| import OAI.Algebra.Finitistic.Localization.FiniteEvaluationTarget |
| import OAI.Algebra.Finitistic.Localization.BoundedDerivedEmbedding |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.QuadraticEncoding |
| open FiniteProjectiveHomotopy |
| universe u v |
| 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 finiteQuotientEvaluationAdditive : (finiteQuotientEvaluation X).Additive := by |
| rw [Localization.functor_additive_iff (boundedDenominators P).Q (boundedDenominators P)] |
| exact Functor.additive_of_iso (finiteQuotientEvaluationFactors X).symm |
| |
| instance finiteQuotientEvaluationLifting : Localization.Lifting |
| (boundedDenominators P).Q (boundedDenominators P) |
| (finiteEvaluation X) (finiteQuotientEvaluation X) := |
| ⟨finiteQuotientEvaluationFactors X⟩ |
| |
| instance finiteQuotientEvaluationShift : (finiteQuotientEvaluation X).CommShift ℤ := |
| Functor.commShiftOfLocalization (boundedDenominators P).Q (boundedDenominators P) ℤ |
| (finiteEvaluation X) (finiteQuotientEvaluation X) |
| |
| instance finiteQuotientEvaluationFactorsShift : |
| NatTrans.CommShift (finiteQuotientEvaluationFactors X).hom ℤ := |
| inferInstanceAs (NatTrans.CommShift |
| (Localization.Lifting.iso (boundedDenominators P).Q (boundedDenominators P) |
| (finiteEvaluation X) (finiteQuotientEvaluation X)).hom ℤ) |
| |
| instance finiteQuotientEvaluationTriangulated : (finiteQuotientEvaluation X).IsTriangulated := by |
| have : (boundedDenominators P).Q.mapArrow.EssSurj := |
| Localization.essSurj_mapArrow (boundedDenominators P).Q (boundedDenominators P) |
| exact Functor.isTriangulated_of_precomp_iso (finiteQuotientEvaluationFactors X) |
| |
| / |
| finite-dimensional complexes. Over a field all its terms are projective. -/ |
| abbrev FiniteVectorDerived := (derivedEmbedding.{u,max u v} k).EssImageSubcategory |
| |
| instance derivedToEssImageShift : ((derivedEmbedding.{u,max u v} k).toEssImage).CommShift ℤ := by |
| dsimp [Functor.toEssImage] |
| infer_instance |
| instance derivedToEssImageTriangulated : ((derivedEmbedding.{u,max u v} k).toEssImage).IsTriangulated := by |
| dsimp [Functor.toEssImage] |
| infer_instance |
| |
| def derivedEvaluation : BoundedK P ⥤ FiniteVectorDerived.{u,v} (k := k) := |
| finiteEvaluation X ⋙ (derivedEmbedding k).toEssImage |
| |
| def derivedQuotientEvaluation : BoundedQ P ⥤ FiniteVectorDerived.{u,v} (k := k) := |
| finiteQuotientEvaluation X ⋙ (derivedEmbedding k).toEssImage |
| |
| def derivedQuotientEvaluationFactors : |
| (boundedDenominators P).Q ⋙ derivedQuotientEvaluation X ≅ derivedEvaluation X := |
| (Functor.associator _ _ _).symm ≪≫ |
| Functor.isoWhiskerRight (finiteQuotientEvaluationFactors X) (derivedEmbedding k).toEssImage |
| |
| instance derivedEvaluationShift : (derivedEvaluation X).CommShift ℤ := by |
| dsimp [derivedEvaluation] |
| infer_instance |
| instance derivedEvaluationTriangulated : (derivedEvaluation X).IsTriangulated := by |
| dsimp [derivedEvaluation] |
| infer_instance |
| instance derivedQuotientEvaluationShift : (derivedQuotientEvaluation X).CommShift ℤ := by |
| dsimp [derivedQuotientEvaluation] |
| infer_instance |
| instance derivedQuotientEvaluationTriangulated : |
| (derivedQuotientEvaluation X).IsTriangulated := by |
| dsimp [derivedQuotientEvaluation] |
| infer_instance |
| end LittleFinitistic.QuadraticEncoding |
| |
| end |
| |
| end OAI |
| |
