Merge branch 'master' of https://github.com/cmu-phil/Spectral
This commit is contained in:
Egbert Rijke
commit
5ac3ce7058
4 changed files with 39 additions and 30 deletions
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@ -90,7 +90,12 @@ end
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definition cohomology_isomorphism_shomotopy_group_sp_cotensor (X : Type*) (Y : spectrum) {n m : ℤ}
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(p : -m = n) : H^n[X, Y] ≃g πₛ[m] (sp_cotensor X Y) :=
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sorry /- TODO FOR SSS -/
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begin
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refine !trunc_ppi_isomorphic_pmap⁻¹ᵍ ⬝g _,
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refine parametrized_cohomology_isomorphism_shomotopy_group_spi (λx, Y) p ⬝g _,
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apply shomotopy_group_isomorphism_of_pequiv, intro k,
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apply pppi_pequiv_ppmap
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end
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definition unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor (X : Type) (Y : spectrum)
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{n m : ℤ} (p : -m = n) : uH^n[X, Y] ≃g πₛ[m] (sp_ucotensor X Y) :=
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@ -99,8 +104,6 @@ begin
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apply shomotopy_group_isomorphism_of_pequiv, intro k, apply ppmap_add_point
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end
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/- functoriality -/
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definition cohomology_functor [constructor] {X X' : Type*} (f : X' →* X) (Y : spectrum)
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@ -138,23 +141,24 @@ definition cohomology_isomorphism_refl (X : Type*) (Y : spectrum) (n : ℤ) (x :
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definition cohomology_isomorphism_right (X : Type*) {Y Y' : spectrum} (e : Πn, Y n ≃* Y' n)
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(n : ℤ) : H^n[X, Y] ≃g H^n[X, Y'] :=
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sorry /- TODO FOR SSS -/
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cohomology_isomorphism_shomotopy_group_sp_cotensor X Y !neg_neg ⬝g
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shomotopy_group_isomorphism_of_pequiv (-n) (λk, pequiv_ppcompose_left (e k)) ⬝g
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(cohomology_isomorphism_shomotopy_group_sp_cotensor X Y' !neg_neg)⁻¹ᵍ
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definition parametrized_cohomology_isomorphism_right {X : Type*} {Y Y' : X → spectrum}
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(e : Πx n, Y x n ≃* Y' x n) (n : ℤ) : pH^n[(x : X), Y x] ≃g pH^n[(x : X), Y' x] :=
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parametrized_cohomology_isomorphism_shomotopy_group_spi Y !neg_neg ⬝g
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shomotopy_group_isomorphism_of_pequiv (-n) (λk, ppi_pequiv_right sorry) ⬝g
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shomotopy_group_isomorphism_of_pequiv (-n) (λk, ppi_pequiv_right (λx, e x k)) ⬝g
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(parametrized_cohomology_isomorphism_shomotopy_group_spi Y' !neg_neg)⁻¹ᵍ
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--sorry /- TODO FOR SSS -/
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definition unreduced_parametrized_cohomology_isomorphism_right {X : Type} {Y Y' : X → spectrum}
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(e : Πx n, Y x n ≃* Y' x n) (n : ℤ) : upH^n[(x : X), Y x] ≃g upH^n[(x : X), Y' x] :=
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sorry /- TODO FOR SSS -/
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parametrized_cohomology_isomorphism_right (λx' k, add_point_over_pequiv (λx, e x k) x') n
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definition unreduced_ordinary_parametrized_cohomology_isomorphism_right {X : Type}
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{G G' : X → AbGroup} (e : Πx, G x ≃g G' x) (n : ℤ) :
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uopH^n[(x : X), G x] ≃g uopH^n[(x : X), G' x] :=
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sorry /- TODO FOR SSS -/
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unreduced_parametrized_cohomology_isomorphism_right (λx, EM_spectrum_pequiv (e x)) n
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definition ordinary_cohomology_isomorphism_right (X : Type*) {G G' : AbGroup} (e : G ≃g G')
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(n : ℤ) : oH^n[X, G] ≃g oH^n[X, G'] :=
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@ -5,8 +5,8 @@ Authors: Egbert Rijke
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-/
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/-
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The goal of this file is to extend the library of pointed types and pointed maps to support the library of prespectra
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/-
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The goal of this file is to extend the library of pointed types and pointed maps to support the library of prespectra
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-/
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@ -37,14 +37,14 @@ end
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definition psquare_of_pid_top_bot {A B : Type*} {fleft : A →* B} {fright : A →* B} (phtpy : fright ~* fleft) : psquare (pid A) (pid B) fleft fright :=
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psquare_of_phomotopy ((pcompose_pid fright) ⬝* phtpy ⬝* (pid_pcompose fleft)⁻¹*)
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print psquare_of_pid_top_bot
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--print psquare_of_pid_top_bot
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--λ phtpy, psquare_of_phomotopy ((pid_pcompose fleft) ⬝* phtpy ⬝* ((pcompose_pid fright)⁻¹*))
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definition psquare_of_pid_left_right {A B : Type*} {ftop : A →* B} {fbot : A →* B} (phtpy : ftop ~* fbot) : psquare ftop fbot (pid A) (pid B) :=
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psquare_of_phomotopy ((pid_pcompose ftop) ⬝* phtpy ⬝* ((pcompose_pid fbot)⁻¹*))
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print psquare_of_pid_left_right
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--print psquare_of_pid_left_right
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definition psquare_hcompose {A B C D E F : Type*} {ftop : A →* B} {fbot : D →* E} {fleft : A →* D} {fright : B →* E} {gtop : B →* C} {gbot : E →* F} {gright : C →* F} (psq_left : psquare ftop fbot fleft fright) (psq_right : psquare gtop gbot fright gright) : psquare (gtop ∘* ftop) (gbot ∘* fbot) fleft gright :=
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begin
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@ -100,7 +100,7 @@ phsquare (pwhisker_left fright phtpy_top) (pwhisker_right fleft phtpy_bot) psq_b
