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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | 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
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