fix definition of atiyah-hirzebruch spectral sequence, define serre spectral sequence
The construction of the Serre spectral sequence is done up to 11 sorry's, all which are marked with 'TODO FOR SSS'. 8 of them are equivalences related to cohomology (6 of which are corollaries of the other 2), 2 of them are calculations on int, and the last is in the definition of a spectrum map.
This commit is contained in:
Floris van Doorn
parent
f6978927b2
commit
90f4acb3f6
8 changed files with 284 additions and 98 deletions
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@ -6,6 +6,7 @@ import ..homotopy.smash
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open bool pointed eq equiv is_equiv sum bool prod unit circle cofiber prod.ops wedge is_trunc
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function red_susp unit sigma
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exit
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namespace smash
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variables {A B C : Type*}
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@ -43,7 +43,7 @@ definition ordinary_parametrized_cohomology [reducible] {X : Type*} (G : X → A
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parametrized_cohomology (λx, EM_spectrum (G x)) n
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definition unreduced_parametrized_cohomology {X : Type} (Y : X → spectrum) (n : ℤ) : AbGroup :=
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@parametrized_cohomology X₊ (add_point_spectrum Y) n
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parametrized_cohomology (add_point_spectrum Y) n
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definition unreduced_ordinary_parametrized_cohomology [reducible] {X : Type} (G : X → AbGroup)
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(n : ℤ) : AbGroup :=
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@ -72,11 +72,19 @@ trunc_equiv_trunc 0 (!pfunext ⬝e loop_pequiv_loop !pfunext ⬝e loopn_pequiv_l
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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
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sorry /- TODO FOR SSS -/
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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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sorry /- TODO FOR SSS -/
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definition parametrized_cohomology_isomorphism_shomotopy_group_spi {X : Type*} (Y : X → spectrum)
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{n m : ℤ} (p : -m = n) : pH^n[(x : X), Y x] ≃g πₛ[m] (spi X Y) :=
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sorry
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sorry /- TODO FOR SSS -/
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definition unreduced_parametrized_cohomology_isomorphism_shomotopy_group_supi {X : Type}
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(Y : X → spectrum) {n m : ℤ} (p : -m = n) : upH^n[(x : X), Y x] ≃g πₛ[m] (supi X Y) :=
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sorry /- TODO FOR SSS -/
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/- functoriality -/
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@ -113,14 +121,22 @@ definition cohomology_isomorphism_refl (X : Type*) (Y : spectrum) (n : ℤ) (x :
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cohomology_isomorphism (pequiv.refl X) Y n x = x :=
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!Group_trunc_pmap_isomorphism_refl
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definition cohomology_isomorphism_right (X : Type*) {Y Y' : spectrum} (e : Πn, Y n ≃* Y' n) (n : ℤ)
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: H^n[X, Y] ≃g H^n[X, Y'] :=
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sorry
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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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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)
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: pH^n[(x : X), Y x] ≃g pH^n[(x : X), Y' x] :=
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sorry
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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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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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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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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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@ -151,79 +151,73 @@ begin
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{ apply phomotopy_of_is_contr_cod, apply is_trunc_trunc }
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end
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definition pfiber_postnikov_smap (A : spectrum) (n : ℤ) (k : ℤ) :
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sfiber (postnikov_smap A n) k ≃* EM_spectrum (πₛ[n] A) (n + k) :=
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definition sfiber_postnikov_smap_pequiv (A : spectrum) (n : ℤ) (k : ℤ) :
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sfiber (postnikov_smap A n) k ≃* ssuspn n (EM_spectrum (πₛ[n] A)) k :=
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proof
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pfiber_pequiv_of_square _ _ (postnikov_smap_postnikov_map A n k (n + k) idp) ⬝e*
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pfiber_postnikov_map_pred' A n k _ idp
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pfiber_postnikov_map_pred' A n k _ idp ⬝e*
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pequiv_ap (EM_spectrum (πₛ[n] A)) (add.comm n k)
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qed
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section atiyah_hirzebruch
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parameters {X : Type*} (Y : X → spectrum) (s₀ : ℤ) (H : Πx, is_strunc s₀ (Y x))
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definition atiyah_hirzebruch_exact_couple : exact_couple rℤ Z2 :=
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@exact_couple_sequence (λs, strunc s (spi X Y)) (postnikov_smap (spi X Y))
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definition atiyah_hirzebruch_ub ⦃s n : ℤ⦄ (Hs : s ≤ n - 1) :
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is_contr (πₛ[n] (strunc s (spi X Y))) :=
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begin
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apply trivial_shomotopy_group_of_is_strunc,
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apply is_strunc_strunc,
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exact lt_of_le_sub_one Hs