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definition ptube_h {A B C D : Type*} {ftop : A →* B} {fbot : C →* D} {fleft fleft' : A →* C} (phtpy_left : fleft ~* fleft') {fright fright' : B →* D} (phtpy_right : fright ~* fright') (psq_back : psquare ftop fbot fleft fright) (psq_front : psquare ftop fbot fleft' fright') : Type :=
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phsquare (pwhisker_right ftop phtpy_right) (pwhisker_left fbot phtpy_left) psq_back psq_front
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print pinv_right_phomotopy_of_phomotopy
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--print pinv_right_phomotopy_of_phomotopy
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definition psquare_inv_top_bot {A B C D : Type*} {ftop : A ≃* B} {fbot : C ≃* D} {fleft : A →* C} {fright : B →* D} (psq : psquare ftop fbot fleft fright) : psquare ftop⁻¹ᵉ* fbot⁻¹ᵉ* fright fleft :=
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begin
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@ -114,25 +114,25 @@ end
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definition p2homotopy_ty_respect_pt {A B : Type*} {f g : A →* B} {H K : f ~* g} (htpy : H ~ K) : Type :=
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begin
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induction H with H p, exact p
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end = whisker_right (respect_pt g) (htpy pt) ⬝
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end = whisker_right (respect_pt g) (htpy pt) ⬝
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begin
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induction K with K q, exact q
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end
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print p2homotopy_ty_respect_pt
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--print p2homotopy_ty_respect_pt
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structure p2homotopy {A B : Type*} {f g : A →* B} (H K : f ~* g) : Type :=
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( to_2htpy : H ~ K)
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( respect_pt : p2homotopy_ty_respect_pt to_2htpy)
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definition ptube_v_phtpy_bot {A B C D : Type*}
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{ftop ftop' : A →* B} {phtpy_top : ftop ~* ftop'}
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definition ptube_v_phtpy_bot {A B C D : Type*}
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{ftop ftop' : A →* B} {phtpy_top : ftop ~* ftop'}
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{fbot fbot' : C →* D} {phtpy_bot phtpy_bot' : fbot ~* fbot'} (ppi_htpy_bot : phtpy_bot ~~* phtpy_bot')
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{fleft : A →* C} {fright : B →* D}
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{psq_back : psquare ftop fbot fleft fright}
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{psq_front : psquare ftop' fbot' fleft fright}
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(ptb : ptube_v phtpy_top phtpy_bot psq_back psq_front)
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: ptube_v phtpy_top phtpy_bot' psq_back psq_front
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{fleft : A →* C} {fright : B →* D}
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{psq_back : psquare ftop fbot fleft fright}
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{psq_front : psquare ftop' fbot' fleft fright}
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(ptb : ptube_v phtpy_top phtpy_bot psq_back psq_front)
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: ptube_v phtpy_top phtpy_bot' psq_back psq_front
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:=
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begin
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induction ppi_htpy_bot using ppi_homotopy_rec_on_idp,
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@ -146,12 +146,12 @@ begin
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exact id,
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end
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definition ptube_v_left_inv {A B C D : Type*} {ftop : A ≃* B} {fbot : C ≃* D} {fleft : A →* C} {fright : B →* D}
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(psq : psquare ftop fbot fleft fright) :
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ptube_v
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(pleft_inv ftop)
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(pleft_inv fbot)
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(psquare_hcompose psq (psquare_inv_top_bot psq))
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definition ptube_v_left_inv {A B C D : Type*} {ftop : A ≃* B} {fbot : C ≃* D} {fleft : A →* C} {fright : B →* D}
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(psq : psquare ftop fbot fleft fright) :
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ptube_v
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(pleft_inv ftop)
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(pleft_inv fbot)
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(psquare_hcompose psq (psquare_inv_top_bot psq))
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(psquare_of_pid_top_bot phomotopy.rfl) :=
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begin
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refine ptube_v_phtpy_bot _ _,
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@ -210,8 +210,8 @@ end
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open option
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definition is_strunc_add_point_spectrum {X : Type} {Y : X → spectrum} {s₀ : ℤ}
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(H : Πx, is_strunc s₀ (Y x)) : Π(x : X₊), is_strunc s₀ (add_point_spectrum Y x)
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| (some x) := H x
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| none := is_strunc_sunit s₀
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| (some x) := proof H x qed
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| none := begin intro k, apply is_trunc_lift, apply is_trunc_unit end
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definition is_strunc_EM_spectrum (G : AbGroup)
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: is_strunc 0 (EM_spectrum G) :=
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@ -231,6 +231,11 @@ namespace pointed
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| (some a) := B a
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| none := plift punit
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definition add_point_over_pequiv {A : Type} {B B' : A → Type*} (e : Πa, B a ≃* B' a) :
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Π(a : A₊), add_point_over B a ≃* add_point_over B' a
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| (some a) := e a
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| none := pequiv.rfl
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definition phomotopy_group_plift_punit.{u} (n : ℕ) [H : is_at_least_two n] :
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πag[n] (plift.{0 u} punit) ≃g trivial_ab_group_lift.{u} :=
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begin
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