * n)
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: ptrunc k A ≃* ptrunc k (Ω[m] (iterate_psusp m A)) :=
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begin
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revert A n k HA H, induction m with m IH: intro A n k HA H,
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{ reflexivity},
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{ have H2 : succ k ≤ 2 * succ n,
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from calc
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succ k ≤ succ (2 * n) : succ_le_succ H
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... ≤ 2 * succ n : self_le_succ,
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exact calc
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ptrunc k A ≃* ptrunc k (Ω (psusp A)) : freudenthal_pequiv A H
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... ≃* Ω (ptrunc (succ k) (psusp A)) : loop_ptrunc_pequiv
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... ≃* Ω (ptrunc (succ k) (Ω[m] (iterate_psusp m (psusp A)))) :
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loop_pequiv_loop (IH (psusp A) (succ n) (succ k) _ H2)
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... ≃* ptrunc k (Ω[succ m] (iterate_psusp m (psusp A))) : loop_ptrunc_pequiv
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... ≃* ptrunc k (Ω[succ m] (iterate_psusp (succ m) A)) :
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ptrunc_pequiv_ptrunc _ (loopn_pequiv_loopn _ !iterate_psusp_succ_pequiv)}
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end
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definition pmap_eq_of_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) : f = g :=
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pmap_eq (to_homotopy p) (to_homotopy_pt p)⁻¹
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definition pmap_equiv_pmap_right {A B : Type*} (C : Type*) (f : A ≃* B) : C →* A ≃ C →* B :=
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begin
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fapply equiv.MK,
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{ exact pcompose f},
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{ exact pcompose f⁻¹ᵉ*},
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{ intro f, apply pmap_eq_of_phomotopy,
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exact !passoc⁻¹* ⬝* pwhisker_right _ !pright_inv ⬝* !pid_comp},
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{ intro f, apply pmap_eq_of_phomotopy,
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exact !passoc⁻¹* ⬝* pwhisker_right _ !pleft_inv ⬝* !pid_comp}
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end
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definition iterate_psusp_adjoint_loopn [constructor] (X Y : Type*) (n : ℕ) :
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iterate_psusp n X →* Y ≃ X →* Ω[n] Y :=
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begin
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revert X Y, induction n with n IH: intro X Y,
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{ reflexivity},
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{ refine !susp_adjoint_loop ⬝e !IH ⬝e _, apply pmap_equiv_pmap_right,
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symmetry, apply pequiv_of_eq, apply loop_space_succ_eq_in}
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end
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end EM
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-- cohomology ∥ X → K(G,n) ∥
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-- reduced cohomology ∥ X →* K(G,n) ∥
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