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end
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@exact_couple_sequence (λs, spi X (λx, strunc s (Y x)))
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(λs, spi_compose_left (λx, postnikov_smap (Y x) s))
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include H
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definition atiyah_hirzebruch_lb ⦃s n : ℤ⦄ (Hs : s ≥ s₀ + 1) :
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is_equiv (postnikov_smap (spi X Y) s n) :=
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definition atiyah_hirzebruch_ub ⦃s n : ℤ⦄ (Hs : s ≤ n - 1) :
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is_contr (πₛ[n] (spi X (λx, strunc s (Y x)))) :=
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begin
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refine trivial_shomotopy_group_of_is_strunc _ _ (lt_of_le_sub_one Hs),
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apply is_strunc_spi, intro x, exact is_strunc_strunc _ _
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end
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definition atiyah_hirzebruch_lb ⦃s n : ℤ⦄ (Hs : s ≥ s₀ + 1) :
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is_equiv (spi_compose_left (λx, postnikov_smap (Y x) s) n) :=
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begin
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have H2 : is_strunc s₀ (spi X Y), from is_strunc_spi _ _ H,
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refine is_equiv_of_equiv_of_homotopy
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(ptrunc_pequiv_ptrunc_of_is_trunc _ _ (H2 n)) _,
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{ apply maxm2_monotone, apply add_le_add_right, exact le.trans !le_add_one Hs },
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{ apply maxm2_monotone, apply add_le_add_right, exact le_sub_one_of_lt Hs },
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refine @trunc.rec _ _ _ _ _,
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{ intro x, apply is_trunc_eq,
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assert H3 : maxm2 (s - 1 + n) ≤ (maxm2 (s + n)).+1,
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{ refine trunc_index.le_succ (maxm2_monotone (le.trans (le_of_eq !add.right_comm)
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!sub_one_le)) },
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exact @is_trunc_of_le _ _ _ H3 !is_trunc_trunc },
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intro a, reflexivity
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(ppi_pequiv_right (λx, ptrunc_pequiv_ptrunc_of_is_trunc _ _ (H x n))) _,
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{ intro x, apply maxm2_monotone, apply add_le_add_right, exact le.trans !le_add_one Hs },
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{ intro x, apply maxm2_monotone, apply add_le_add_right, exact le_sub_one_of_lt Hs },
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intro f, apply eq_of_ppi_homotopy,
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apply pmap_compose_ppi_phomotopy_left, intro x,
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fapply phomotopy.mk,
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{ refine @trunc.rec _ _ _ _ _,
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{ intro x, apply is_trunc_eq,
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assert H3 : maxm2 (s - 1 + n) ≤ (maxm2 (s + n)).+1,
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{ refine trunc_index.le_succ (maxm2_monotone (le.trans (le_of_eq !add.right_comm)
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!sub_one_le)) },
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exact @is_trunc_of_le _ _ _ H3 !is_trunc_trunc },
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intro a, reflexivity },
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reflexivity
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end
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definition is_bounded_atiyah_hirzebruch : is_bounded atiyah_hirzebruch_exact_couple :=
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is_bounded_sequence _ s₀ (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
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definition atiyah_hirzebruch_convergence' :
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(λn s, πₛ[n] (sfiber (postnikov_smap (spi X Y) s))) ⟹ᵍ (λn, πₛ[n] (strunc s₀ (spi X Y))) :=
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(λn s, πₛ[n] (sfiber (spi_compose_left (λx, postnikov_smap (Y x) s)))) ⟹ᵍ
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(λn, πₛ[n] (spi X (λx, strunc s₀ (Y x)))) :=
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converges_to_sequence _ s₀ (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
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lemma spi_EM_spectrum (n : ℤ) : Π(k : ℤ),
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EM_spectrum (πₛ[n] (spi X Y)) (n + k) ≃* spi X (λx, EM_spectrum (πₛ[n] (Y x))) k :=
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begin
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exact sorry
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-- apply spectrum_pequiv_of_nat_add 2, intro k,
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-- fapply EMadd1_pequiv (k+1),
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-- { exact sorry },
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-- { exact sorry },
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-- { apply is_trunc_ppi, rotate 1, intro x, },
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end
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set_option formatter.hide_full_terms false
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definition atiyah_hirzebruch_convergence :
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(λn s, opH^-n[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
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(λn s, opH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
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converges_to_g_isomorphism atiyah_hirzebruch_convergence'
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begin
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intro n s,
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refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
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apply shomotopy_group_isomorphism_of_pequiv, intro k,
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refine pfiber_postnikov_smap (spi X Y) s k ⬝e* _,
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apply spi_EM_spectrum
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refine _ ⬝g !shomotopy_group_ssuspn,
