| 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 |
|
|
