| import OAI.Algebra.Finitistic.Selected.SelectedTensorComparison |
| import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor |
| import Mathlib.Algebra.Homology.DerivedCategory.TStructure |
| import Mathlib.Algebra.Homology.DerivedCategory.FullyFaithful |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.SelectedTargetHomology |
| open QuadraticEncoding SelectedNativeEvaluation FiniteProjectiveHomotopy |
| universe v |
| variable {Y : Type v} [AddCommGroup Y] [Module ℂ Y] [FiniteDimensional ℂ Y] |
| [Module SelectionRing Y] [IsScalarTower ℂ SelectionRing Y] |
| abbrev C := ModuleCat.{v} ℂ |
| abbrev I := derivedTargetInclusion.{0,v} (k := ℂ) |
| abbrev cohomology (n : ℤ) := I ⋙ DerivedCategory.homologyFunctor C n |
| instance (n : ℤ) : (cohomology.{v} n).Additive := by dsimp [cohomology]; infer_instance |
| instance (n : ℤ) : PreservesBinaryBiproducts (cohomology.{v} n) := |
| preservesBinaryBiproducts_of_preservesBiproducts _ |
| |
| def stalkIso : I.obj (HDerived (Y := Y)) ≅ |
| (DerivedCategory.singleFunctor C 0).obj (H (Y := Y)) := Iso.refl _ |
| def shiftedIso : I.obj ((HDerived (Y := Y))⟦(3 : ℤ)⟧) ≅ |
| (DerivedCategory.singleFunctor C (-3)).obj (H (Y := Y)) := |
| (I.commShiftIso (3 : ℤ)).app _ ≪≫ |
| (shiftFunctor _ (3 : ℤ)).mapIso (stalkIso (Y := Y)) ≪≫ |
| ((DerivedCategory.singleFunctors C).shiftIso 3 (-3) 0 (by norm_num)).app _ |
| |
| def stalkHomologyIso : (cohomology 0).obj (HDerived (Y := Y)) ≅ H (Y := Y) := |
| (DerivedCategory.homologyFunctor C 0).mapIso stalkIso ≪≫ |
| (DerivedCategory.singleFunctorCompHomologyFunctorIso C 0).app _ |
| def shiftedHomologyIso : (cohomology (-3)).obj ((HDerived (Y := Y))⟦(3 : ℤ)⟧) ≅ H (Y := Y) := |
| (DerivedCategory.homologyFunctor C (-3)).mapIso shiftedIso ≪≫ |
| (DerivedCategory.singleFunctorCompHomologyFunctorIso C (-3)).app _ |
| |
| lemma single_zero (M : C) (i n : ℤ) (hn : n ≠ i) : |
| IsZero ((DerivedCategory.homologyFunctor C n).obj |
| ((DerivedCategory.singleFunctor C i).obj M)) := by |
| rcases lt_or_gt_of_ne hn with h | h |
| · exact DerivedCategory.isZero_of_isGE _ i n h |
| · exact DerivedCategory.isZero_of_isLE _ i n h |
| lemma stalk_zero (n : ℤ) (hn : n ≠ 0) : |
| IsZero ((cohomology n).obj (HDerived (Y := Y))) := |
| ((DerivedCategory.homologyFunctor C n).mapIso (stalkIso (Y := Y))).isZero_iff.mpr |
| (single_zero _ _ _ hn) |
| lemma shifted_zero (n : ℤ) (hn : n ≠ -3) : |
| IsZero ((cohomology n).obj ((HDerived (Y := Y))⟦(3 : ℤ)⟧)) := |
| ((DerivedCategory.homologyFunctor C n).mapIso (shiftedIso (Y := Y))).isZero_iff.mpr |
| (single_zero _ _ _ hn) |
| |
| def targetH0 : (cohomology 0).obj (target (Y := Y)) ≅ H (Y := Y) := |
| (cohomology 0).mapBiprod _ _ ≪≫ |
| (isoBiprodZero (shifted_zero (Y := Y) 0 (by norm_num))).symm ≪≫ stalkHomologyIso |
| |
| def targetHm3 : (cohomology (-3)).obj (target (Y := Y)) ≅ H (Y := Y) := |
| (cohomology (-3)).mapBiprod _ _ ≪≫ |
| (isoZeroBiprod (stalk_zero (Y := Y) (-3) (by norm_num))).symm ≪≫ shiftedHomologyIso |
| |
| lemma target_zero (n : ℤ) (h0 : n ≠ 0) (hm : n ≠ -3) : |
| IsZero ((cohomology n).obj (target (Y := Y))) := |
| ((cohomology n).mapBiprod _ _).isZero_iff.mpr |
| ((biprod_isZero_iff _ _).mpr ⟨stalk_zero n h0,shifted_zero n hm⟩) |
| |
| lemma stalkHomologyIso_hom : (stalkHomologyIso (Y := Y)).hom = |
| ((DerivedCategory.singleFunctorCompHomologyFunctorIso C 0).app (H (Y := Y))).hom := by |
| change (DerivedCategory.homologyFunctor C 0).map |
| (𝟙 ((DerivedCategory.singleFunctor C 0).obj (H (Y := Y)))) ≫ _ = _ |
| rw [CategoryTheory.Functor.map_id,Category.id_comp] |
| |
| lemma stalk_action (r : SelectionRing) : |
| (cohomology 0).map (HDerivedAction (Y := Y) r) ≫ (stalkHomologyIso (Y := Y)).hom = |
