250 lines
12 KiB
Text
250 lines
12 KiB
Text
import .basic ..colimit.pointed
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open eq pointed succ_str is_equiv equiv spectrum.smap seq_colim nat
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namespace spectrum
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/- Prespectrification -/
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definition is_sconnected {N : succ_str} {X Y : gen_prespectrum N} (h : X →ₛ Y) : Type :=
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Π (E : gen_spectrum N), is_equiv (λ g : Y →ₛ E, g ∘ₛ h)
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-- We introduce a prespectrification operation X ↦ prespectrification X, with the goal of implementing spectrification as the sequential colimit of iterated applications of the prespectrification operation.
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definition prespectrification [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_prespectrum N :=
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gen_prespectrum.mk (λ n, Ω (X (S n))) (λ n, Ω→ (glue X (S n)))
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-- To define the unit η : X → prespectrification X, we need its underlying family of pointed maps
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definition to_prespectrification_pfun {N : succ_str} (X : gen_prespectrum N) (n : N) : X n →* prespectrification X n :=
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glue X n
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-- And similarly we need the pointed homotopies witnessing that the squares commute
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definition to_prespectrification_psq {N : succ_str} (X : gen_prespectrum N) (n : N) :
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psquare (to_prespectrification_pfun X n) (Ω→ (to_prespectrification_pfun X (S n))) (glue X n)
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(glue (prespectrification X) n) :=
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psquare_of_phomotopy phomotopy.rfl
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-- We bundle the previous two definitions into a morphism of prespectra
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definition to_prespectrification {N : succ_str} (X : gen_prespectrum N) : X →ₛ prespectrification X :=
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begin
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fapply smap.mk,
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exact to_prespectrification_pfun X,
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exact to_prespectrification_psq X,
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end
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-- When E is a spectrum, then the map η : E → prespectrification E is an equivalence.
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definition is_sequiv_smap_to_prespectrification_of_is_spectrum {N : succ_str} (E : gen_prespectrum N) (H : is_spectrum E) : is_sequiv_smap (to_prespectrification E) :=
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begin
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intro n, induction E, induction H, exact is_equiv_glue n,
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end
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-- If η : E → prespectrification E is an equivalence, then E is a spectrum.
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definition is_spectrum_of_is_sequiv_smap_to_prespectrification {N : succ_str} (E : gen_prespectrum N) (H : is_sequiv_smap (to_prespectrification E)) : is_spectrum E :=
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begin
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fapply is_spectrum.mk,
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exact H
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end
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-- Our next goal is to show that if X is a prespectrum and E is a spectrum, then maps X →ₛ E extend uniquely along η : X → prespectrification X.
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-- In the following we define the underlying family of pointed maps of such an extension
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definition is_sconnected_to_prespectrification_inv_pfun {N : succ_str} {X : gen_prespectrum N} {E : gen_spectrum N} : (X →ₛ E) → Π (n : N), prespectrification X n →* E n :=
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λ (f : X →ₛ E) n, (equiv_glue E n)⁻¹ᵉ* ∘* Ω→ (f (S n))
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-- In the following we define the commuting squares of the extension
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definition is_sconnected_to_prespectrification_inv_psq {N : succ_str} {X : gen_prespectrum N} {E : gen_spectrum N} (f : X →ₛ E) (n : N) :
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psquare (is_sconnected_to_prespectrification_inv_pfun f n)
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(Ω→ (is_sconnected_to_prespectrification_inv_pfun f (S n)))
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(glue (prespectrification X) n)
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(glue (gen_spectrum.to_prespectrum E) n)
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:=
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begin
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fapply psquare_of_phomotopy,
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refine (passoc (glue (gen_spectrum.to_prespectrum E) n) (pequiv.to_pmap
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(equiv_glue (gen_spectrum.to_prespectrum E) n)⁻¹ᵉ*) (Ω→ (to_fun f (S n))))⁻¹* ⬝* _,
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refine pwhisker_right (Ω→ (to_fun f (S n))) (pright_inv (equiv_glue E n)) ⬝* _,
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refine _ ⬝* pwhisker_right (glue (prespectrification X) n) ((ap1_pcompose (pequiv.to_pmap (equiv_glue (gen_spectrum.to_prespectrum E) (S n))⁻¹ᵉ*) (Ω→ (to_fun f (S (S n)))))⁻¹*),
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refine pid_pcompose (Ω→ (f (S n))) ⬝* _,
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repeat exact sorry
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end
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-- Now we bundle the definition into a map (X →ₛ E) → (prespectrification X →ₛ E)
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definition is_sconnected_to_prespectrification_inv {N : succ_str} (X : gen_prespectrum N) (E : gen_spectrum N)
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: (X →ₛ E) → (prespectrification X →ₛ E) :=
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begin
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intro f,
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fapply smap.mk,
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exact is_sconnected_to_prespectrification_inv_pfun f,
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exact is_sconnected_to_prespectrification_inv_psq f
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end
