add strunc file for truncatedness/truncations of spectra
This commit is contained in:
Floris van Doorn
parent
635b10821f
commit
d814c472ab
5 changed files with 177 additions and 48 deletions
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@ -485,13 +485,14 @@ namespace pointed
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open prod prod.ops fiber
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parameters {A : ℤ → Type*[1]} (f : Π(n : ℤ), A n →* A (n - 1)) [Hf : Πn, is_conn_fun 1 (f n)]
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include Hf
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definition I [constructor] : Set := trunctype.mk (ℤ × ℤ) !is_trunc_prod
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protected definition I [constructor] : Set := trunctype.mk (gℤ ×g gℤ) !is_trunc_prod
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local abbreviation I := @pointed.I
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definition D_sequence : graded_module rℤ I :=
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λv, LeftModule_int_of_AbGroup (πc[v.2] (A (v.1)))
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-- definition D_sequence : graded_module rℤ I :=
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-- λv, LeftModule_int_of_AbGroup (πc[v.2] (A (v.1)))
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definition E_sequence : graded_module rℤ I :=
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λv, LeftModule_int_of_AbGroup (πc[v.2] (pconntype.mk (pfiber (f (v.1))) !Hf pt))
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-- definition E_sequence : graded_module rℤ I :=
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-- λv, LeftModule_int_of_AbGroup (πc[v.2] (pconntype.mk (pfiber (f (v.1))) !Hf pt))
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/- first need LES of these connected homotopy groups -/
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@ -509,7 +510,8 @@ namespace spectrum
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parameters {A : ℤ → spectrum} (f : Π(s : ℤ), A s →ₛ A (s - 1))
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definition I [constructor] : Set := trunctype.mk (gℤ ×g gℤ) !is_trunc_prod
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protected definition I [constructor] : Set := trunctype.mk (gℤ ×g gℤ) !is_trunc_prod
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local abbreviation I := @spectrum.I
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definition D_sequence : graded_module rℤ I :=
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λv, LeftModule_int_of_AbGroup (πₛ[v.1] (A (v.2)))
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@ -151,13 +151,14 @@ namespace spectrum
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-- I guess this manual eta-expansion is necessary because structures lack definitional eta?
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:= phomotopy.mk (glue_square f n) (to_homotopy_pt (glue_square f n))
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definition sid {N : succ_str} (E : gen_prespectrum N) : E →ₛ E :=
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definition sid [constructor] [refl] {N : succ_str} (E : gen_prespectrum N) : E →ₛ E :=
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smap.mk (λn, pid (E n))
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(λn, calc glue E n ∘* pid (E n) ~* glue E n : pcompose_pid
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... ~* pid (Ω(E (S n))) ∘* glue E n : pid_pcompose
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... ~* Ω→(pid (E (S n))) ∘* glue E n : pwhisker_right (glue E n) ap1_pid⁻¹*)
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definition scompose {N : succ_str} {X Y Z : gen_prespectrum N} (g : Y →ₛ Z) (f : X →ₛ Y) : X →ₛ Z :=
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definition scompose [trans] {N : succ_str} {X Y Z : gen_prespectrum N}
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(g : Y →ₛ Z) (f : X →ₛ Y) : X →ₛ Z :=
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smap.mk (λn, g n ∘* f n)
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(λn, calc glue Z n ∘* to_fun g n ∘* to_fun f n
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~* (glue Z n ∘* to_fun g n) ∘* to_fun f n : passoc
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52
homotopy/strunc.hlean
Normal file
52
homotopy/strunc.hlean
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@ -0,0 +1,52 @@
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import .spectrum .EM
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open int trunc eq is_trunc lift unit pointed equiv is_equiv algebra EM
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namespace spectrum
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definition trunc_int.{u} (k : ℤ) (X : Type.{u}) : Type.{u} :=
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begin
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induction k with k k, exact trunc k X,
