290 lines
11 KiB
Text
290 lines
11 KiB
Text
/-
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Copyright (c) 2017 Floris van Doorn and Ulrik Buchholtz. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Ulrik Buchholtz
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Truncatedness and truncation of spectra
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-/
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import .basic
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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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/- interactions of ptrunc / trunc with maxm2 -/
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definition ptrunc_maxm2_change_int {k l : ℤ} (p : k = l) (X : Type*)
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: ptrunc (maxm2 k) X ≃* ptrunc (maxm2 l) X :=
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ptrunc_change_index (ap maxm2 p) X
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definition is_trunc_maxm2_change_int {k l : ℤ} (X : pType) (p : k = l)
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: is_trunc (maxm2 k) X → is_trunc (maxm2 l) X :=
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by induction p; exact id
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definition is_trunc_maxm2_loop {k : ℤ} {A : Type*} (H : is_trunc (maxm2 (k+1)) A) :
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is_trunc (maxm2 k) (Ω A) :=
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begin
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induction k with k k,
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apply is_trunc_loop, exact H,
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apply is_contr_loop,
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cases k with k,
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{ exact H },
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{ apply is_trunc_succ, apply is_trunc_succ, exact H }
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end
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definition loop_ptrunc_maxm2_pequiv {k : ℤ} {l : ℕ₋₂} (p : maxm2 (k+1) = l) (X : Type*) :
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Ω (ptrunc l X) ≃* ptrunc (maxm2 k) (Ω X) :=
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begin
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induction p,
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induction k with k k,
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{ exact loop_ptrunc_pequiv k X },
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{ refine pequiv_of_is_contr _ _ _ !is_trunc_trunc,
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apply is_contr_loop,
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cases k with k,
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{ change is_set (trunc 0 X), apply _ },
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{ change is_set (trunc -2 X), apply _ }}
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end
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definition loop_ptrunc_maxm2_pequiv_ptrunc_elim' {k : ℤ} {l : ℕ₋₂} (p : maxm2 (k+1) = l)
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{A B : Type*} (f : A →* B) {H : is_trunc l B} :
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Ω→ (ptrunc.elim l f) ∘* (loop_ptrunc_maxm2_pequiv p A)⁻¹ᵉ* ~*
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@ptrunc.elim (maxm2 k) _ _ (is_trunc_maxm2_loop (is_trunc_of_eq p⁻¹ H)) (Ω→ f) :=
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begin
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induction p, induction k with k k,
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{ refine pwhisker_right _ (ap1_phomotopy _) ⬝* @(ap1_ptrunc_elim k f) H,
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apply ptrunc_elim_phomotopy2, reflexivity },
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{ apply phomotopy_of_is_contr_cod, exact is_trunc_maxm2_loop H }
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end
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definition loop_ptrunc_maxm2_pequiv_ptrunc_elim {k : ℤ} {l : ℕ₋₂} (p : maxm2 (k+1) = l)
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{A B : Type*} (f : A →* B) {H1 : is_trunc ((maxm2 k).+1) B } {H2 : is_trunc l B} :
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Ω→ (ptrunc.elim l f) ∘* (loop_ptrunc_maxm2_pequiv p A)⁻¹ᵉ* ~* ptrunc.elim (maxm2 k) (Ω→ f) :=
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begin
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induction p, induction k with k k: esimp at H1,
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{ refine pwhisker_right _ (ap1_phomotopy _) ⬝* ap1_ptrunc_elim k f,
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apply ptrunc_elim_phomotopy2, reflexivity },
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{ apply phomotopy_of_is_contr_cod }
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end
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definition loop_ptrunc_maxm2_pequiv_ptr {k : ℤ} {l : ℕ₋₂} (p : maxm2 (k+1) = l) (A : Type*) :
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Ω→ (ptr l A) ~* (loop_ptrunc_maxm2_pequiv p A)⁻¹ᵉ* ∘* ptr (maxm2 k) (Ω A) :=
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begin
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induction p, induction k with k k,
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{ exact ap1_ptr k A },
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{ apply phomotopy_pinv_left_of_phomotopy, apply phomotopy_of_is_contr_cod, apply is_trunc_trunc }
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end
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definition is_trunc_of_is_trunc_maxm2 (k : ℤ) (X : Type)
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: is_trunc (maxm2 k) X → is_trunc (max0 k) X :=
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λ H, @is_trunc_of_le X _ _ (maxm2_le_maxm0 k) H
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definition ptrunc_maxm2_pred {n m : ℤ} (A : Type*) (p : n - 1 = m) :
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ptrunc (maxm2 m) A ≃* ptrunc (trunc_index.pred (maxm2 n)) A :=
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begin
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cases n with n, cases n with n, apply pequiv_of_is_contr,
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induction p, apply is_trunc_trunc,
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apply is_contr_ptrunc_minus_one,
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exact ptrunc_change_index (ap maxm2 (p⁻¹ ⬝ !add_sub_cancel)) A,
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exact ptrunc_change_index (ap maxm2 p⁻¹) A
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end
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definition ptrunc_maxm2_pred_nat {n : ℕ} {m l : ℤ} (A : Type*)
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(p : nat.succ n = l) (q : pred l = m) (r : maxm2 m = trunc_index.pred (maxm2 (nat.succ n))) :
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@ptrunc_maxm2_pred (nat.succ n) m A (ap pred p ⬝ q) ~* ptrunc_change_index r A :=