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apply shomotopy_group_isomorphism_of_pequiv n, intro k,
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refine !pfiber_ppi_compose_left ⬝e* _,
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exact ppi_pequiv_right (λx, sfiber_postnikov_smap_pequiv (Y x) s k)
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end
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begin
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intro n,
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refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
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apply shomotopy_group_isomorphism_of_pequiv, intro k,
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apply ptrunc_pequiv, exact is_strunc_spi s₀ Y H k,
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exact ppi_pequiv_right (λx, ptrunc_pequiv (maxm2 (s₀ + k)) (Y x k)),
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end
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end atiyah_hirzebruch
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@ -232,7 +226,7 @@ section unreduced_atiyah_hirzebruch
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definition unreduced_atiyah_hirzebruch_convergence {X : Type} (Y : X → spectrum) (s₀ : ℤ)
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(H : Πx, is_strunc s₀ (Y x)) :
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(λn s, uopH^-n[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, upH^-n[(x : X), Y x]) :=
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(λn s, uopH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, upH^-n[(x : X), Y x]) :=
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converges_to_g_isomorphism
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(@atiyah_hirzebruch_convergence X₊ (add_point_spectrum Y) s₀ (is_strunc_add_point_spectrum H))
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begin
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@ -249,36 +243,28 @@ end unreduced_atiyah_hirzebruch
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section serre
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variables {X : Type} (F : X → Type) (Y : spectrum) (s₀ : ℤ) (H : is_strunc s₀ Y)
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open option
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definition add_point_over {X : Type} (F : X → Type) (x : option X) : Type* :=
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(option.cases_on x (lift empty) F)₊
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postfix `₊ₒ`:(max+1) := add_point_over
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include H
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-- definition serre_convergence :
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-- (λn s, uopH^-n[(x : X), uH^-s[F x, Y]]) ⟹ᵍ (λn, uH^-n[Σ(x : X), F x, Y]) :=
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-- proof
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-- converges_to_g_isomorphism
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-- (unreduced_atiyah_hirzebruch_convergence
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-- (λx, sp_cotensor (F x) Y) s₀
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-- (λx, is_strunc_sp_cotensor s₀ (F x) H))
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-- begin
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-- exact sorry
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-- -- intro n s,
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-- -- apply ordinary_parametrized_cohomology_isomorphism_right,
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-- -- intro x,
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-- -- exact (cohomology_isomorphism_shomotopy_group_sp_cotensor _ _ idp)⁻¹ᵍ,
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-- end
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-- begin
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-- intro n, exact sorry
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-- -- refine parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp ⬝g _,
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-- -- refine _ ⬝g (cohomology_isomorphism_shomotopy_group_sp_cotensor _ _ idp)⁻¹ᵍ,
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-- -- apply shomotopy_group_isomorphism_of_pequiv, intro k,
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-- end
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-- qed
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definition serre_convergence :
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(λn s, uopH^-(n-s)[(x : X), uH^-s[F x, Y]]) ⟹ᵍ (λn, uH^-n[Σ(x : X), F x, Y]) :=
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proof
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converges_to_g_isomorphism
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(unreduced_atiyah_hirzebruch_convergence
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(λx, sp_ucotensor (F x) Y) s₀
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(λx, is_strunc_sp_ucotensor s₀ (F x) H))
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begin
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intro n s,
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refine unreduced_ordinary_parametrized_cohomology_isomorphism_right _ (-(n-s)),
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intro x,
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exact (unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor _ _ idp)⁻¹ᵍ
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end
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begin
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intro n,
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refine unreduced_parametrized_cohomology_isomorphism_shomotopy_group_supi _ idp ⬝g _,
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refine _ ⬝g (unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor _ _ idp)⁻¹ᵍ,
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apply shomotopy_group_isomorphism_of_pequiv, intro k,
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exact (sigma_pumap F (Y k))⁻¹ᵉ*
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end
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qed
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end serre
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/- TODO: πₛ[n] (strunc 0 (spi X Y)) ≃g H^n[X, λx, Y x] -/
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end spectrum
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@ -639,17 +639,35 @@ set_option pp.coercions true
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-- Makes sense for any indexing succ_str. Could be done for
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-- prespectra too, but as with truncation, why bother?
