| import OAI.Algebra.Finitistic.Localization.VectorIdempotentCompleteness |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Idempotents |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.QuadraticEncoding |
| open FiniteProjectiveHomotopy VectorComplexFormality |
| universe u v w z |
| section General |
| variable {C : Type u} [Category.{v} C] [Preadditive C] |
| variable {D : Type w} [Category.{z} D] [Preadditive D] |
| variable (F : C ⥤ D) [F.Additive] |
| |
| instance karoubiExtensionAdditive : ((functorExtension₂ C D).obj F).Additive where |
| map_add := by |
| intro A B f g |
| apply Karoubi.Hom.ext |
| exact F.map_add |
|
|
| variable [IsIdempotentComplete D] |
| |
| instance completedExtensionAdditive : ((functorExtension C D).obj F).Additive := |
| inferInstanceAs (((functorExtension₂ C D).obj F ⋙ (toKaroubiEquivalence D).inverse).Additive) |
| |
| / |
| def completedExtensionFactors : |
| toKaroubi C ⋙ (functorExtension C D).obj F ≅ F := |
| (Functor.associator _ _ _).symm ≪≫ |
| Functor.isoWhiskerRight |
| ((functorExtension₂CompWhiskeringLeftToKaroubiIso C D).app F) |
| (toKaroubiEquivalence D).inverse ≪≫ |
| Functor.associator _ _ _ ≪≫ |
| Functor.isoWhiskerLeft F (toKaroubiEquivalence D).unitIso.symm ≪≫ |
| Functor.rightUnitor F |
| / |
| def completedExtensionImageIso (U : Karoubi C) (I : D) |
| (π : F.obj U.X ⟶ I) (ι : I ⟶ F.obj U.X) |
| (hπι : π ≫ ι = F.map U.p) (hιπ : ι ≫ π = 𝟙 I) : |
| ((functorExtension C D).obj F).obj U ≅ I := |
| (toKaroubiEquivalence D).inverse.mapIso |
| (splitKaroubiIso (F.map U.p) (by rw [← F.map_comp, U.idem]) π ι hπι hιπ) ≪≫ |
| (toKaroubiEquivalence D).unitIso.symm.app I |
| |
| end General |
| |
| variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t} |
| variable {Y : Type v} [AddCommGroup Y] [Module k Y] [FiniteDimensional k Y] |
| variable (X : Representation P Y) |
| |
| instance derivedEmbeddingAdditive : (derivedEmbedding.{u,max u v} k).Additive := |
| inferInstanceAs ((inclusion k ⋙ DerivedCategory.Qh).Additive) |
| |
| instance derivedToEssImageAdditive : ((derivedEmbedding.{u,max u v} k).toEssImage).Additive := by |
| dsimp [Functor.toEssImage] |
| infer_instance |
| |
| instance derivedQuotientEvaluationAdditive : (derivedQuotientEvaluation X).Additive := by |
| dsimp [derivedQuotientEvaluation] |
| infer_instance |
| |
| / |
| values in the actual bounded finite-dimensional derived category. -/ |
| def completedDerivedEvaluation : Karoubi (BoundedQ P) ⥤ FiniteVectorDerived.{u,v} (k := k) := |
| (functorExtension _ _).obj (derivedQuotientEvaluation X) |
| |
| instance completedDerivedEvaluationAdditive : (completedDerivedEvaluation X).Additive := |
| completedExtensionAdditive (derivedQuotientEvaluation X) |
| |
| def completedDerivedEvaluationFactors : |
| toKaroubi (BoundedQ P) ⋙ completedDerivedEvaluation X ≅ derivedQuotientEvaluation X := |
| completedExtensionFactors (derivedQuotientEvaluation X) |
| end LittleFinitistic.QuadraticEncoding |
| |
| end |
| |
| end OAI |
| |