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definition is_sconnected_to_prespectrification_isretr_pfun {N : succ_str} {X : gen_prespectrum N} {E : gen_spectrum N} (f : prespectrification X →ₛ E) (n : N) : to_fun (is_sconnected_to_prespectrification_inv X E (f ∘ₛ to_prespectrification X)) n ~* to_fun f n := begin exact sorry end
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definition is_sconnected_to_prespectrification_isretr_psq {N : succ_str} {X : gen_prespectrum N} {E : gen_spectrum N} (f : prespectrification X →ₛ E) (n : N) :
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ptube_v (is_sconnected_to_prespectrification_isretr_pfun f n)
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(Ω⇒ (is_sconnected_to_prespectrification_isretr_pfun f (S n)))
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(glue_square (is_sconnected_to_prespectrification_inv X E (f ∘ₛ to_prespectrification X)) n)
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(glue_square f n)
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:=
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begin
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repeat exact sorry
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end
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definition is_sconnected_to_prespectrification_isretr {N : succ_str} (X : gen_prespectrum N) (E : gen_spectrum N) (f : prespectrification X →ₛ E) : is_sconnected_to_prespectrification_inv X E (f ∘ₛ to_prespectrification X) = f :=
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begin
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fapply eq_of_shomotopy,
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fapply shomotopy.mk,
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exact is_sconnected_to_prespectrification_isretr_pfun f,
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exact is_sconnected_to_prespectrification_isretr_psq f,
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end
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definition is_sconnected_to_prespectrification_issec_pfun {N : succ_str} {X : gen_prespectrum N} {E : gen_spectrum N} (g : X →ₛ E) (n : N) :
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to_fun (is_sconnected_to_prespectrification_inv X E g ∘ₛ to_prespectrification X) n ~* to_fun g n
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:=
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begin
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repeat exact sorry
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end
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definition is_sconnected_to_prespectrification_issec_psq {N : succ_str} {X : gen_prespectrum N} {E : gen_spectrum N} (g : X →ₛ E) (n : N) :
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ptube_v (is_sconnected_to_prespectrification_issec_pfun g n)
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(Ω⇒ (is_sconnected_to_prespectrification_issec_pfun g (S n)))
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(glue_square (is_sconnected_to_prespectrification_inv X E g ∘ₛ to_prespectrification X) n)
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(glue_square g n)
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:=
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begin
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repeat exact sorry
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end
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definition is_sconnected_to_prespectrification_issec {N : succ_str} (X : gen_prespectrum N) (E : gen_spectrum N) (g : X →ₛ E) :
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is_sconnected_to_prespectrification_inv X E g ∘ₛ to_prespectrification X = g :=
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begin
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fapply eq_of_shomotopy,
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fapply shomotopy.mk,
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exact is_sconnected_to_prespectrification_issec_pfun g,
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exact is_sconnected_to_prespectrification_issec_psq g,
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end
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definition is_sconnected_to_prespectrify {N : succ_str} (X : gen_prespectrum N) :
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is_sconnected (to_prespectrification X) :=
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begin
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intro E,
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fapply adjointify,
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exact is_sconnected_to_prespectrification_inv X E,
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exact is_sconnected_to_prespectrification_issec X E,
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exact is_sconnected_to_prespectrification_isretr X E
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end
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-- Conjecture
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definition is_spectrum_of_local (X : gen_prespectrum +ℕ) (Hyp : is_equiv (λ (f : prespectrification (psp_sphere) →ₛ X), f ∘ₛ to_prespectrification (psp_sphere))) : is_spectrum X :=
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begin
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fapply is_spectrum.mk,
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exact sorry
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end
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/- Spectrification -/
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open chain_complex
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definition spectrify_type_term {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) : Type* :=
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Ω[k] (X (n +' k))
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definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
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Ω[k] (X n) →* Ω[k+1] (X (S n)) :=
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!loopn_succ_in⁻¹ᵉ* ∘* Ω→[k] (glue X n)
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definition spectrify_type_fun {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
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spectrify_type_term X n k →* spectrify_type_term X n (k+1) :=
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spectrify_type_fun' X (n +' k) k
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definition spectrify_type_fun_zero {N : succ_str} (X : gen_prespectrum N) (n : N) :
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spectrify_type_fun X n 0 ~* glue X n :=
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!pid_pcompose
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definition spectrify_type {N : succ_str} (X : gen_prespectrum N) (n : N) : Type* :=
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pseq_colim (spectrify_type_fun X n)
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/-
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Let Y = spectify X ≡ colim_k Ω^k X (n + k). Then
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Ω Y (n+1) ≡ Ω colim_k Ω^k X ((n + 1) + k)