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definition sp_cotensor [constructor] {N : succ_str} (A : Type*) (B : gen_spectrum N) : gen_spectrum N :=
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definition sp_cotensor [constructor] {N : succ_str} (A : Type*) (B : gen_spectrum N) :
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gen_spectrum N :=
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spectrum.MK (λn, ppmap A (B n))
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(λn, (loop_ppmap_commute A (B (S n)))⁻¹ᵉ* ∘*ᵉ (pequiv_ppcompose_left (equiv_glue B n)))
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/- unpointed cotensor -/
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definition sp_ucotensor [constructor] {N : succ_str} (A : Type) (B : gen_spectrum N) :
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gen_spectrum N :=
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spectrum.MK (λn, A →ᵘ* B n)
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(λn, pumap_pequiv_right A (equiv_glue B n) ⬝e* (loop_pumap A (B (S n)))⁻¹ᵉ*)
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----------------------------------------
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-- Sections of parametrized spectra
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----------------------------------------
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definition spi [constructor] {N : succ_str} (A : Type*) (E : A -> gen_spectrum N) : gen_spectrum N :=
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spectrum.MK (λn, Π*a, E a n)
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(λn, !ppi_loop_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
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definition spi [constructor] {N : succ_str} (A : Type*) (E : A → gen_spectrum N) :
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gen_spectrum N :=
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spectrum.MK (λn, Π*a, E a n)
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(λn, !ppi_loop_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
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definition spi_compose_left [constructor] {N : succ_str} {A : Type*} {E F : A -> gen_spectrum N}
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(f : Πa, E a →ₛ F a) : spi A E →ₛ spi A F :=
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smap.mk (λn, ppi_compose_left (λa, f a n)) sorry /- TODO FOR SSS -/
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-- unpointed spi
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definition supi [constructor] {N : succ_str} (A : Type) (E : A → gen_spectrum N) :
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gen_spectrum N :=
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spectrum.MK (λn, Πᵘ*a, E a n)
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(λn, pupi_pequiv_right (λa, equiv_glue (E a) n) ⬝e* (loop_pupi (λa, E a (S n)))⁻¹ᵉ*)
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/-----------------------------------------
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Fibers and long exact sequences
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@ -1064,6 +1082,28 @@ set_option pp.coercions true
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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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/-
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suspension of a spectrum
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this is just a shift. A shift in the other direction is loopn,
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but we might not want to define that yet.
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-/
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definition ssusp [constructor] {N : succ_str} (X : gen_spectrum N) : gen_spectrum N :=
