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ae00908 | 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 | import OAI.Algebra.Finitistic.Localization.UnderlyingRepresentatives
import OAI.Algebra.Finitistic.Selected.SelectedChainDiagram
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
attribute [local instance] HasDerivedCategory.standard
namespace LittleFinitistic.SelectedTensorComparison
open QuadraticEncoding 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] (X : Representation P Y)
abbrev tensor (D : SourceVerdier.K P) :=
((BalancedTensor.tensorFunctor k (B P) X.Evaluation).mapHomologicalComplex (.up ℤ)).obj
(UnderlyingRepresentatives.complex P D)
/-- The chosen representative's actual tensor is the source evaluation object. -/
def homotopyIso (D : SourceVerdier.K P) :
(HomotopyCategory.quotient _ _).obj (tensor X D) ≅ (SourceEvaluation.evaluation X).obj D :=
Iso.refl _
variable [FiniteDimensional k Y]
def finiteIso (D : SourceVerdier.K P) :
(HomotopyCategory.quotient _ _).obj (tensor X D) ≅
(inclusion k).obj ((SourceEvaluation.finite X).obj D) := Iso.refl _
abbrev tensorMap {D E : SourceVerdier.K P}
(f : ConcreteChainRepresentatives.representative P D ⟶
ConcreteChainRepresentatives.representative P E) :=
((BalancedTensor.tensorFunctor k (B P) X.Evaluation).mapHomologicalComplex (.up ℤ)).map
(((UnderlyingRepresentatives.forget P).mapHomologicalComplex (.up ℤ)).map f)
lemma map_comparison {D E : SourceVerdier.K P}
(f : ConcreteChainRepresentatives.representative P D ⟶
ConcreteChainRepresentatives.representative P E)
(a : D ⟶ E)
(ha : (HomotopyCategory.quotient _ _).map f = a.hom) :
(HomotopyCategory.quotient _ _).map (tensorMap X f) ≫ (finiteIso X E).hom =
(finiteIso X D).hom ≫ (inclusion k).map ((SourceEvaluation.finite X).map a) := by
dsimp only [finiteIso,Iso.refl_hom]
erw [Category.comp_id,Category.id_comp]
change _ = ((BalancedTensor.tensorFunctor k (B P) X.Evaluation).mapHomotopyCategory (.up ℤ)).map
(((UnderlyingRepresentatives.forget P).mapHomotopyCategory (.up ℤ)).map a.hom)
erw [← ha,Functor.mapHomotopyCategory_map,Functor.mapHomotopyCategory_map]
end LittleFinitistic.SelectedTensorComparison
end
end OAI
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