File size: 3,556 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 79 | 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)
/-- The full subcategory of the canonical derived category represented by bounded
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
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