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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 | import OAI.Algebra.Finitistic.Bar.IntegerCountShift
namespace OAI
open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt
open scoped TensorProduct DirectSum
noncomputable section
namespace DerivedBar.ParityBridge
open DerivedBar.Multibar DerivedBar.RecursiveTotal
universe u
variable (k D X : Type u) [Field k] [Ring D] [Algebra k D]
[AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X]
local instance (V : ModuleCat D) : Module k V := BimoduleTensor.scalarModule k D V
local instance (V : ModuleCat D) : IsScalarTower k D V :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
local instance derivedCountShiftIndexDecidableEq (r n : ℕ) :
DecidableEq (Index r n) := Classical.decEq _
local instance derivedCountShiftShapeDecidableEq (n r : ℕ) :
DecidableEq (Multibar.Indexed.Shape n r) := Classical.decEq _
variable (N : ModuleCat D)
local notation "A" => TrivSqZeroExt D X
local notation "XB" => BimoduleTensor.twoSidedBarFunctor k D X
local notation "Z" => Functor.obj (ChainComplex.single₀ (ModuleCat D)) N
local notation "B" => (fun _ : ℕ => XB)
local notation "c" => ComplexShape.down ℕ
local instance : Module A N := inflatedModule D X N
local instance : IsScalarTower k A N := IsScalarTower.of_algebraMap_smul
(fun a v => IsScalarTower.algebraMap_smul D a v)
local notation "C" => ModuleCat D
local notation "E" => ComplexShape.embeddingDownNat
local notation "K" => Multibar.Indexed.signedBlockComplex k D X N (inflated_ideal_zero D X N)
/-- Identification with the original count subcomplex, using the actual integer
cochain shift and the already verified component sign involution. -/
def integerIndexedWeightShiftIso (r : ℕ) :
((integerExtension D).obj (indexedMultibar k D X N r))⟦(r:ℤ)⟧ ≅
(integerExtension D).obj (SquareZero.twoSidedWeightComplex k D X N (inflated_ideal_zero D X N) r) :=
integerIndexedCountShiftIso k D X N r ≪≫
(integerExtension D).mapIso (Multibar.Indexed.signedBlockIso k D X N (inflated_ideal_zero D X N) r)
local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) :=
HasDerivedCategory.standard (ModuleCat.{u} D)
/-- Each original count summand is the genuine [r] shift of the constructed
finite derived tensor iterate. -/
def derivedIterateWeightShiftIso (r : ℕ) :
((chainToIntegerDerived D).obj
((derivedTensorIterate k D X r).obj ((chainQ D).obj Z)))⟦(r:ℤ)⟧ ≅
(integerQ D).obj (SquareZero.twoSidedWeightComplex k D X N (inflated_ideal_zero D X N) r) :=
(CategoryTheory.shiftFunctor (DerivedCategory (ModuleCat.{u} D)) (r:ℤ)).mapIso
(integerDerivedIterateIndexedIso D k X N r) ≪≫
((DerivedCategory.Q.commShiftIso (r:ℤ)).symm.app
((integerExtension D).obj (indexedMultibar k D X N r))) ≪≫
DerivedCategory.Q.mapIso (integerIndexedWeightShiftIso k D X N r)
end DerivedBar.ParityBridge
end
end OAI
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