| import OAI.Algebra.Finitistic.Localization.ModuleActionCompatibility | |
| import OAI.Algebra.Finitistic.Localization.FiniteDerivedEvaluation | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| 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 finiteUnderlyingLifting : Localization.Lifting | |
| (boundedDenominators P).Q (boundedDenominators P) (boundedEvaluation X) | |
| (finiteQuotientEvaluation X ⋙ inclusion k) where | |
| iso := (Functor.associator _ _ _).symm ≪≫ | |
| Functor.isoWhiskerRight (finiteQuotientEvaluationFactors X) (inclusion k) | |
| /-- Both constructions of evaluation through Q agree by uniqueness of localization. -/ | |
| def finiteUnderlyingComparison : finiteQuotientEvaluation X ⋙ inclusion k ≅ | |
| boundedQuotientEvaluation X := | |
| Localization.liftNatIso (boundedDenominators P).Q (boundedDenominators P) | |
| (boundedEvaluation X) (boundedEvaluation X) _ _ (Iso.refl _) | |
| /-- The existing finite derived evaluation agrees with canonical derived localization | |
| of the explicitly computed homotopy evaluation. -/ | |
| def derivedUnderlyingComparison : | |
| derivedQuotientEvaluation X ⋙ (derivedEmbedding k).essImage.ι ≅ | |
| boundedQuotientEvaluation X ⋙ DerivedCategory.Qh := | |
| Functor.isoWhiskerRight (finiteUnderlyingComparison X) DerivedCategory.Qh | |
| def derivedVertexIso (i : Fin 3) : | |
| ((derivedQuotientEvaluation X ⋙ (derivedEmbedding k).essImage.ι).obj (vertexQ P i)) ≅ | |
| DerivedCategory.Qh.obj ((vectorStalk (k := k)).obj (ModuleCat.of k (ULift.{u} Y))) := | |
| (derivedUnderlyingComparison X).app (vertexQ P i) ≪≫ | |
| DerivedCategory.Qh.mapIso (quotientVertexIso X i) | |
| /-- The full action square in the actual derived category, for the previously | |
| constructed bounded finite-dimensional evaluation functor. -/ | |
| theorem derived_quotientAction_square (r : PresentedAlgebra P) : | |
| (derivedQuotientEvaluation X ⋙ (derivedEmbedding k).essImage.ι).map (quotientAction P r) ≫ | |
| (derivedVertexIso X 0).hom = | |
| (derivedVertexIso X 0).hom ≫ DerivedCategory.Qh.map | |
| ((vectorStalk (k := k)).map (liftedEndHom (givenAction X r))) := by | |
| have hn := (derivedUnderlyingComparison X).hom.naturality (quotientAction P r) | |
| have hm := congrArg DerivedCategory.Qh.map (quotientAction_square X r) | |
| simp only [Functor.map_comp] at hm | |
| change _ ≫ ((derivedUnderlyingComparison X).hom.app (vertexQ P 0) ≫ | |
| DerivedCategory.Qh.map (quotientVertexIso X 0).hom) = | |
| ((derivedUnderlyingComparison X).hom.app (vertexQ P 0) ≫ | |
| DerivedCategory.Qh.map (quotientVertexIso X 0).hom) ≫ _ | |
| rw [← Category.assoc, hn, Category.assoc] | |
| change (derivedUnderlyingComparison X).hom.app (vertexQ P 0) ≫ | |
| (DerivedCategory.Qh.map ((boundedQuotientEvaluation X).map (quotientAction P r)) ≫ _) = _ | |
| rw [hm, ← Category.assoc] | |
| end LittleFinitistic.QuadraticEncoding | |
| end | |
| end OAI | |