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cases k with k, exact trunc -1 X,
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exact lift unit
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end
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definition ptrunc_int.{u} (k : ℤ) (X : pType.{u}) : pType.{u} :=
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begin
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induction k with k k, exact ptrunc k X,
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exact plift punit
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end
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definition ptrunc_int_pequiv_ptrunc_int (k : ℤ) {X Y : Type*} (e : X ≃* Y) :
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ptrunc_int k X ≃* ptrunc_int k Y :=
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begin
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induction k with k k,
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exact ptrunc_pequiv_ptrunc k e,
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exact !pequiv_plift⁻¹ᵉ* ⬝e* !pequiv_plift
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end
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definition ptrunc_int_change_int {k l : ℤ} (X : Type*) (p : k = l) :
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ptrunc_int k X ≃* ptrunc_int l X :=
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pequiv_ap (λn, ptrunc_int n X) p
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definition loop_ptrunc_int_pequiv (k : ℤ) (X : Type*) :
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Ω (ptrunc_int (k+1) X) ≃* ptrunc_int k (Ω X) :=
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begin
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induction k with k k,
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exact loop_ptrunc_pequiv k X,
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cases k with k,
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change Ω (ptrunc 0 X) ≃* plift punit,
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exact !loop_pequiv_punit_of_is_set ⬝e* !pequiv_plift,
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exact loop_pequiv_loop !pequiv_plift⁻¹ᵉ* ⬝e* !loop_punit ⬝e* !pequiv_plift
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end
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definition strunc_int [constructor] (k : ℤ) (E : spectrum) : spectrum :=
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spectrum.MK (λ(n : ℤ), ptrunc_int (k + n) (E n))
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(λ(n : ℤ), ptrunc_int_pequiv_ptrunc_int (k + n) (equiv_glue E n) ⬝e*
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(loop_ptrunc_int_pequiv (k + n) (E (n+1)))⁻¹ᵉ* ⬝e*
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loop_pequiv_loop (ptrunc_int_change_int _ (add.assoc k n 1)))
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definition strunc_int_change_int [constructor] {k l : ℤ} (E : spectrum) (p : k = l) :
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strunc_int k E →ₛ strunc_int l E :=
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begin induction p, reflexivity end
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end spectrum
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@ -182,4 +182,25 @@ namespace pointed
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sorry
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end psquare
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definition ap1_pequiv_ap {A : Type} (B : A → Type*) {a a' : A} (p : a = a') :
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Ω→ (pequiv_ap B p) ~* pequiv_ap (Ω ∘ B) p :=
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begin induction p, apply ap1_pid end
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definition pequiv_ap_natural {A : Type} (B C : A → Type*) {a a' : A} (p : a = a')
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(f : Πa, B a →* C a) :
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psquare (pequiv_ap B p) (pequiv_ap C p) (f a) (f a') :=
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begin induction p, exact phrfl end
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definition pequiv_ap_natural2 {A : Type} (B C : A → Type*) {a a' : A} (p : a = a')
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(f : Πa, B a →* C a) :
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psquare (pequiv_ap B p) (pequiv_ap C p) (f a) (f a') :=
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begin induction p, exact phrfl end
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definition loop_pequiv_punit_of_is_set (X : Type*) [is_set X] : Ω X ≃* punit :=
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pequiv_punit_of_is_contr _ (is_contr_of_inhabited_prop pt)
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definition loop_punit : Ω punit ≃* punit :=
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loop_pequiv_punit_of_is_set punit
