| import OAI.Algebra.Finitistic.Localization.FixedDiagram |
| import OAI.Algebra.Finitistic.Localization.SourceShiftLinear |
| import OAI.Algebra.Finitistic.Encoding.RelationBasisAction |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Idempotents |
| namespace LittleFinitistic.ConcreteFixedDiagram |
| open SelectionEncoding RelationBasisCoefficients RelationBasisAction FormalSummandAction |
| universe u v |
| variable {k : Type u} [Field k] {R : Type v} [Ring R] [Algebra k R] |
| variable (E : Encoding k R) |
| local instance : HasFiniteBiproducts (Karoubi (SourceVerdier.Q E.relations)) := |
| Karoubi.karoubi_hasFiniteBiproducts |
| local instance : HasBinaryBiproducts (Karoubi (SourceVerdier.Q E.relations)) := |
| hasBinaryBiproducts_of_finite_biproducts _ |
|
|
| /-- The fixed odd-double diagram in the actual Kᵇ(proj-B) and its actual Verdier |
| quotient. Every independent relation of the constructed path algebra is included. -/ |
| theorem fixed_diagram {L : SourceVerdier.Q E.relations} (θ : R →ₐ[k] End L) |
| (e : R) (he : e*e=e) (hec : ∀ r : R, e*r=r*e) (α : R →ₐ[k] R) : |
| ∃ (V₀ : SourceVerdier.Q E.relations) (ρ₀ : R →ₐ[k] End V₀) |
| (i : (toKaroubi (SourceVerdier.Q E.relations)).obj V₀ ≅ |
| summand θ e he ⊞ OddDouble.formalShift (summand θ e he) 3) |
| (D₀ D₁ D₂ : SourceVerdier.K E.relations) |
| (φ₀ : (SourceVerdier.q E.relations).obj D₀ ≅ V₀) |
| (φ₁ : (SourceVerdier.q E.relations).obj D₁ ≅ V₀) |
| (φ₂ : (SourceVerdier.q E.relations).obj D₂ ≅ V₀) |
| (f : Fin (E.generators+1) → (D₀ ⟶ D₁)) |
| (g : Fin (E.generators+1) → (D₁ ⟶ D₂)), |
| (∀ r : R, (toKaroubi (SourceVerdier.Q E.relations)).map (ρ₀ r) ≫ i.hom = |
| i.hom ≫ oddDoubleAction θ e he hec α r) ∧ |
| (∀ a, (SourceVerdier.q E.relations).map (f a) ≫ φ₁.hom = |
| φ₀.hom ≫ ρ₀ (arrow (generator E) a)) ∧ |
| (∀ b, (SourceVerdier.q E.relations).map (g b) ≫ φ₂.hom = |
| φ₁.hom ≫ ρ₀ (arrow (generator E) b)) ∧ |
| (∀ r : RelationIndex E.relations, |
| ∑ a, ∑ b, basisCoefficient E.relations r a b • (f a ≫ g b) = 0) := |
| FixedDiagram.fixed_diagram (SourceVerdier.q E.relations) (SourceVerdier.Sigma E.relations) |
| θ e he hec α (E.generators+1) (E.generators+1) |
| (Module.finrank k (QuadraticEncoding.relationSpace E.relations)) |
| (arrow (generator E)) (arrow (generator E)) (basisCoefficient E.relations) |
| (encoding_basis_action E) |
| end LittleFinitistic.ConcreteFixedDiagram |
|
|
| end |
|
|
| end OAI |
|
|
