116 lines
4.6 KiB
Text
116 lines
4.6 KiB
Text
import ..algebra.module_exact_couple .strunc
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open eq spectrum trunc is_trunc pointed int EM algebra left_module fiber lift
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/- Eilenberg MacLane spaces are the fibers of the Postnikov system of a type -/
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definition postnikov_map [constructor] (A : Type*) (n : ℕ₋₂) : ptrunc (n.+1) A →* ptrunc n A :=
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ptrunc.elim (n.+1) !ptr
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definition ptrunc_functor_postnikov_map {A B : Type*} (n : ℕ₋₂) (f : A →* B) :
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ptrunc_functor n f ∘* postnikov_map A n ~* ptrunc.elim (n.+1) (!ptr ∘* f) :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x with a, reflexivity },
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{ reflexivity }
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end
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section
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open nat is_conn group
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definition pfiber_postnikov_map (A : Type*) (n : ℕ) :
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pfiber (postnikov_map A n) ≃* EM_type A (n+1) :=
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begin
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symmetry, apply EM_type_pequiv,
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{ symmetry, refine _ ⬝g ghomotopy_group_ptrunc (n+1) A,
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exact chain_complex.LES_isomorphism_of_trivial_cod _ _
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(trivial_homotopy_group_of_is_trunc _ (self_lt_succ n))
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(trivial_homotopy_group_of_is_trunc _ (le_succ _)) },
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{ apply is_conn_fun_trunc_elim, apply is_conn_fun_tr },
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{ have is_trunc (n+1) (ptrunc n.+1 A), from !is_trunc_trunc,
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have is_trunc ((n+1).+1) (ptrunc n A), by do 2 apply is_trunc_succ, apply is_trunc_trunc,
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apply is_trunc_pfiber }
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end
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end
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definition postnikov_map_natural {A B : Type*} (f : A →* B) (n : ℕ₋₂) :
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psquare (postnikov_map A n) (postnikov_map B n)
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(ptrunc_functor (n.+1) f) (ptrunc_functor n f) :=
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!ptrunc_functor_postnikov_map ⬝* !ptrunc_elim_ptrunc_functor⁻¹*
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definition encode_ap1_gen_tr (n : ℕ₋₂) {A : Type*} {a a' : A} (p : a = a') :
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trunc.encode (ap1_gen tr idp idp p) = tr p :> trunc n (a = a') :=
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by induction p; reflexivity
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definition ap1_postnikov_map (A : Type*) (n : ℕ₋₂) :
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psquare (Ω→ (postnikov_map A (n.+1))) (postnikov_map (Ω A) n)
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(loop_ptrunc_pequiv (n.+1) A) (loop_ptrunc_pequiv n A) :=
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have psquare (postnikov_map (Ω A) n) (Ω→ (postnikov_map A (n.+1)))
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(loop_ptrunc_pequiv (n.+1) A)⁻¹ᵉ* (loop_ptrunc_pequiv n A)⁻¹ᵉ*,
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begin
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refine _ ⬝* !ap1_ptrunc_elim⁻¹*, apply pinv_left_phomotopy_of_phomotopy,
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fapply phomotopy.mk,
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{ intro x, induction x with p, exact !encode_ap1_gen_tr⁻¹ },
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{ reflexivity }
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end,
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this⁻¹ᵛ*
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-- definition postnikov_map_int (X : Type*) (k : ℤ) :
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-- ptrunc (maxm2 (k + 1)) X →* ptrunc (maxm2 k) X :=
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-- begin
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-- induction k with k k,
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-- exact postnikov_map X k,
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-- exact !pconst
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-- end
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-- definition postnikov_map_int_natural {A B : Type*} (f : A →* B) (k : ℤ) :
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-- psquare (postnikov_map_int A k) (postnikov_map_int B k)
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-- (ptrunc_int_functor (k+1) f) (ptrunc_int_functor k f) :=
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-- begin
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-- induction k with k k,
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-- exact postnikov_map_natural f k,
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-- apply psquare_of_phomotopy, exact !pcompose_pconst ⬝* !pconst_pcompose⁻¹*
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-- end
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-- definition postnikov_map_int_natural_pequiv {A B : Type*} (f : A ≃* B) (k : ℤ) :
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-- psquare (postnikov_map_int A k) (postnikov_map_int B k)
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-- (ptrunc_int_pequiv_ptrunc_int (k+1) f) (ptrunc_int_pequiv_ptrunc_int k f) :=
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-- begin
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-- induction k with k k,
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-- exact postnikov_map_natural f k,
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-- apply psquare_of_phomotopy, exact !pcompose_pconst ⬝* !pconst_pcompose⁻¹*
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-- end
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-- definition pequiv_ap_natural2 {X A : Type} (B C : X → A → Type*) {a a' : X → A}
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-- (p : a ~ a')
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-- (s : X → X) (f : Πx a, B x a →* C (s x) a) (x : X) :
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-- psquare (pequiv_ap (B x) (p x)) (pequiv_ap (C (s x)) (p x)) (f x (a x)) (f x (a' x)) :=
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-- begin induction p using homotopy.rec_on_idp, exact phrfl end
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-- definition pequiv_ap_natural3 {X A : Type} (B C : X → A → Type*) {a a' : A}
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-- (p : a = a')
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-- (s : X → X) (x : X) (f : Πx a, B x a →* C (s x) a) :
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-- psquare (pequiv_ap (B x) p) (pequiv_ap (C (s x)) p) (f x a) (f x a') :=
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-- begin induction p, exact phrfl end
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-- definition postnikov_smap (X : spectrum) (k : ℤ) :
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-- strunc (k+1) X →ₛ strunc k X :=
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-- smap.mk (λn, postnikov_map_int (X n) (k + n) ∘* ptrunc_int_change_int _ !norm_num.add_comm_middle)
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-- (λn, begin
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-- exact sorry
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-- -- exact (_ ⬝vp* !ap1_pequiv_ap) ⬝h* (!postnikov_map_int_natural_pequiv ⬝v* _ ⬝v* _) ⬝vp* !ap1_pcompose,
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-- end)
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-- section atiyah_hirzebruch
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-- parameters {X : Type*} (Y : X → spectrum)
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-- definition atiyah_hirzebruch_exact_couple : exact_couple rℤ spectrum.I :=
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-- @exact_couple_sequence (λs, strunc s (spi X (λx, Y x)))
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-- (λs, !postnikov_smap ∘ₛ strunc_change_int _ !succ_pred⁻¹)
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-- -- parameters (k : ℕ) (H : Π(x : X) (n : ℕ), is_trunc (k + n) (Y x n))
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-- end atiyah_hirzebruch
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