openAI-math-dataset / lean /OAI /Algebra /Finitistic /Localization /ConcreteChainRepresentatives.lean
| import OAI.Algebra.Finitistic.Localization.BoundedRepresentatives | |
| import OAI.Algebra.Finitistic.Localization.SourceLinear | |
| import OAI.Algebra.Finitistic.Localization.ChainRepresentatives | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| namespace LittleFinitistic.ConcreteChainRepresentatives | |
| open SourceVerdier | |
| universe u | |
| variable {k : Type u} [Field k] {d t : ℕ} (P : QuadraticEncoding.Relations k d t) | |
| abbrev representative (X : K P) : CochainComplex (Proj P) ℤ := X.obj.as | |
| theorem representative_bounded (X : K P) : | |
| BoundedHomotopy.boundedComplex (Proj P) (representative P X) := | |
| (BoundedHomotopy.bounded_quotient_obj_iff _ _).1 X.property | |
| /-- The lifted source diagram has genuine bounded projective chain representatives | |
| and a specified null-homotopy for every relation; no chain representatives are assumed. -/ | |
| theorem quadratic_representatives (na nb nr : ℕ) (X₀ X₁ X₂ : K P) | |
| (a : Fin na → (X₀ ⟶ X₁)) (b : Fin nb → (X₁ ⟶ X₂)) | |
| (c : Fin nr → Fin na → Fin nb → k) | |
| (hc : ∀ r, ∑ i, ∑ j, c r i j • (a i ≫ b j) = 0) : | |
| ∃ (f : Fin na → (representative P X₀ ⟶ representative P X₁)) | |
| (g : Fin nb → (representative P X₁ ⟶ representative P X₂)), | |
| (∀ i, (HomotopyCategory.quotient (Proj P) (.up ℤ)).map (f i) = (a i).hom) ∧ | |
| (∀ j, (HomotopyCategory.quotient (Proj P) (.up ℤ)).map (g j) = (b j).hom) ∧ | |
| Nonempty (∀ r, Homotopy (∑ i, ∑ j, c r i j • (f i ≫ g j)) 0) := by | |
| let I := (BoundedHomotopy.bounded (Proj P)).ι | |
| apply ChainRepresentatives.quadratic_chain_representatives na nb nr | |
| (representative P X₀) (representative P X₁) (representative P X₂) | |
| (fun i => I.map (a i)) (fun j => I.map (b j)) c | |
| intro r | |
| have h := congrArg I.map (hc r) | |
| simp only [Functor.map_sum,Functor.map_smul,Functor.map_comp,Functor.map_zero] at h | |
| exact h | |
| end LittleFinitistic.ConcreteChainRepresentatives | |
| end | |
| end OAI | |