| 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 |
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|
