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working on spectral equivalences

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This commit is contained in:
Egbert Rijke 2017-07-01 13:06:47 +01:00
parent bf3a132e99
commit dffe842061
1 changed files with 37 additions and 1 deletions
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38
homotopy/spectrum.hlean
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@ -244,8 +244,44 @@ namespace spectrum
infix ` ≃ₛ ` : 25 := sequiv
definition is_sequiv_of_smap {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) (H : Π (n : N), is_equiv (f n)) : is_sequiv f :=
definition is_sequiv_smap {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) : Type := Π (n: N), is_equiv (f n)
definition is_sequiv_of_smap_pequiv {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) (H : is_sequiv_smap f) (n : N) : E n ≃* F n :=
begin
fapply pequiv_of_pmap,
exact f n,
fapply H,
end
definition is_sequiv_of_smap_inv {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) (H : is_sequiv_smap f) : F →ₛ E :=
begin
fapply smap.mk,
intro n,
exact (is_sequiv_of_smap_pequiv f H n)⁻¹ᵉ*,
intro n,
refine _ ⬝vp* (to_pinv_loopn_pequiv_loopn 1 (is_sequiv_of_smap_pequiv f H (S n)))⁻¹*,
fapply phinverse,
exact glue_square f n,
end
local postfix `⁻¹ˢ` : (max + 1) := is_sequiv_of_smap_inv
definition is_sequiv_of_smap_isretr {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) (H : is_sequiv_smap f) : is_sequiv_of_smap_inv f H ∘ₛ f ~ₛ sid E :=
begin
fapply shomotopy.mk,
intro n,
fapply pleft_inv,
intro n,
refine _ ⬝hp** _,
repeat exact sorry,
end
definition is_sequiv_of_smap {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) : is_sequiv_smap f → is_sequiv f :=
begin
intro H,
fapply is_sequiv.mk,
fapply is_sequiv_of_smap_inv f H,
fapply is_sequiv_of_smap_isretr f H,
repeat exact sorry
end