| import OAI.Algebra.Finitistic.Localization.CompletedImageEvaluation |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Idempotents |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.QuadraticEncoding |
| 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] [IsIdempotentComplete D] |
| variable (F : C ⥤ D) [F.Additive] |
|
|
| /-- Commutation with any endofunctor extends uniquely across idempotent completion. -/ |
| def completedCommutationIso (S : C ⥤ C) (T : D ⥤ D) (e : S ⋙ F ≅ F ⋙ T) : |
| (functorExtension₂ C C).obj S ⋙ (functorExtension C D).obj F ≅ |
| (functorExtension C D).obj F ⋙ T := by |
| let G := (functorExtension C D).obj F |
| let S' := (functorExtension₂ C C).obj S |
| let h : toKaroubi C ⋙ (S' ⋙ G) ≅ toKaroubi C ⋙ (G ⋙ T) := |
| (Functor.associator _ _ _).symm ≪≫ |
| Functor.isoWhiskerRight ((functorExtension₂CompWhiskeringLeftToKaroubiIso C C).app S) G ≪≫ |
| Functor.associator _ _ _ ≪≫ |
| Functor.isoWhiskerLeft S (completedExtensionFactors F) ≪≫ e ≪≫ |
| Functor.isoWhiskerRight (completedExtensionFactors F).symm T ≪≫ |
| Functor.associator _ _ _ |
| exact (whiskeringLeftObjToKaroubiFullyFaithful (C := C) (D := D)).preimageIso h |
| 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) |
|
|
| /-- This is the formal shift used by the native odd-double construction: shift |
| the underlying object and its idempotent. No shift coherence is assumed. -/ |
| def completedDerivedShiftIso (n : ℤ) : |
| (functorExtension₂ (BoundedQ P) (BoundedQ P)).obj (shiftFunctor (BoundedQ P) n) ⋙ |
| completedDerivedEvaluation X ≅ |
| completedDerivedEvaluation X ⋙ shiftFunctor (FiniteVectorDerived.{u,v} (k := k)) n := |
| completedCommutationIso (derivedQuotientEvaluation X) _ _ |
| ((derivedQuotientEvaluation X).commShiftIso n) |
|
|
| /-- In particular, every formal shift of U evaluates to the same shift of eY. -/ |
| def completedShiftedSummandImageIso (r : PresentedAlgebra P) (hr : r*r=r) (n : ℤ) : |
| (completedDerivedEvaluation X).obj |
| (((functorExtension₂ (BoundedQ P) (BoundedQ P)).obj (shiftFunctor (BoundedQ P) n)).obj |
| (quotientSummand r hr)) ≅ (evaluatedImageDerived X r)⟦n⟧ := |
| (completedDerivedShiftIso X n).app (quotientSummand r hr) ≪≫ |
| (shiftFunctor _ n).mapIso (completedSummandImageIso X r hr) |
| end LittleFinitistic.QuadraticEncoding |
|
|
| end |
|
|
| end OAI |
|
|
