| import OAI.Algebra.Finitistic.Localization.FormalSummandAction |
| import OAI.Algebra.Finitistic.Localization.DiagramLifting |
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|
| namespace OAI |
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| namespace LittleFinitistic.FixedDiagram |
| open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Idempotents |
| open FormalSummandAction |
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| variable {k R : Type*} [CommRing k] [Ring R] [Algebra k R] |
| variable {C Q : Type*} [Category C] [Category Q] [Preadditive C] [Preadditive Q] |
| [Linear k C] [Linear k Q] |
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| / |
| theorem action_relations {X : Q} (ρ : R →ₐ[k] End X) (na nb nr : ℕ) |
| (x : Fin na → R) (y : Fin nb → R) (c : Fin nr → Fin na → Fin nb → k) |
| (hc : ∀ t, ∑ a, ∑ b, c t a b • (y b * x a) = 0) : |
| ∀ t, ∑ a, ∑ b, c t a b • ((ρ (x a) : X ⟶ X) ≫ ρ (y b)) = 0 := by |
| intro t |
| have h := congrArg ρ (hc t) |
| simpa only [map_sum,map_smul,map_mul,map_zero,End.mul_def] using h |
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| variable (q : C ⥤ Q) (W : MorphismProperty C) [q.IsLocalization W] |
| [W.HasRightCalculusOfFractions] [q.Additive] [q.Linear k] |
| variable [HasShift Q ℤ] [HasZeroObject Q] |
| [∀ n : ℤ, (shiftFunctor Q n).Additive] [Pretriangulated Q] |
| [∀ n : ℤ, (shiftFunctor Q n).Linear k] |
| local instance : HasFiniteBiproducts (Karoubi Q) := Karoubi.karoubi_hasFiniteBiproducts |
| local instance : HasBinaryBiproducts (Karoubi Q) := |
| hasBinaryBiproducts_of_finite_biproducts (Karoubi Q) |
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| include W in |
| / |
| represent the odd double of the selecting action in the localization. -/ |
| theorem fixed_diagram {L : Q} (θ : R →ₐ[k] End L) |
| (e : R) (he : e*e=e) (hec : ∀ r : R, e*r=r*e) (α : R →ₐ[k] R) |
| (na nb nr : ℕ) (x : Fin na → R) (y : Fin nb → R) |
| (c : Fin nr → Fin na → Fin nb → k) |
| (hc : ∀ t, ∑ a, ∑ b, c t a b • (y b * x a) = 0) : |
| ∃ (V₀ : Q) (ρ₀ : R →ₐ[k] End V₀) |
| (i : (toKaroubi Q).obj V₀ ≅ |
| summand θ e he ⊞ OddDouble.formalShift (summand θ e he) 3) |
| (D₀ D₁ D₂ : C) (φ₀ : q.obj D₀ ≅ V₀) (φ₁ : q.obj D₁ ≅ V₀) |
| (φ₂ : q.obj D₂ ≅ V₀) (f : Fin na → (D₀ ⟶ D₁)) (g : Fin nb → (D₁ ⟶ D₂)), |
| (∀ r : R, (toKaroubi Q).map (ρ₀ r) ≫ i.hom = |
| i.hom ≫ oddDoubleAction θ e he hec α r) ∧ |
| (∀ a, q.map (f a) ≫ φ₁.hom = φ₀.hom ≫ ρ₀ (x a)) ∧ |
| (∀ b, q.map (g b) ≫ φ₂.hom = φ₁.hom ≫ ρ₀ (y b)) ∧ |
| (∀ t, ∑ a, ∑ b, c t a b • (f a ≫ g b) = 0) := by |
| obtain ⟨V₀,ρ₀,i,hi⟩ := fixed_old_object_with_action θ e he hec α |
| obtain ⟨D₀,D₁,D₂,φ₀,φ₁,φ₂,f,g,hf,hg,hrel⟩ := |
| DiagramLifting.finite_quadratic_diagram_lifting q W na nb nr V₀ V₀ V₀ |
| (fun a => ρ₀ (x a)) (fun b => ρ₀ (y b)) c (action_relations ρ₀ na nb nr x y c hc) |
| exact ⟨V₀,ρ₀,i,D₀,D₁,D₂,φ₀,φ₁,φ₂,f,g,hi,hf,hg,hrel⟩ |
| end LittleFinitistic.FixedDiagram |
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| end OAI |
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