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spectrify_pequiv

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spiceghello 2017-06-08 10:26:21 -06:00
parent 3ab890c464
commit abe62d1f61
1 changed files with 14 additions and 6 deletions
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20
homotopy/spectrum.hlean
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@ -373,13 +373,13 @@ namespace spectrum
definition spectrify_type_term {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) : Type* := definition spectrify_type_term {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) : Type* :=
Ω[k] (X (n +' k)) Ω[k] (X (n +' k))
definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (k : ℕ) (n : N) : definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
Ω[k] (X n) →* Ω[k+1] (X (S n)) := Ω[k] (X n) →* Ω[k+1] (X (S n)) :=
!loopn_succ_in⁻¹ᵉ* ∘* Ω→[k] (glue X n) !loopn_succ_in⁻¹ᵉ* ∘* Ω→[k] (glue X n)
definition spectrify_type_fun {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) : definition spectrify_type_fun {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
spectrify_type_term X n k →* spectrify_type_term X n (k+1) := spectrify_type_term X n k →* spectrify_type_term X n (k+1) :=
spectrify_type_fun' X k (n +' k) spectrify_type_fun' X (n +' k) k
definition spectrify_type_fun_zero {N : succ_str} (X : gen_prespectrum N) (n : N) : definition spectrify_type_fun_zero {N : succ_str} (X : gen_prespectrum N) (n : N) :
spectrify_type_fun X n 0 ~* glue X n := spectrify_type_fun X n 0 ~* glue X n :=
@ -397,20 +397,28 @@ namespace spectrum
... ≡ Y n ... ≡ Y n
-/ -/
definition spectrify_type_fun'_succ {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
spectrify_type_fun' X n (succ k) ~* Ω→ (spectrify_type_fun' X n k) :=
begin
refine _ ⬝* !ap1_pcompose⁻¹*,
apply !pwhisker_right,
refine !to_pinv_pequiv_MK2
end
definition spectrify_pequiv {N : succ_str} (X : gen_prespectrum N) (n : N) : definition spectrify_pequiv {N : succ_str} (X : gen_prespectrum N) (n : N) :
spectrify_type X n ≃* Ω (spectrify_type X (S n)) := spectrify_type X n ≃* Ω (spectrify_type X (S n)) :=
begin begin
refine !pshift_equiv ⬝e* _, refine !pshift_equiv ⬝e* _,
transitivity pseq_colim (λk, spectrify_type_fun' X (succ k) (S n +' k)), transitivity pseq_colim (λk, spectrify_type_fun' X (S n +' k) (succ k)),
fapply pseq_colim_pequiv', fapply pseq_colim_pequiv,
{ intro n, apply loopn_pequiv_loopn, apply pequiv_ap X, apply succ_str.add_succ }, { intro n, apply loopn_pequiv_loopn, apply pequiv_ap X, apply succ_str.add_succ },
{ exact abstract begin intro k, apply to_homotopy, { exact abstract begin intro k,
refine !passoc⁻¹* ⬝* _, refine pwhisker_right _ (loopn_succ_in_inv_natural (succ k) _) ⬝* _, refine !passoc⁻¹* ⬝* _, refine pwhisker_right _ (loopn_succ_in_inv_natural (succ k) _) ⬝* _,
refine !passoc ⬝* _ ⬝* !passoc⁻¹*, apply pwhisker_left, refine !passoc ⬝* _ ⬝* !passoc⁻¹*, apply pwhisker_left,
refine !apn_pcompose⁻¹* ⬝* _ ⬝* !apn_pcompose, apply apn_phomotopy, refine !apn_pcompose⁻¹* ⬝* _ ⬝* !apn_pcompose, apply apn_phomotopy,
exact !glue_ptransport⁻¹* end end }, exact !glue_ptransport⁻¹* end end },
refine _ ⬝e* !pseq_colim_loop⁻¹ᵉ*, refine _ ⬝e* !pseq_colim_loop⁻¹ᵉ*,
exact pseq_colim_equiv_constant (λn, begin unfold [spectrify_type_fun], refine sorry ⬝* !ap1_pcompose⁻¹* end), exact pseq_colim_equiv_constant (λn, !spectrify_type_fun'_succ),
end end
definition spectrify [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_spectrum N := definition spectrify [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_spectrum N :=