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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 | import OAI.Algebra.Finitistic.Bar.IntegerDerivedMultibar
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.IntegerExtensionFunctor
universe u
variable {B : Type u} [Ring B]
variable (F : ModuleCat.{u} B ⥤ ModuleCat.{u} B) [F.Additive]
variable (Z : ChainComplex (ModuleCat.{u} B) ℕ)
local notation "c" => ComplexShape.down ℕ
local notation "e" => ComplexShape.embeddingDownNat
/-- Additive functors preserve the zero components added by integer extension. -/
def termIso : (o : Option ℕ) → F.obj (HomologicalComplex.extend.X Z o) ≅
HomologicalComplex.extend.X ((F.mapHomologicalComplex c).obj Z) o
| none => (F.map_isZero (isZero_zero _)).iso (isZero_zero _)
| some _ => Iso.refl _
theorem termIso_d (o o' : Option ℕ) :
(termIso F Z o).hom ≫ HomologicalComplex.extend.d ((F.mapHomologicalComplex c).obj Z) o o' =
F.map (HomologicalComplex.extend.d Z o o') ≫ (termIso F Z o').hom := by
cases o <;> cases o' <;>
simp [termIso,HomologicalComplex.extend.d,HomologicalComplex.extend.X] <;>
erw [Functor.map_zero] <;> (try erw [comp_zero]) <;> (try erw [zero_comp])
/-- Applying an additive module functor commutes with the actual n ↦ -n zero extension. -/
def complexIso : (F.mapHomologicalComplex (.up ℤ)).obj ((DerivedBar.integerExtension B).obj Z) ≅
(DerivedBar.integerExtension B).obj ((F.mapHomologicalComplex c).obj Z) :=
HomologicalComplex.Hom.isoOfComponents (fun n => termIso F Z (ComplexShape.embeddingDownNat.r n))
(by intro n m h; exact termIso_d F Z (ComplexShape.embeddingDownNat.r n) (ComplexShape.embeddingDownNat.r m))
variable {Z} {Z' : ChainComplex (ModuleCat.{u} B) ℕ}
theorem termIso_natural (f : Z ⟶ Z') (o : Option ℕ) :
F.map (HomologicalComplex.extend.mapX f o) ≫ (termIso F Z' o).hom =
(termIso F Z o).hom ≫
HomologicalComplex.extend.mapX ((F.mapHomologicalComplex c).map f) o := by
cases o <;> simp [termIso,HomologicalComplex.extend.mapX,HomologicalComplex.extend.X]
erw [Functor.map_zero,comp_zero,zero_comp]
/-- Naturality holds on every chain map, including the added zero degrees. -/
def naturalIso : DerivedBar.integerExtension B ⋙ F.mapHomologicalComplex (.up ℤ) ≅
F.mapHomologicalComplex c ⋙ DerivedBar.integerExtension B :=
NatIso.ofComponents (complexIso F) (by
intro Z Z' f
apply HomologicalComplex.Hom.ext
funext n
exact termIso_natural F f (ComplexShape.embeddingDownNat.r n))
end LittleFinitistic.IntegerExtensionFunctor
end
end OAI
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