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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 56 57 58 59 60 61 62 63 | 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
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