| (stalkHomologyIso (Y := Y)).hom ≫ HAction (Y := Y) r := by |
| rw [stalkHomologyIso_hom] |
| change (DerivedCategory.homologyFunctor C 0).map |
| (DerivedCategory.Qh.map ((inclusion ℂ).map (HKAction (Y := Y) r))) ≫ _ = _ |
| dsimp only [HKAction] |
| erw [Functor.map_preimage] |
| change (DerivedCategory.homologyFunctor C 0).map |
| ((DerivedCategory.singleFunctor C 0).map (HAction (Y := Y) r)) ≫ _ = _ |
| exact (DerivedCategory.singleFunctorCompHomologyFunctorIso C 0).hom.naturality |
| (HAction (Y := Y) r) |
| |
| lemma shifted_action (r : SelectionRing) : |
| (cohomology (-3)).map ((HDerivedAction (Y := Y) r)⟦(3 : ℤ)⟧') ≫ |
| (shiftedHomologyIso (Y := Y)).hom = |
| (shiftedHomologyIso (Y := Y)).hom ≫ HAction (Y := Y) r := by |
| have hst : I.map (HDerivedAction (Y := Y) r) = |
| (DerivedCategory.singleFunctor C 0).map (HAction (Y := Y) r) := by |
| change DerivedCategory.Qh.map ((inclusion ℂ).map (HKAction (Y := Y) r)) = _ |
| dsimp only [HKAction] |
| erw [Functor.map_preimage] |
| rfl |
| have hshift : I.map ((HDerivedAction (Y := Y) r)⟦(3 : ℤ)⟧') ≫ (shiftedIso (Y := Y)).hom = |
| (shiftedIso (Y := Y)).hom ≫ |
| (DerivedCategory.singleFunctor C (-3)).map (HAction (Y := Y) r) := by |
| have h1 := (I.commShiftIso (3 : ℤ)).hom.naturality (HDerivedAction (Y := Y) r) |
| have h2 := ((DerivedCategory.singleFunctors C).shiftIso 3 (-3) 0 (by norm_num)).hom.naturality |
| (HAction (Y := Y) r) |
| dsimp only [Functor.comp_map] at h1 h2 |
| rw [hst] at h1 |
| have hh := (congrArg (fun f => f ≫ |
| (((DerivedCategory.singleFunctors C).shiftIso 3 (-3) 0 (by norm_num)).app |
| (H (Y := Y))).hom) h1).trans |
| ((Category.assoc _ _ _).trans |
| (congrArg (fun g => (I.commShiftIso (3 : ℤ)).hom.app (HDerived (Y := Y)) ≫ g) h2)) |
| dsimp only [shiftedIso, stalkIso, Iso.trans_hom, Functor.mapIso_hom, Iso.refl_hom] |
| erw [CategoryTheory.Functor.mapIso_hom, Iso.refl_hom, |
| CategoryTheory.Functor.map_id, Category.id_comp] |
| exact (Category.assoc _ _ _).symm.trans (hh.trans (Category.assoc _ _ _).symm) |
| |
| have h := congrArg (DerivedCategory.homologyFunctor C (-3)).map hshift |
| simp only [Functor.map_comp] at h |
| have hn := (DerivedCategory.singleFunctorCompHomologyFunctorIso C (-3)).hom.naturality |
| (HAction (Y := Y) r) |
| dsimp only [Functor.comp_map,Functor.id_map] at hn |
| dsimp only [shiftedHomologyIso,Iso.trans_hom,Functor.mapIso_hom,cohomology,Functor.comp_map] |
| rw [← Category.assoc,h,Category.assoc] |
| exact (congrArg (fun g => (DerivedCategory.homologyFunctor C (-3)).map |
| (shiftedIso (Y := Y)).hom ≫ g) hn).trans (Category.assoc _ _ _).symm |
| |
| lemma targetH0_hom : (targetH0 (Y := Y)).hom = |
| (cohomology 0).map biprod.fst ≫ (stalkHomologyIso (Y := Y)).hom := by |
| simp [targetH0,isoBiprodZero] |
| erw [biprod.lift_fst_assoc] |
| lemma targetHm3_hom : (targetHm3 (Y := Y)).hom = |
| (cohomology (-3)).map biprod.snd ≫ (shiftedHomologyIso (Y := Y)).hom := by |
| simp [targetHm3,isoZeroBiprod] |
| erw [biprod.lift_snd_assoc] |
| |
| lemma targetH0_action (r : SelectionRing) : |
| (cohomology 0).map (targetAction (Y := Y) r) ≫ (targetH0 (Y := Y)).hom = |
| (targetH0 (Y := Y)).hom ≫ HAction (Y := Y) r := by |
| rw [targetH0_hom,← Category.assoc,← Functor.map_comp] |
| simp only [targetAction,biprod.map_fst,Functor.map_comp,Category.assoc,stalk_action] |
| lemma targetHm3_action (r : SelectionRing) : |
| (cohomology (-3)).map (targetAction (Y := Y) r) ≫ (targetHm3 (Y := Y)).hom = |
| (targetHm3 (Y := Y)).hom ≫ HAction (Y := Y) r := by |
| rw [targetHm3_hom,← Category.assoc,← Functor.map_comp] |
| simp only [targetAction,biprod.map_snd,Functor.map_comp,Category.assoc,shifted_action] |
| end LittleFinitistic.SelectedTargetHomology |
| |
| end |
| |
| end OAI |
| |