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end pointed
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133
pointed_pi.hlean
133
pointed_pi.hlean
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@ -63,21 +63,21 @@ namespace pointed
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variable (k)
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protected definition ppi_homotopy.refl : k ~~* k :=
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sorry
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ppi_homotopy.mk homotopy.rfl !idp_con
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variable {k}
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protected definition ppi_homotopy.rfl [refl] : k ~~* k :=
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ppi_homotopy.refl k
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protected definition ppi_homotopy.symm [symm] (p : k ~~* l) : l ~~* k :=
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sorry
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ppi_homotopy.mk p⁻¹ʰᵗʸ (inv_con_eq_of_eq_con (ppi_to_homotopy_pt p)⁻¹)
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protected definition ppi_homotopy.trans [trans] (p : k ~~* l) (q : l ~~* m) : k ~~* m :=
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sorry
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ppi_homotopy.mk (λa, p a ⬝ q a) (!con.assoc ⬝ whisker_left (p pt) (ppi_to_homotopy_pt q) ⬝ ppi_to_homotopy_pt p)
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infix ` ⬝*' `:75 := ppi_homotopy.trans
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postfix `⁻¹*'`:(max+1) := ppi_homotopy.symm
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definition ppi_equiv_pmap (A B : Type*) : (Π*(a : A), B) ≃ (A →* B) :=
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definition ppi_equiv_pmap [constructor] (A B : Type*) : (Π*(a : A), B) ≃ (A →* B) :=
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begin
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fapply equiv.MK,
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{ intro k, induction k with k p, exact pmap.mk k p },
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@ -86,9 +86,19 @@ namespace pointed
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{ intro k, induction k with k p, reflexivity }
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end
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definition pppi_pequiv_ppmap (A B : Type*) : (Π*(a : A), B) ≃* ppmap A B :=
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definition pppi_pequiv_ppmap [constructor] (A B : Type*) : (Π*(a : A), B) ≃* ppmap A B :=
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pequiv_of_equiv (ppi_equiv_pmap A B) idp
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protected definition ppi_gen.sigma_char [constructor] {A : Type*} (B : A → Type) (b₀ : B pt) :
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ppi_gen B b₀ ≃ Σ(k : Πa, B a), k pt = b₀ :=
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begin
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fapply equiv.MK: intro x,
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{ constructor, exact ppi_gen.resp_pt x },
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{ induction x, constructor, assumption },
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{ induction x, reflexivity },
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{ induction x, reflexivity }
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end
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definition ppi.sigma_char [constructor] {A : Type*} (B : A → Type*)
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: (Π*(a : A), B a) ≃ Σ(k : (Π (a : A), B a)), k pt = pt :=
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begin
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@ -99,15 +109,6 @@ namespace pointed
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all_goals reflexivity
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end
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protected definition ppi_gen.sigma_char [constructor] {A : Type*} (B : A → Type) (b₀ : B pt) :
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ppi_gen B b₀ ≃ Σ(k : Πa, B a), k pt = b₀ :=
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begin
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fapply equiv.MK: intro x,
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{ constructor, exact ppi_gen.resp_pt x },
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{ induction x, constructor, assumption },
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{ induction x, reflexivity },
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{ induction x, reflexivity }
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end
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variables (k l)
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@ -206,7 +207,7 @@ namespace pointed
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definition pmap_compose_ppi_const_left (f : Π*(a : A), P a) :