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begin
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have ap maxm2 ((ap pred p ⬝ q)⁻¹ ⬝ add_sub_cancel n 1) = r, from !is_set.elim,
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induction this, reflexivity
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end
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/- truncatedness of spectra -/
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definition is_strunc [reducible] (k : ℤ) (E : spectrum) : Type :=
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Π (n : ℤ), is_trunc (maxm2 (k + n)) (E n)
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definition is_strunc_change_int {k l : ℤ} (E : spectrum) (p : k = l)
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: is_strunc k E → is_strunc l E :=
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begin induction p, exact id end
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definition is_strunc_of_le {k l : ℤ} (E : spectrum) (H : k ≤ l)
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: is_strunc k E → is_strunc l E :=
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begin
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intro T, intro n, exact is_trunc_of_le (E n)
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(maxm2_monotone (algebra.add_le_add_right H n))
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end
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definition is_strunc_pequiv_closed {k : ℤ} {E F : spectrum} (H : Πn, E n ≃* F n)
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(H2 : is_strunc k E) : is_strunc k F :=
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λn, is_trunc_equiv_closed (maxm2 (k + n)) (H n)
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definition is_strunc_sunit (n : ℤ) : is_strunc n sunit :=
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begin
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intro k, apply is_trunc_lift, apply is_trunc_unit
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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) := 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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begin
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intro n, induction n with n n,
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{ -- case ≥ 0
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apply is_trunc_maxm2_change_int (EM G n) (zero_add n)⁻¹,
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apply is_trunc_EM },
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{ change is_contr (EM_spectrum G (-[1+n])),
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induction n with n IH,
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{ -- case = -1
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apply is_contr_loop, exact is_trunc_EM G 0 },
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{ -- case < -1
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apply is_trunc_loop, apply is_trunc_succ, exact IH }}
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end
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definition trivial_shomotopy_group_of_is_strunc (E : spectrum)
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{n k : ℤ} (K : is_strunc n E) (H : n < k)
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: is_contr (πₛ[k] E) :=
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let m := n + (2 - k) in
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have I : m < 2, from
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calc
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m = (2 - k) + n : int.add_comm n (2 - k)
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... < (2 - k) + k : add_lt_add_left H (2 - k)
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... = 2 : sub_add_cancel 2 k,
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@trivial_homotopy_group_of_is_trunc (E (2 - k)) (max0 m) 2
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(is_trunc_of_is_trunc_maxm2 m (E (2 - k)) (K (2 - k)))
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(nat.succ_le_succ (max0_le_of_le (le_sub_one_of_lt I)))
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/- truncation of spectra -/
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definition strunc [constructor] (k : ℤ) (E : spectrum) : spectrum :=
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spectrum.MK (λ(n : ℤ), ptrunc (maxm2 (k + n)) (E n))
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(λ(n : ℤ), ptrunc_pequiv_ptrunc (maxm2 (k + n)) (equiv_glue E n)
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⬝e* (loop_ptrunc_maxm2_pequiv (ap maxm2 (add.assoc k n 1)) (E (n+1)))⁻¹ᵉ*)
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definition strunc_change_int [constructor] {k l : ℤ} (E : spectrum) (p : k = l) :
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strunc k E →ₛ strunc l E :=
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begin induction p, reflexivity end
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definition is_strunc_strunc (k : ℤ) (E : spectrum)
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: is_strunc k (strunc k E) :=
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λ n, is_trunc_trunc (maxm2 (k + n)) (E n)
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definition is_strunc_strunc_pred (X : spectrum) (k : ℤ) : is_strunc k (strunc (k - 1) X) :=
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λn, @(is_trunc_of_le _ (maxm2_monotone (add_le_add_right (sub_one_le k) n))) !is_strunc_strunc
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definition is_strunc_strunc_of_is_strunc (k : ℤ) {l : ℤ} {E : spectrum} (H : is_strunc l E)
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: is_strunc l (strunc k E) :=
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λ n, !is_trunc_trunc_of_is_trunc
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definition str [constructor] (k : ℤ) (E : spectrum) : E →ₛ strunc k E :=
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smap.mk (λ n, ptr (maxm2 (k + n)) (E n))
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abstract begin
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intro n,
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apply psquare_of_phomotopy,
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refine !passoc ⬝* pwhisker_left _ !ptr_natural ⬝* _,
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refine !passoc⁻¹* ⬝* pwhisker_right _ !loop_ptrunc_maxm2_pequiv_ptr⁻¹*,
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end end
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definition strunc_elim [constructor] {k : ℤ} {E F : spectrum} (f : E →ₛ F)
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(H : is_strunc k F) : strunc k E →ₛ F :=
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smap.mk (λn, ptrunc.elim (maxm2 (k + n)) (f n))
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abstract begin
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intro n,
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apply psquare_of_phomotopy,
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symmetry,
