reorganizing the prespectrification section
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Egbert Rijke
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997d75cbf3
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58b007c873
1 changed files with 44 additions and 13 deletions
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@ -332,18 +332,21 @@ namespace spectrum
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repeat exact sorry,
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end
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definition is_sequiv_of_smap_issec {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) (H : is_sequiv_smap f) : f ∘ₛ is_sequiv_of_smap_inv f H ~ₛ sid F :=
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begin
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repeat exact sorry
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end
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definition is_sequiv_of_smap {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) : is_sequiv_smap f → is_sequiv f :=
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begin
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intro H,
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fapply is_sequiv.mk,
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fapply is_sequiv_of_smap_inv f H,
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fapply is_sequiv_of_smap_isretr f H,
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repeat exact sorry
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fapply is_sequiv_of_smap_inv f H,
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fapply is_sequiv_of_smap_issec f H,
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end
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-- definition is_sequiv_psimple {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) : Type :=
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-- Π (n : N), is_pequiv
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------------------------------
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-- Suspension prespectra
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------------------------------
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@ -566,18 +569,31 @@ namespace spectrum
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-- note: see also cotensor above
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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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definition prespectrify [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_prespectrum N :=
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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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definition to_prespectrify {N : succ_str} (X : gen_prespectrum N) : X →ₛ prespectrify X :=
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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 glue X,
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intro n, fapply psquare_of_phomotopy, reflexivity
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end
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definition is_leftmap_to_prespectrify_inv {N : succ_str} (X : gen_prespectrum N) (E : gen_spectrum N) : X →ₛ gen_spectrum.to_prespectrum E → prespectrify X →ₛ gen_spectrum.to_prespectrum E :=
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definition is_sequiv_smap_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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repeat exact sorry
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end
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definition is_sequiv_of_spectrum {N : succ_str} (E : gen_spectrum N) : is_sequiv_smap (to_prespectrification E) :=
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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_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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@ -586,21 +602,36 @@ namespace spectrum
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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 (prespectrify 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 _ ⬝* 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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definition is_leftmap_to_prespectrify {N : succ_str} (X : gen_prespectrum N) (E : gen_spectrum N) :
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is_equiv (λ (f : prespectrify X →ₛ E), f ∘ₛ to_prespectrify X) :=
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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 adjointify,
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exact is_leftmap_to_prespectrify_inv X E,
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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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repeat exact sorry
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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 (E : gen_spectrum +ℕ) (Hyp : is_equiv (λ (f : prespectrify (psp_sphere) →ₛ E), f ∘ₛ to_prespectrify (psp_sphere))) : is_spectrum E :=
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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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