File size: 2,392 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 | import OAI.Algebra.Finitistic.Bar.DerivedCountShift
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 integerCountScalarModule (V : ModuleCat D) : Module k V :=
BimoduleTensor.scalarModule k D V
local instance integerCountScalarTower (V : ModuleCat D) : IsScalarTower k D V :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
local instance integerCountIndexDecidableEq (r n : ℕ) : DecidableEq (Index r n) :=
Classical.decEq _
local instance integerCountShapeDecidableEq (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 integerCountInflatedModule : Module A N := inflatedModule D X N
local instance integerCountInflatedScalarTower : 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)
/-- The original r-count object, with exactly the original scalar and inflated actions. -/
abbrev countObject (r : ℕ) :=
(integerQ D).obj (SquareZero.twoSidedWeightComplex k D X N (inflated_ideal_zero D X N) r)
local instance integerCountHasDerivedCategory : HasDerivedCategory.{u+1} (ModuleCat.{u} D) :=
HasDerivedCategory.standard (ModuleCat.{u} D)
/-- The proved genuine shift comparison, with its target named for downstream use. -/
def countObjectComparison (r : ℕ) :
((chainToIntegerDerived D).obj
((derivedTensorIterate k D X r).obj ((chainQ D).obj Z)))⟦(r:ℤ)⟧ ≅ countObject k D X N r :=
derivedIterateWeightShiftIso k D X N r
end DerivedBar.ParityBridge
end
end OAI
|