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pmap_compose_ppi (λa, pconst (P a) (Q a)) f ~~* ppi_const Q :=
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sorry
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ppi_homotopy.mk homotopy.rfl !ap_constant⁻¹
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definition ppi_compose_left [constructor] (g : Π(a : A), ppmap (P a) (Q a)) :
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(Π*(a : A), P a) →* Π*(a : A), Q a :=
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@ -214,16 +215,19 @@ namespace pointed
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definition pmap_compose_ppi_phomotopy_left [constructor] {g g' : Π(a : A), ppmap (P a) (Q a)}
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(f : Π*(a : A), P a) (p : Πa, g a ~* g' a) : pmap_compose_ppi g f ~~* pmap_compose_ppi g' f :=
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sorry
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ppi_homotopy.mk (λa, p a (f a))
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abstract !con.assoc⁻¹ ⬝ whisker_right _ !ap_con_eq_con_ap⁻¹ ⬝ !con.assoc ⬝
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whisker_left _ (to_homotopy_pt (p pt)) end
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definition pmap_compose_ppi_pid_left [constructor]
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(f : Π*(a : A), P a) : pmap_compose_ppi (λa, pid (P a)) f ~~* f :=
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sorry
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ppi_homotopy.mk homotopy.rfl idp
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definition pmap_compose_pmap_compose_ppi [constructor] (h : Π(a : A), ppmap (Q a) (R a))
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definition pmap_compose_ppi_pcompose [constructor] (h : Π(a : A), ppmap (Q a) (R a))
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(g : Π(a : A), ppmap (P a) (Q a)) :
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pmap_compose_ppi h (pmap_compose_ppi g f) ~~* pmap_compose_ppi (λa, h a ∘* g a) f :=
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sorry
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pmap_compose_ppi (λa, h a ∘* g a) f ~~* pmap_compose_ppi h (pmap_compose_ppi g f) :=
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ppi_homotopy.mk homotopy.rfl
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abstract !idp_con ⬝ whisker_right _ (!ap_con ⬝ whisker_right _ !ap_compose'⁻¹) ⬝ !con.assoc end
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definition ppi_pequiv_right [constructor] (g : Π(a : A), P a ≃* Q a) :
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(Π*(a : A), P a) ≃* Π*(a : A), Q a :=
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@ -231,11 +235,11 @@ namespace pointed
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apply pequiv_of_pmap (ppi_compose_left g),
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apply adjointify _ (ppi_compose_left (λa, (g a)⁻¹ᵉ*)),
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{ intro f, apply ppi_eq,
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refine !pmap_compose_pmap_compose_ppi ⬝*' _,
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refine !pmap_compose_ppi_pcompose⁻¹*' ⬝*' _,
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refine pmap_compose_ppi_phomotopy_left _ (λa, !pright_inv) ⬝*' _,
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apply pmap_compose_ppi_pid_left },
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{ intro f, apply ppi_eq,
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refine !pmap_compose_pmap_compose_ppi ⬝*' _,
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refine !pmap_compose_ppi_pcompose⁻¹*' ⬝*' _,
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refine pmap_compose_ppi_phomotopy_left _ (λa, !pleft_inv) ⬝*' _,
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apply pmap_compose_ppi_pid_left }
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end
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@ -255,28 +259,41 @@ namespace pointed
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pequiv_of_equiv (fiber.sigma_char f pt) idp
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/- the pointed type of unpointed (nondependent) maps -/
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definition parrow [constructor] (A : Type) (B : Type*) : Type* :=
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definition upmap [constructor] (A : Type) (B : Type*) : Type* :=
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pointed.MK (A → B) (const A pt)
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/- the pointed type of unpointed dependent maps -/
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definition p_pi [constructor] {A : Type} (B : A → Type*) : Type* :=
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definition uppi [constructor] {A : Type} (B : A → Type*) : Type* :=
