File size: 3,698 Bytes
216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import OAI.Algebra.Finitistic.Localization.LocalizedEvaluationStructure
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits CategoryTheory.Pretriangulated
namespace LittleFinitistic.QuadraticEncoding
open FiniteProjectiveHomotopy ThickEvaluation
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
/-- The actual degree-zero right vertex projective in K. -/
def vertexK (i : Fin 3) : BoundedK P :=
⟨(HomotopyCategory.quotient _ _).obj ((degreeZero P).obj (vertexModule P i)),
single_property (vertexModule P i) 0⟩
def arrowK01 (h : Fin (d+1)) : vertexK P 0 ⟶ vertexK P 1 :=
(inclusion (B P)ᵐᵒᵖ).preimage
((HomotopyCategory.quotient _ _).map ((degreeZero P).map (arrowModule01 P h)))
def arrowK12 (h : Fin (d+1)) : vertexK P 1 ⟶ vertexK P 2 :=
(inclusion (B P)ᵐᵒᵖ).preimage
((HomotopyCategory.quotient _ _).map ((degreeZero P).map (arrowModule12 P h)))
theorem s_denominator : boundedDenominators P (arrowK01 P 0) := by
obtain ⟨Z,g,h,hT⟩ := distinguished_cocone_triangle (arrowK01 P 0)
let I := inclusion (B P)ᵐᵒᵖ
have e : I.obj Z ≅ I.obj (boundedConeS P) :=
Triangle.π₃.mapIso (isoTriangleOfIso₁₂ _ _ (I.map_distinguished _ hT)
(HomotopyCategory.mappingCone_triangleh_distinguished
((degreeZero P).map (arrowModule01 P 0))) (Iso.refl _) (Iso.refl _) (by
change I.map (arrowK01 P 0) ≫ 𝟙 _ = 𝟙 _ ≫
(HomotopyCategory.quotient _ _).map ((degreeZero P).map (arrowModule01 P 0))
unfold arrowK01
erw [Functor.map_preimage]
simp
exact Category.comp_id _))
have eK : Z ≅ boundedConeS P := I.preimageIso e
exact ⟨Z,g,h,hT,Thick.iso eK.symm (Thick.generator (Or.inl rfl))⟩
instance quotient_s_isIso : IsIso ((boundedDenominators P).Q.map (arrowK01 P 0)) :=
Localization.inverts (boundedDenominators P).Q (boundedDenominators P) _ (s_denominator P)
theorem s'_denominator : boundedDenominators P (arrowK12 P 0) := by
obtain ⟨Z,g,h,hT⟩ := distinguished_cocone_triangle (arrowK12 P 0)
let I := inclusion (B P)ᵐᵒᵖ
have e : I.obj Z ≅ I.obj (boundedConeS' P) :=
Triangle.π₃.mapIso (isoTriangleOfIso₁₂ _ _ (I.map_distinguished _ hT)
(HomotopyCategory.mappingCone_triangleh_distinguished
((degreeZero P).map (arrowModule12 P 0))) (Iso.refl _) (Iso.refl _) (by
change I.map (arrowK12 P 0) ≫ 𝟙 _ = 𝟙 _ ≫
(HomotopyCategory.quotient _ _).map ((degreeZero P).map (arrowModule12 P 0))
unfold arrowK12
erw [Functor.map_preimage]
simp
exact Category.comp_id _))
have eK : Z ≅ boundedConeS' P := I.preimageIso e
exact ⟨Z,g,h,hT,Thick.iso eK.symm (Thick.generator (Or.inr rfl))⟩
instance quotient_s'_isIso : IsIso ((boundedDenominators P).Q.map (arrowK12 P 0)) :=
Localization.inverts (boundedDenominators P).Q (boundedDenominators P) _ (s'_denominator P)
/-- The three actual images in the fixed quotient. -/
def vertexQ (i : Fin 3) : BoundedQ P := (boundedDenominators P).Q.obj (vertexK P i)
def arrowQ01 (h : Fin (d+1)) : vertexQ P 0 ⟶ vertexQ P 1 :=
(boundedDenominators P).Q.map (arrowK01 P h)
def arrowQ12 (h : Fin (d+1)) : vertexQ P 1 ⟶ vertexQ P 2 :=
(boundedDenominators P).Q.map (arrowK12 P h)
instance arrowQ01_zero_isIso : IsIso (arrowQ01 P 0) := quotient_s_isIso P
instance arrowQ12_zero_isIso : IsIso (arrowQ12 P 0) := quotient_s'_isIso P
/-- The normalized generator s^{-1} a_h on L=E_0 in Q. -/
def normalizedGenerator (h : Fin d) : End (vertexQ P 0) :=
arrowQ01 P h.succ ≫ inv (arrowQ01 P 0)
end LittleFinitistic.QuadraticEncoding
end
end OAI
|