File size: 3,186 Bytes
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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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)
/-- The extension restricts to the original functor under the canonical embedding. -/
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
/-- An explicitly split evaluated idempotent identifies the value of the completion. -/
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
/-- The manuscript's additive extension of evaluation to all of Kar(Q), with
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
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