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spectrum.MK (λn, X (S n)) (λn, equiv_glue X (S n))
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definition ssuspn [constructor] (k : ℤ) (X : spectrum) : spectrum :=
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spectrum.MK (λn, X (n + k))
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(λn, equiv_glue X (n + k) ⬝e* loop_pequiv_loop (pequiv_ap X !add.right_comm))
|
||||
|
||||
definition shomotopy_group_ssuspn (k : ℤ) (X : spectrum) (n : ℤ) :
|
||||
πₛ[k] (ssuspn n X) ≃g πₛ[k - n] X :=
|
||||
have k - n + (2 - k + n) = 2, from
|
||||
!add.comm ⬝
|
||||
ap (λx, x + (k - n)) (!add.assoc ⬝ ap (λx, 2 + x) (ap (λx, -k + x) !neg_neg⁻¹ ⬝ !neg_add⁻¹)) ⬝
|
||||
sub_add_cancel 2 (k - n),
|
||||
(shomotopy_group_isomorphism_homotopy_group X this)⁻¹ᵍ
|
||||
|
||||
/- Tensor by spaces -/
|
||||
|
||||
/- Smash product of spectra -/
|
||||
|
|
|
|||
|
|
@ -8,12 +8,12 @@ namespace int
|
|||
private definition maxm2_le.lemma₁ {n k : ℕ} : n+(1:int) + -[1+ k] ≤ n :=
|
||||
le.intro (
|
||||
calc n + 1 + -[1+ k] + k = n + 1 - (k + 1) + k : by reflexivity
|
||||
... = n : sorry)
|
||||
... = n : sorry) /- TODO FOR SSS -/
|
||||
|
||||
private definition maxm2_le.lemma₂ {n : ℕ} {k : ℤ} : -[1+ n] + 1 + k ≤ k :=
|
||||
le.intro (
|
||||
calc -[1+ n] + 1 + k + n = - (n + 1) + 1 + k + n : by reflexivity
|
||||
... = k : sorry)
|
||||
... = k : sorry) /- TODO FOR SSS -/
|
||||
|
||||
definition maxm2_le (n k : ℤ) : maxm2 (n+1+k) ≤ (maxm1m1 n).+1+2+(maxm1m1 k) :=
|
||||
begin
|
||||
|
|
@ -142,6 +142,10 @@ definition is_strunc_strunc (k : ℤ) (E : spectrum)
|
|||
: is_strunc k (strunc k E) :=
|
||||
λ n, is_trunc_trunc (maxm2 (k + n)) (E n)
|
||||
|
||||
definition is_strunc_strunc_of_is_strunc (k : ℤ) {l : ℤ} {E : spectrum} (H : is_strunc l E)
|
||||
: is_strunc l (strunc k E) :=
|
||||
λ n, !is_trunc_trunc_of_is_trunc
|
||||
|
||||
definition str [constructor] (k : ℤ) (E : spectrum) : E →ₛ strunc k E :=
|
||||
smap.mk (λ n, ptr (maxm2 (k + n)) (E n))
|
||||
abstract begin
|
||||
|
|
@ -283,4 +287,8 @@ definition is_strunc_sp_cotensor (n : ℤ) (A : Type*) {Y : spectrum} (H : is_st
|
|||
: is_strunc n (sp_cotensor A Y) :=
|
||||
is_strunc_pequiv_closed (λn, !pppi_pequiv_ppmap) (is_strunc_spi n (λa, Y) (λa, H))
|
||||
|
||||
definition is_strunc_sp_ucotensor (n : ℤ) (A : Type) {Y : spectrum} (H : is_strunc n Y)
|
||||
: is_strunc n (sp_ucotensor A Y) :=
|
||||
λk, !pi.is_trunc_arrow
|
||||
|
||||
end spectrum
|
||||
|
|
|
|||
|
|
@ -7,6 +7,16 @@ open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc pi gr
|
|||
|
||||
namespace eq
|
||||
|
||||
definition apd10_prepostcompose_nondep {A B C D : Type} (h : C → D) {g g' : B → C} (p : g = g')
|
||||
(f : A → B) (a : A) : apd10 (ap (λg a, h (g (f a))) p) a = ap h (apd10 p (f a)) :=
|
||||
begin induction p, reflexivity end
|
||||
|
||||
definition apd10_prepostcompose {A B : Type} {C : B → Type} {D : A → Type}
|
||||
(f : A → B) (h : Πa, C (f a) → D a) {g g' : Πb, C b}
|
||||
(p : g = g') (a : A) :
|
||||
apd10 (ap (λg a, h a (g (f a))) p) a = ap (h a) (apd10 p (f a)) :=
|
||||
begin induction p, reflexivity end
|
||||
|
||||
definition eq.rec_to {A : Type} {a₀ : A} {P : Π⦃a₁⦄, a₀ = a₁ → Type}
|
||||
{a₁ : A} (p₀ : a₀ = a₁) (H : P p₀) ⦃a₂ : A⦄ (p : a₀ = a₂) : P p :=
|
||||
begin
|
||||
|
|
|
|||
|
|
@ -157,6 +157,14 @@ namespace pointed
|
|||
h a' ∘* ptransport X (ap f p) ~* ptransport Y (ap g p) ∘* h a :=
|
||||
by induction p; exact !pcompose_pid ⬝* !pid_pcompose⁻¹*
|
||||
|
||||
definition ptransport_ap {A B : Type} (X : B → Type*) (f : A → B) {a a' : A} (p : a = a') :
|
||||
ptransport X (ap f p) ~* ptransport (X ∘ f) p :=
|
||||
by induction p; reflexivity
|
||||
|
||||
definition ptransport_constant (A : Type) (B : Type*) {a a' : A} (p : a = a') :
|
||||
ptransport (λ(a : A), B) p ~* pid B :=
|
||||
by induction p; reflexivity
|
||||
|
||||
definition ptransport_natural {A : Type} (X : A → Type*) (Y : A → Type*)
|
||||
(h : Πa, X a →* Y a) {a a' : A} (p : a = a') :
