53 lines
1.7 KiB
Text
53 lines
1.7 KiB
Text
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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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