129 lines
4 KiB
Text
129 lines
4 KiB
Text
import .spectrum .EM
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-- TODO move this
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open trunc_index nat
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namespace int
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definition maxm2 : ℤ → ℕ₋₂ :=
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λ n, int.cases_on n trunc_index.of_nat
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(λ m, nat.cases_on m -1 (λ a, -2))
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attribute maxm2 [unfold 1]
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definition maxm2_le_maxm0 (n : ℤ) : maxm2 n ≤ max0 n :=
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begin
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induction n with n n,
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{ exact le.tr_refl n },
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{ cases n with n,
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{ exact le.step (le.tr_refl -1) },
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{ exact minus_two_le 0 } }
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end
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definition max0_le_of_le {n : ℤ} {m : ℕ} (H : n ≤ of_nat m)
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: nat.le (max0 n) m :=
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begin
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induction n with n n,
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{ exact le_of_of_nat_le_of_nat H },
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{ exact nat.zero_le m }
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end
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end int
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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 ptrunc_maxm2_change_int {k l : ℤ} (X : Type*) (p : k = l)
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: ptrunc (maxm2 k) X ≃* ptrunc (maxm2 l) X :=
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pequiv_ap (λ n, ptrunc (maxm2 n) X) p
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definition loop_ptrunc_maxm2_pequiv (k : ℤ) (X : Type*) :
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Ω (ptrunc (maxm2 (k+1)) X) ≃* ptrunc (maxm2 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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{ exact loop_ptrunc_pequiv -1 X },
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{ cases k with k,
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{ exact loop_ptrunc_pequiv -2 X },
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{ exact loop_pequiv_punit_of_is_set (pType.mk (trunc -2 X) (tr pt))
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⬝e* (pequiv_punit_of_is_contr
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(pType.mk (trunc -2 (Point X = Point X)) (tr idp))
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(is_trunc_trunc -2 (Point X = Point X)))⁻¹ᵉ* } } }
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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 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 (k + n) (E (n+1)))⁻¹ᵉ*
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⬝e* (loop_pequiv_loop
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(ptrunc_maxm2_change_int _ (add.assoc k n 1))))
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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_trunc_maxm2_loop (A : pType) (k : ℤ)
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: is_trunc (maxm2 (k + 1)) A → is_trunc (maxm2 k) (Ω A) :=
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begin
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intro H, induction k with k k,
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{ apply is_trunc_loop, exact H },
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{ cases k with k,
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{ apply is_trunc_loop, exact H},
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{ apply is_trunc_loop, cases k with k,
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{ exact H },
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{ apply is_trunc_succ, exact H } } }
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end
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definition is_strunc (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_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_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_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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{ induction n with n IH,
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{ -- case = -1
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apply is_trunc_loop, exact ab_group.is_set_carrier G },
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{ -- case < -1
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apply is_trunc_maxm2_loop, 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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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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(λ n, sorry)
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end spectrum
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