|
||||
h a' ∘* ptransport X p ~* ptransport Y p ∘* h a :=
|
||||
|
|
|
|||
137
pointed_pi.hlean
137
pointed_pi.hlean
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Authors: Ulrik Buchholtz, Floris van Doorn
|
||||
-/
|
||||
|
||||
import homotopy.connectedness types.pointed2 .move_to_lib
|
||||
import homotopy.connectedness types.pointed2 .move_to_lib .pointed
|
||||
|
||||
open eq pointed equiv sigma is_equiv trunc
|
||||
|
||||
|
|
@ -173,7 +173,7 @@ namespace pointed
|
|||
definition eq_ppi_homotopy_refl_ppi_homotopy_of_eq_refl : ppi_homotopy.refl k = ppi_homotopy_of_eq (refl k) :=
|
||||
begin
|
||||
induction k with k p,
|
||||
induction p, reflexivity
|
||||
induction p, reflexivity
|
||||
end
|
||||
|
||||
variable {k}
|
||||
|
|
@ -281,15 +281,15 @@ namespace pointed
|
|||
pequiv_of_equiv (fiber.sigma_char f pt) idp
|
||||
|
||||
/- the pointed type of unpointed (nondependent) maps -/
|
||||
definition upmap [constructor] (A : Type) (B : Type*) : Type* :=
|
||||
definition pumap [constructor] (A : Type) (B : Type*) : Type* :=
|
||||
pointed.MK (A → B) (const A pt)
|
||||
|
||||
/- the pointed type of unpointed dependent maps -/
|
||||
definition uppi [constructor] {A : Type} (B : A → Type*) : Type* :=
|
||||
definition pupi [constructor] {A : Type} (B : A → Type*) : Type* :=
|
||||
pointed.MK (Πa, B a) (λa, pt)
|
||||
|
||||
notation `Πᵘ*` binders `, ` r:(scoped P, uppi P) := r
|
||||
infix ` →ᵘ* `:30 := upmap
|
||||
notation `Πᵘ*` binders `, ` r:(scoped P, pupi P) := r
|
||||
infix ` →ᵘ* `:30 := pumap
|
||||
|
||||
definition ppmap.sigma_char [constructor] (A B : Type*) :
|
||||
ppmap A B ≃* @psigma_gen (A →ᵘ* B) (λf, f pt = pt) idp :=
|
||||
|
|
@ -377,12 +377,12 @@ namespace pointed
|
|||
|
||||
open sigma.ops
|
||||
|
||||
definition psigma_gen_pi_pequiv_uppi_psigma_gen [constructor] {A : Type*} {B : A → Type*}
|
||||
definition psigma_gen_pi_pequiv_pupi_psigma_gen [constructor] {A : Type*} {B : A → Type*}
|
||||
(C : Πa, B a → Type) (c : Πa, C a pt) :
|
||||
@psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c ≃* Πᵘ*a, psigma_gen (C a) (c a) :=
|
||||
pequiv_of_equiv sigma_pi_equiv_pi_sigma idp
|
||||
|
||||
definition uppi_psigma_gen_pequiv_psigma_gen_pi [constructor] {A : Type*} {B : A → Type*}
|
||||
definition pupi_psigma_gen_pequiv_psigma_gen_pi [constructor] {A : Type*} {B : A → Type*}
|
||||
(C : Πa, B a → Type) (c : Πa, C a pt) :
|
||||
(Πᵘ*a, psigma_gen (C a) (c a)) ≃* @psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c :=
|
||||
pequiv_of_equiv sigma_pi_equiv_pi_sigma⁻¹ᵉ idp
|
||||
|
|
@ -408,7 +408,7 @@ namespace pointed
|
|||
≃* @psigma_gen (Πᵘ*a, psigma_gen (C a) (c a)) (λf, f pt = pt) idp : pppi.sigma_char
|
||||
... ≃* @psigma_gen (@psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c)
|
||||
(λv, Σ(p : v.1 pt = pt), v.2 pt =[p] c pt) ⟨idp, idpo⟩ :
|
||||
by exact psigma_gen_pequiv_psigma_gen (uppi_psigma_gen_pequiv_psigma_gen_pi C c)
|
||||
by exact psigma_gen_pequiv_psigma_gen (pupi_psigma_gen_pequiv_psigma_gen_pi C c)
|
||||
(λf, sigma_eq_equiv _ _) idpo
|
||||
... ≃* @psigma_gen (@psigma_gen (Πᵘ*a, B a) (λf, f pt = pt) idp)
|
||||
(λv, Σ(g : Πa, C a (v.1 a)), g pt =[v.2] c pt) ⟨c, idpo⟩ :
|
||||
|
|
@ -456,7 +456,7 @@ namespace pointed
|
|||
... ≃* ppmap A (pfiber f) : by exact pequiv_ppcompose_left !pfiber.sigma_char'⁻¹ᵉ*
|
||||
|
||||
|
||||
definition pfiber_ppcompose_left_dep {B C : A → Type*} (f : Πa, B a →* C a) :
|
||||
definition pfiber_ppi_compose_left {B C : A → Type*} (f : Πa, B a →* C a) :
|
||||
pfiber (ppi_compose_left f) ≃* Π*(a : A), pfiber (f a) :=
|
||||
calc
|
||||
pfiber (ppi_compose_left f) ≃*
|
||||
|
|
@ -485,6 +485,123 @@ namespace pointed
|
|||
-- definition pppi_ppmap {A C : Type*} {B : A → Type*} :
|
||||
-- ppmap (/- dependent smash of B -/) C ≃* Π*(a : A), ppmap (B a) C :=
|
||||
|
||||
-- TODO: homotopy_of_eq and apd10 should be the same
|
||||
-- TODO: there is also apd10_eq_of_homotopy in both pi and eq(?)