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pointed.MK (Πa, B a) (λa, pt)
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definition ppmap.sigma_char (A B : Type*) :
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ppmap A B ≃* @psigma_gen (parrow A B) (λf, f pt = pt) idp :=
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notation `Πᵘ*` binders `, ` r:(scoped P, uppi P) := r
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infix ` →ᵘ* `:30 := upmap
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definition ppmap.sigma_char [constructor] (A B : Type*) :
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ppmap A B ≃* @psigma_gen (A →ᵘ* B) (λf, f pt = pt) idp :=
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pequiv_of_equiv pmap.sigma_char idp
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definition pppi.sigma_char {A : Type*} {B : A → Type*} :
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(Π*(a : A), B a) ≃* @psigma_gen (p_pi B) (λf, f pt = pt) idp :=
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definition pppi.sigma_char [constructor] {A : Type*} (B : A → Type*) :
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(Π*(a : A), B a) ≃* @psigma_gen (Πᵘ*a, B a) (λf, f pt = pt) idp :=
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proof pequiv_of_equiv !ppi.sigma_char idp qed
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definition psigma_gen_pequiv_psigma_gen_right {A : Type*} {B B' : A → Type}
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definition psigma_gen_pequiv_psigma_gen [constructor] {A A' : Type*} {B : A → Type}
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{B' : A' → Type} {b : B pt} {b' : B' pt} (f : A ≃* A')
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(g : Πa, B a ≃ B' (f a)) (p : g pt b =[respect_pt f] b') : psigma_gen B b ≃* psigma_gen B' b' :=
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pequiv_of_equiv (sigma_equiv_sigma f g) (sigma_eq (respect_pt f) p)
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definition psigma_gen_pequiv_psigma_gen_left [constructor] {A A' : Type*} {B : A' → Type}
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{b : B pt} (f : A ≃* A') {b' : B (f pt)} (q : b' =[respect_pt f] b) :
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psigma_gen (B ∘ f) b' ≃* psigma_gen B b :=
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psigma_gen_pequiv_psigma_gen f (λa, erfl) q
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definition psigma_gen_pequiv_psigma_gen_right [constructor] {A : Type*} {B B' : A → Type}
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{b : B pt} {b' : B' pt} (f : Πa, B a ≃ B' a) (p : f pt b = b') :
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psigma_gen B b ≃* psigma_gen B' b' :=
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pequiv_of_equiv (sigma_equiv_sigma_right f) (ap (dpair pt) p)
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psigma_gen_pequiv_psigma_gen pequiv.rfl f (pathover_idp_of_eq p)
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definition psigma_gen_pequiv_psigma_gen_basepoint {A : Type*} {B : A → Type} {b b' : B pt}
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(p : b = b') : psigma_gen B b ≃* psigma_gen B b' :=
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definition psigma_gen_pequiv_psigma_gen_basepoint [constructor] {A : Type*} {B : A → Type}
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{b b' : B pt} (p : b = b') : psigma_gen B b ≃* psigma_gen B b' :=
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psigma_gen_pequiv_psigma_gen_right (λa, erfl) p
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definition ppi_gen_functor_right [constructor] {A : Type*} {B B' : A → Type}
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@ -336,23 +353,59 @@ namespace pointed
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(p : b = b') : ppi_gen B b ≃ ppi_gen B b' :=
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ppi_gen_equiv_ppi_gen_right (λa, erfl) p
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definition ppi_psigma {A : Type*} {B : A → Type*} (C : Πa, B a → Type) (c : Πa, C a pt) :
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(Π*(a : A), (psigma_gen (C a) (c a))) ≃*
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open sigma.ops
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definition psigma_gen_pi_pequiv_uppi_psigma_gen [constructor] {A : Type*} {B : A → Type*}
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(C : Πa, B a → Type) (c : Πa, C a pt) :
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@psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c ≃* Πᵘ*a, psigma_gen (C a) (c a) :=
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pequiv_of_equiv sigma_pi_equiv_pi_sigma idp
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definition uppi_psigma_gen_pequiv_psigma_gen_pi [constructor] {A : Type*} {B : A → Type*}
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(C : Πa, B a → Type) (c : Πa, C a pt) :