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refine !passoc⁻¹* ⬝* pwhisker_right _ !loop_ptrunc_maxm2_pequiv_ptrunc_elim' ⬝* _,
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refine @(ptrunc_elim_ptrunc_functor _ _ _) _ ⬝* _,
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refine _ ⬝* @(ptrunc_elim_pcompose _ _ _) _ _,
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apply is_trunc_maxm2_loop,
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refine is_trunc_of_eq _ (H (n+1)), exact ap maxm2 (add.assoc k n 1)⁻¹,
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apply ptrunc_elim_phomotopy2,
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apply phomotopy_of_psquare,
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apply ptranspose,
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apply smap.glue_square
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end end
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definition strunc_functor [constructor] (k : ℤ) {E F : spectrum} (f : E →ₛ F) :
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strunc k E →ₛ strunc k F :=
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strunc_elim (str k F ∘ₛ f) (is_strunc_strunc k F)
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/- truncated spectra -/
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structure truncspectrum (n : ℤ) :=
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(carrier : spectrum)
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(struct : is_strunc n carrier)
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notation n `-spectrum` := truncspectrum n
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attribute truncspectrum.carrier [coercion]
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definition genspectrum_of_truncspectrum [coercion] (n : ℤ) : n-spectrum → gen_spectrum +ℤ :=
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λ E, truncspectrum.carrier E
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/- Comment (by Floris):
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I think we should really not bundle truncated spectra up,
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unless we are interested in the type of truncated spectra (e.g. when proving n-spectrum ≃ ...).
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Properties of truncated a spectrum should just be stated with two assumptions
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(X : spectrum) (H : is_strunc n X)
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-/
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/- truncatedness of spi and sp_cotensor assuming the domain has a level of connectedness -/
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section
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open is_conn
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definition is_conn_maxm1_of_maxm2 (A : Type*) (n : ℤ)
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: is_conn (maxm2 n) A → is_conn (maxm1m1 n).+1 A :=
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begin
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intro H, induction n with n n,
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{ exact H },
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{ exact is_conn_minus_one A (tr pt) }
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end
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definition is_trunc_maxm2_of_maxm1 (A : Type*) (n : ℤ)
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: is_trunc (maxm1m1 n).+1 A → is_trunc (maxm2 n) A :=
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begin
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intro H, induction n with n n,
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{ exact H},
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{ apply is_contr_of_merely_prop,
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{ exact H },
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{ exact tr pt } }
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end
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variables (A : Type*) (n : ℤ) [H : is_conn (maxm2 n) A]
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include H
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definition is_trunc_maxm2_ppi (k l : ℤ) (H3 : l ≤ n+1+k) (P : A → Type*)
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(H2 : Πa, is_trunc (maxm2 l) (P a)) : is_trunc (maxm2 k) (Π*(a : A), P a) :=
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is_trunc_maxm2_of_maxm1 (Π*(a : A), P a) k
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(@is_trunc_ppi_of_is_conn A (maxm1m1 n) (is_conn_maxm1_of_maxm2 A n H) (maxm1m1 k) _
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(le.trans (maxm2_monotone H3) (maxm2_le n k)) P H2)
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definition is_strunc_spi_of_is_conn (k l : ℤ) (H3 : l ≤ n+1+k) (P : A → spectrum)
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(H2 : Πa, is_strunc l (P a)) : is_strunc k (spi A P) :=
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begin
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intro m, unfold spi,
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exact is_trunc_maxm2_ppi A n (k+m) _ (le.trans (add_le_add_right H3 _)
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(le_of_eq (add.assoc (n+1) k m))) (λ a, P a m) (λa, H2 a m)
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end
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end
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definition is_strunc_spi_of_le {A : Type*} (k n : ℤ) (H : n ≤ k) (P : A → spectrum)
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(H2 : Πa, is_strunc n (P a)) : is_strunc k (spi A P) :=
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begin
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assert K : n ≤ -[1+ 0] + 1 + k,
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{ krewrite (int.zero_add k), exact H },
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{ exact @is_strunc_spi_of_is_conn A (-[1+ 0]) (is_conn.is_conn_minus_two A) k _ K P H2 }
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end
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definition is_strunc_spi {A : Type*} (n : ℤ) (P : A → spectrum) (H : Πa, is_strunc n (P a))
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: is_strunc n (spi A P) :=
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is_strunc_spi_of_le n n !le.refl P H
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definition is_strunc_sp_cotensor (n : ℤ) (A : Type*) {Y : spectrum} (H : is_strunc n Y)
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: is_strunc n (sp_cotensor A Y) :=
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is_strunc_pequiv_closed (λn, !pppi_pequiv_ppmap) (is_strunc_spi n (λa, Y) (λa, H))
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definition is_strunc_sp_ucotensor (n : ℤ) (A : Type) {Y : spectrum} (H : is_strunc n Y)
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: is_strunc n (sp_ucotensor A Y) :=
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λk, !pi.is_trunc_arrow
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end spectrum
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