|
||||
|
||||
/- stuff about the pointed type of unpointed maps (dependent and non-dependent) -/
|
||||
definition sigma_pumap {A : Type} (B : A → Type) (C : Type*) :
|
||||
((Σa, B a) →ᵘ* C) ≃* Πᵘ*a, B a →ᵘ* C :=
|
||||
pequiv_of_equiv (equiv_sigma_rec _)⁻¹ᵉ idp
|
||||
|
||||
definition loop_pupi [constructor] {A : Type} (B : A → Type*) : Ω (Πᵘ*a, B a) ≃* Πᵘ*a, Ω (B a) :=
|
||||
pequiv_of_equiv eq_equiv_homotopy idp
|
||||
|
||||
definition phomotopy_mk_pupi [constructor] {A : Type*} {B : Type} {C : B → Type*}
|
||||
{f g : A →* (Πᵘ*b, C b)} (p : f ~2 g)
|
||||
(q : p pt ⬝hty apd10 (respect_pt g) ~ apd10 (respect_pt f)) : f ~* g :=
|
||||
begin
|
||||
apply phomotopy.mk (λa, eq_of_homotopy (p a)),
|
||||
apply eq_of_fn_eq_fn eq_equiv_homotopy,
|
||||
apply eq_of_homotopy, intro b,
|
||||
refine !apd10_con ⬝ _,
|
||||
refine whisker_right _ !pi.apd10_eq_of_homotopy ⬝ q b
|
||||
end
|
||||
|
||||
definition pupi_functor [constructor] {A A' : Type} {B : A → Type*} {B' : A' → Type*}
|
||||
(f : A' → A) (g : Πa, B (f a) →* B' a) : (Πᵘ*a, B a) →* (Πᵘ*a', B' a') :=
|
||||
pmap.mk (λh a, g a (h (f a))) (eq_of_homotopy (λa, respect_pt (g a)))
|
||||
|
||||
definition pupi_functor_compose {A A' A'' : Type}
|
||||
{B : A → Type*} {B' : A' → Type*} {B'' : A'' → Type*} (f : A'' → A') (f' : A' → A)
|
||||
(g' : Πa, B' (f a) →* B'' a) (g : Πa, B (f' a) →* B' a) :
|
||||
pupi_functor (f' ∘ f) (λa, g' a ∘* g (f a)) ~* pupi_functor f g' ∘* pupi_functor f' g :=
|
||||
begin
|
||||
fapply phomotopy_mk_pupi,
|
||||
{ intro h a, reflexivity },
|
||||
{ intro a, refine !idp_con ⬝ _, refine !apd10_con ⬝ _ ⬝ !pi.apd10_eq_of_homotopy⁻¹, esimp,
|
||||
refine (!apd10_prepostcompose ⬝ ap02 (g' a) !pi.apd10_eq_of_homotopy) ◾
|
||||
!pi.apd10_eq_of_homotopy }
|
||||
end
|
||||
|
||||
definition pupi_functor_pid (A : Type) (B : A → Type*) :
|
||||
pupi_functor id (λa, pid (B a)) ~* pid (Πᵘ*a, B a) :=
|
||||
begin
|
||||
fapply phomotopy_mk_pupi,
|
||||
{ intro h a, reflexivity },
|
||||
{ intro a, refine !idp_con ⬝ !pi.apd10_eq_of_homotopy⁻¹ }
|
||||
end
|
||||
|
||||
definition pupi_functor_phomotopy {A A' : Type} {B : A → Type*} {B' : A' → Type*}
|
||||
{f f' : A' → A} {g : Πa, B (f a) →* B' a} {g' : Πa, B (f' a) →* B' a}
|
||||
(p : f ~ f') (q : Πa, g a ~* g' a ∘* ptransport B (p a)) :
|
||||
pupi_functor f g ~* pupi_functor f' g' :=
|
||||
begin
|
||||
fapply phomotopy_mk_pupi,
|
||||
{ intro h a, exact q a (h (f a)) ⬝ ap (g' a) (apdt h (p a)) },
|
||||
{ intro a, esimp,
|
||||
exact whisker_left _ !pi.apd10_eq_of_homotopy ⬝ !con.assoc ⬝
|
||||
to_homotopy_pt (q a) ⬝ !pi.apd10_eq_of_homotopy⁻¹ }
|
||||
end
|
||||
|
||||
definition pupi_pequiv [constructor] {A A' : Type} {B : A → Type*} {B' : A' → Type*}
|
||||
(e : A' ≃ A) (f : Πa, B (e a) ≃* B' a) :
|
||||
(Πᵘ*a, B a) ≃* (Πᵘ*a', B' a') :=