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... = colim_k Ω^{k+1} X ((n + 1) + k)
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... = colim_k Ω^{k+1} X (n + (k + 1))
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... = colim_k Ω^k X(n + k)
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... ≡ Y n
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-/
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definition spectrify_type_fun'_succ {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
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spectrify_type_fun' X n (succ k) ~* Ω→ (spectrify_type_fun' X n k) :=
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begin
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refine !ap1_pcompose⁻¹*
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end
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definition spectrify_pequiv {N : succ_str} (X : gen_prespectrum N) (n : N) :
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spectrify_type X n ≃* Ω (spectrify_type X (S n)) :=
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begin
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refine !pshift_equiv ⬝e* _,
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transitivity pseq_colim (λk, spectrify_type_fun' X (S n +' k) (succ k)),
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fapply pseq_colim_pequiv,
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{ intro n, apply loopn_pequiv_loopn, apply pequiv_ap X, apply succ_str.add_succ },
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{ exact abstract begin intro k,
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refine !passoc⁻¹* ⬝* _, refine pwhisker_right _ (loopn_succ_in_inv_natural (succ k) _) ⬝* _,
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refine !passoc ⬝* _ ⬝* !passoc⁻¹*, apply pwhisker_left,
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refine !apn_pcompose⁻¹* ⬝* _ ⬝* !apn_pcompose, apply apn_phomotopy,
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exact !glue_ptransport⁻¹* end end },
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refine _ ⬝e* !pseq_colim_loop⁻¹ᵉ*,
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exact pseq_colim_equiv_constant (λn, !spectrify_type_fun'_succ),
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end
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definition spectrify [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_spectrum N :=
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spectrum.MK (spectrify_type X) (spectrify_pequiv X)
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definition spectrify_map {N : succ_str} {X : gen_prespectrum N} : X →ₛ spectrify X :=
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begin
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fapply smap.mk,
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{ intro n, exact pinclusion _ 0 },
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{ intro n, apply phomotopy_of_psquare,
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refine !pid_pcompose⁻¹* ⬝ph* _,
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refine !passoc ⬝* pwhisker_left _ (pshift_equiv_pinclusion (spectrify_type_fun X n) 0) ⬝* _,
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refine !passoc⁻¹* ⬝* _,
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refine _ ◾* (spectrify_type_fun_zero X n ⬝* !pid_pcompose⁻¹*),
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refine !passoc ⬝* pwhisker_left _ !pseq_colim_pequiv_pinclusion ⬝* _,
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refine pwhisker_left _ (pwhisker_left _ (ap1_pid) ⬝* !pcompose_pid) ⬝* _,
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refine !passoc ⬝* pwhisker_left _ !seq_colim_equiv_constant_pinclusion ⬝* _,
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apply pinv_left_phomotopy_of_phomotopy,
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exact !pseq_colim_loop_pinclusion⁻¹*
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}
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end
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definition spectrify.elim_n {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
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(f : X →ₛ Y) (n : N) : (spectrify X) n →* Y n :=
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begin
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fapply pseq_colim.elim,
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{ intro k, refine !equiv_gluen⁻¹ᵉ* ∘* apn k (f (n +' k)) },
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{ intro k, refine !passoc ⬝* pwhisker_right _ !equiv_gluen_inv_succ ⬝* _,
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refine !passoc ⬝* _, apply pwhisker_left,
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refine !passoc ⬝* _,
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refine pwhisker_left _ ((passoc _ _ (_ ∘* _))⁻¹*) ⬝* _,
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refine pwhisker_left _ !passoc⁻¹* ⬝* _,
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refine pwhisker_left _ (pwhisker_right _ (phomotopy_pinv_right_of_phomotopy (!loopn_succ_in_natural))⁻¹*) ⬝* _,
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refine pwhisker_right _ !apn_pinv ⬝* _,
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refine (phomotopy_pinv_left_of_phomotopy _)⁻¹*,
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refine apn_psquare k _,
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refine psquare_of_phomotopy !smap.glue_square }
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end
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definition spectrify.elim {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
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(f : X →ₛ Y) : spectrify X →ₛ Y :=
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begin
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fapply smap.mk,
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{ intro n, exact spectrify.elim_n f n },
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{ intro n, exact sorry }
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end
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definition phomotopy_spectrify.elim {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
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(f : X →ₛ Y) (n : N) : spectrify.elim_n f n ∘* spectrify_map n ~* f n :=
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begin
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refine pseq_colim.elim_pinclusion _ _ 0 ⬝* _,
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exact !pid_pcompose
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end
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definition spectrify_fun {N : succ_str} {X Y : gen_prespectrum N} (f : X →ₛ Y) : spectrify X →ₛ spectrify Y :=
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spectrify.elim ((@spectrify_map _ Y) ∘ₛ f)
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end spectrum
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