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(Πᵘ*a, psigma_gen (C a) (c a)) ≃* @psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c :=
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pequiv_of_equiv sigma_pi_equiv_pi_sigma⁻¹ᵉ idp
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definition psigma_gen_assoc [constructor] {A : Type*} {B : A → Type} (C : Πa, B a → Type)
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(b₀ : B pt) (c₀ : C pt b₀) :
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psigma_gen (λa, Σb, C a b) ⟨b₀, c₀⟩ ≃* @psigma_gen (psigma_gen B b₀) (λv, C v.1 v.2) c₀ :=
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pequiv_of_equiv !sigma_assoc_equiv idp
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definition psigma_gen_swap [constructor] {A : Type*} {B B' : A → Type}
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(C : Π⦃a⦄, B a → B' a → Type) (b₀ : B pt) (b₀' : B' pt) (c₀ : C b₀ b₀') :
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@psigma_gen (psigma_gen B b₀ ) (λv, Σb', C v.2 b') ⟨b₀', c₀⟩ ≃*
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@psigma_gen (psigma_gen B' b₀') (λv, Σb , C b v.2) ⟨b₀ , c₀⟩ :=
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!psigma_gen_assoc⁻¹ᵉ* ⬝e* psigma_gen_pequiv_psigma_gen_right (λa, !sigma_comm_equiv) idp ⬝e*
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!psigma_gen_assoc
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definition ppi_psigma.{u v w} {A : pType.{u}} {B : A → pType.{v}} (C : Πa, B a → Type.{w})
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(c : Πa, C a pt) : (Π*(a : A), (psigma_gen (C a) (c a))) ≃*
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psigma_gen (λ(f : Π*(a : A), B a), ppi_gen (λa, C a (f a))
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(transport (C pt) (ppi.resp_pt f)⁻¹ (c pt)))
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(ppi_const _) :=
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(transport (C pt) (ppi.resp_pt f)⁻¹ (c pt))) (ppi_const _) :=
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proof
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calc (Π*(a : A), psigma_gen (C a) (c a))
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≃* @psigma_gen (p_pi (λa, psigma_gen (C a) (c a))) (λf, f pt = pt) idp : pppi.sigma_char
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≃* @psigma_gen (Πᵘ*a, psigma_gen (C a) (c a)) (λf, f pt = pt) idp : pppi.sigma_char
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... ≃* @psigma_gen (@psigma_gen (Πᵘ*a, B a) (λf, Πa, C a (f a)) c)
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(λv, Σ(p : v.1 pt = pt), v.2 pt =[p] c pt) ⟨idp, idpo⟩ :
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by exact psigma_gen_pequiv_psigma_gen (uppi_psigma_gen_pequiv_psigma_gen_pi C c)
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(λf, sigma_eq_equiv _ _) idpo
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... ≃* @psigma_gen (@psigma_gen (Πᵘ*a, B a) (λf, f pt = pt) idp)
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(λv, Σ(g : Πa, C a (v.1 a)), g pt =[v.2] c pt) ⟨c, idpo⟩ :
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by apply psigma_gen_swap
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... ≃* psigma_gen (λ(f : Π*(a : A), B a), ppi_gen (λa, C a (f a))
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(transport (C pt) (ppi.resp_pt f)⁻¹ (c pt)))
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(ppi_const _) : sorry
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(ppi_const _) :
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by exact (psigma_gen_pequiv_psigma_gen (pppi.sigma_char B)
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(λf, !ppi_gen.sigma_char ⬝e sigma_equiv_sigma_right (λg, !pathover_equiv_eq_tr⁻¹ᵉ))
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idpo)⁻¹ᵉ*
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qed
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definition pmap_psigma {A B : Type*} (C : B → Type) (c : C pt) :
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ppmap A (psigma_gen C c) ≃*
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psigma_gen (λ(f : ppmap A B), ppi_gen (C ∘ f) (transport C (respect_pt f)⁻¹ c))
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(ppi_const _) :=
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!pppi_pequiv_ppmap⁻¹ᵉ* ⬝e* !ppi_psigma ⬝e* sorry
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!pppi_pequiv_ppmap⁻¹ᵉ* ⬝e* !ppi_psigma ⬝e*
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sorry
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-- psigma_gen_pequiv_psigma_gen (pppi_pequiv_ppmap A B) (λf, begin esimp, exact ppi_gen_equiv_ppi_gen_right (λa, _) _ end) _
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definition pfiber_ppcompose_left (f : B →* C) :
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pfiber (@ppcompose_left A B C f) ≃* ppmap A (pfiber f) :=
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|
|
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Loading…
Reference in a new issue