|
||||
pequiv.MK (pupi_functor e f)
|
||||
(pupi_functor e⁻¹ᵉ (λa, ptransport B (right_inv e a) ∘* (f (e⁻¹ᵉ a))⁻¹ᵉ*))
|
||||
abstract begin
|
||||
refine !pupi_functor_compose⁻¹* ⬝* pupi_functor_phomotopy (to_right_inv e) _ ⬝*
|
||||
!pupi_functor_pid,
|
||||
intro a, exact !pinv_pcompose_cancel_right ⬝* !pid_pcompose⁻¹*
|
||||
end end
|
||||
abstract begin
|
||||
refine !pupi_functor_compose⁻¹* ⬝* pupi_functor_phomotopy (to_left_inv e) _ ⬝*
|
||||
!pupi_functor_pid,
|
||||
intro a, refine !passoc⁻¹* ⬝* pinv_right_phomotopy_of_phomotopy _ ⬝* !pid_pcompose⁻¹*,
|
||||
refine pwhisker_left _ _ ⬝* !ptransport_natural,
|
||||
exact ptransport_change_eq _ (adj e a) ⬝* ptransport_ap B e (to_left_inv e a)
|
||||
end end
|
||||
|
||||
definition pupi_pequiv_right [constructor] {A : Type} {B B' : A → Type*} (f : Πa, B a ≃* B' a) :
|
||||
(Πᵘ*a, B a) ≃* (Πᵘ*a, B' a) :=
|
||||
pupi_pequiv erfl f
|
||||
|
||||
definition loop_pumap [constructor] (A : Type) (B : Type*) : Ω (A →ᵘ* B) ≃* A →ᵘ* Ω B :=
|
||||
loop_pupi (λa, B)
|
||||
|
||||
definition phomotopy_mk_pumap [constructor] {A C : Type*} {B : Type} {f g : A →* (B →ᵘ* C)}
|
||||
(p : f ~2 g) (q : p pt ⬝hty apd10 (respect_pt g) ~ apd10 (respect_pt f))
|
||||
: f ~* g :=
|
||||
phomotopy_mk_pupi p q
|
||||
|
||||
definition pumap_functor [constructor] {A A' : Type} {B B' : Type*} (f : A' → A) (g : B →* B') :
|
||||
(A →ᵘ* B) →* (A' →ᵘ* B') :=
|
||||
pupi_functor f (λa, g)
|
||||
|
||||
definition pumap_functor_compose {A A' A'' : Type} {B B' B'' : Type*}
|
||||
(f : A'' → A') (f' : A' → A) (g' : B' →* B'') (g : B →* B') :
|
||||
pumap_functor (f' ∘ f) (g' ∘* g) ~* pumap_functor f g' ∘* pumap_functor f' g :=
|
||||
pupi_functor_compose f f' (λa, g') (λa, g)
|
||||
|
||||
definition pumap_functor_pid (A : Type) (B : Type*) :
|
||||
pumap_functor id (pid B) ~* pid (A →ᵘ* B) :=
|
||||
pupi_functor_pid A (λa, B)
|
||||
|
||||
definition pumap_functor_phomotopy {A A' : Type} {B B' : Type*} {f f' : A' → A} {g g' : B →* B'}
|
||||
(p : f ~ f') (q : g ~* g') : pumap_functor f g ~* pumap_functor f' g' :=
|
||||
pupi_functor_phomotopy p (λa, q ⬝* !pcompose_pid⁻¹* ⬝* pwhisker_left _ !ptransport_constant⁻¹*)
|
||||
|
||||
definition pumap_pequiv [constructor] {A A' : Type} {B B' : Type*} (e : A ≃ A') (f : B ≃* B') :
|
||||
(A →ᵘ* B) ≃* (A' →ᵘ* B') :=
|
||||
pupi_pequiv e⁻¹ᵉ (λa, f)
|
||||
|
||||
definition pumap_pequiv_right [constructor] (A : Type) {B B' : Type*} (f : B ≃* B') :
|
||||
(A →ᵘ* B) ≃* (A →ᵘ* B') :=
|
||||
pumap_pequiv erfl f
|
||||
|
||||
definition pumap_pequiv_left [constructor] {A A' : Type} (B : Type*) (f : A ≃ A') :
|
||||
(A →ᵘ* B) ≃* (A' →ᵘ* B) :=
|
||||
pumap_pequiv f pequiv.rfl
|
||||
|
||||
end pointed open pointed
|
||||
|
||||
open is_trunc is_conn
|
||||
|
|
|
|||
Loading…
Reference in a new issue