| import OAI.Algebra.Finitistic.Localization.QuotientRelations |
| import Mathlib.Algebra.Module.ULift |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.QuadraticEncoding |
| open BalancedTensor FiniteProjectiveHomotopy |
| universe u v |
| variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t} |
| variable {Y : Type v} [AddCommGroup Y] [Module k Y] (X : Representation P Y) |
|
|
| abbrev liftedY := ULift.{u} Y |
|
|
| def liftedVertexEquiv (i : Fin 3) : |
| (tensorFunctor k (B P) X.Evaluation).obj (vertexModule P i) ≃ₗ[k] ULift.{u} Y := |
| (evaluationVertexEquiv X i).trans ULift.moduleEquiv.symm |
|
|
| def actionArrow (h : Fin (d+1)) : ModuleCat.of k (ULift.{u} Y) ⟶ ModuleCat.of k (ULift.{u} Y) := |
| ModuleCat.ofHom (ULift.moduleEquiv.symm.toLinearMap.comp |
| ((Fin.cases (motive := fun _ => Module.End k Y) 1 X.generator h).comp |
| ULift.moduleEquiv.toLinearMap)) |
|
|
| lemma tensor_arrow01_square (h : Fin (d+1)) : |
| (tensorFunctor k (B P) X.Evaluation).map (arrowModule01 P h) ≫ |
| (liftedVertexEquiv X 1).toModuleIso.hom = |
| (liftedVertexEquiv X 0).toModuleIso.hom ≫ actionArrow X h := by |
| ext z |
| have h := evaluation_arrow01 X h (evaluationVertexEquiv X 0 z) |
| simpa [liftedVertexEquiv, actionArrow] using h |
|
|
| lemma tensor_arrow12_square (h : Fin (d+1)) : |
| (tensorFunctor k (B P) X.Evaluation).map (arrowModule12 P h) ≫ |
| (liftedVertexEquiv X 2).toModuleIso.hom = |
| (liftedVertexEquiv X 1).toModuleIso.hom ≫ actionArrow X h := by |
| ext z |
| have h := evaluation_arrow12 X h (evaluationVertexEquiv X 1 z) |
| simpa [liftedVertexEquiv, actionArrow] using h |
|
|
| abbrev vectorStalk := HomotopyCategory.singleFunctor (ModuleCat.{max u v} k) 0 |
|
|
| /-- Evaluation of each concrete degree-zero vertex object is the stalk complex Y. -/ |
| def homotopyVertexIso (i : Fin 3) : |
| (boundedEvaluation X).obj (vertexK P i) ≅ (vectorStalk (k := k)).obj (ModuleCat.of k (ULift.{u} Y)) := |
| (HomotopyCategory.quotient _ _).mapIso |
| ((HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).app (vertexModule P i)) ≪≫ |
| (vectorStalk (k := k)).mapIso (liftedVertexEquiv X i).toModuleIso |
|
|
| lemma homotopy_arrow01_square (h : Fin (d+1)) : |
| (boundedEvaluation X).map (arrowK01 P h) ≫ (homotopyVertexIso X 1).hom = |
| (homotopyVertexIso X 0).hom ≫ (vectorStalk (k := k)).map (actionArrow X h) := by |
| change (homotopyEvaluation k (B P) X.Evaluation).map |
| ((inclusion (B P)ᵐᵒᵖ).map (arrowK01 P h)) ≫ _ = _ |
| unfold arrowK01 |
| erw [Functor.map_preimage] |
| change ((tensorFunctor k (B P) X.Evaluation).mapHomotopyCategory (.up ℤ)).map |
| ((HomotopyCategory.quotient _ _).map ((degreeZero P).map (arrowModule01 P h))) ≫ |
| ((HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.app (vertexModule P 1)) ≫ |
| (HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map (liftedVertexEquiv X 1).toModuleIso.hom)) = |
| ((HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.app (vertexModule P 0)) ≫ |
| (HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map (liftedVertexEquiv X 0).toModuleIso.hom)) ≫ |
| (HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map (actionArrow X h)) |
| simp only [Functor.mapHomotopyCategory_map] |
| have hn := (HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.naturality (arrowModule01 P h) |
| have hm := congrArg (fun f => (HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map f) |
| (tensor_arrow01_square X h) |
| simp only [Functor.map_comp] at hm |
| have hc := (congrArg (fun f => f ≫ |
| (HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map |
| (liftedVertexEquiv X 1).toModuleIso.hom) hn).trans (by |
| simpa only [Category.assoc, Functor.comp_map, degreeZero] using congrArg (fun f => |
| (HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.app (vertexModule P 0) ≫ f) hm) |
| simpa only [Functor.map_comp, Category.assoc, Functor.comp_map, degreeZero] using |
| congrArg (fun f => (HomotopyCategory.quotient (ModuleCat.{max u v} k) (.up ℤ)).map f) hc |
|
|
| lemma homotopy_arrow12_square (h : Fin (d+1)) : |
| (boundedEvaluation X).map (arrowK12 P h) ≫ (homotopyVertexIso X 2).hom = |
| (homotopyVertexIso X 1).hom ≫ (vectorStalk (k := k)).map (actionArrow X h) := by |
| change (homotopyEvaluation k (B P) X.Evaluation).map |
| ((inclusion (B P)ᵐᵒᵖ).map (arrowK12 P h)) ≫ _ = _ |
| unfold arrowK12 |
| erw [Functor.map_preimage] |
| change ((tensorFunctor k (B P) X.Evaluation).mapHomotopyCategory (.up ℤ)).map |
| ((HomotopyCategory.quotient _ _).map ((degreeZero P).map (arrowModule12 P h))) ≫ |
| ((HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.app (vertexModule P 2)) ≫ |
| (HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map (liftedVertexEquiv X 2).toModuleIso.hom)) = |
| ((HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.app (vertexModule P 1)) ≫ |
| (HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map (liftedVertexEquiv X 1).toModuleIso.hom)) ≫ |
| (HomotopyCategory.quotient _ _).map |
| ((HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map (actionArrow X h)) |
| simp only [Functor.mapHomotopyCategory_map] |
| have hn := (HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.naturality (arrowModule12 P h) |
| have hm := congrArg (fun f => (HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map f) |
| (tensor_arrow12_square X h) |
| simp only [Functor.map_comp] at hm |
| have hc := (congrArg (fun f => f ≫ |
| (HomologicalComplex.single (ModuleCat.{max u v} k) (.up ℤ) (0 : ℤ)).map |
| (liftedVertexEquiv X 2).toModuleIso.hom) hn).trans (by |
| simpa only [Category.assoc, Functor.comp_map, degreeZero] using congrArg (fun f => |
| (HomologicalComplex.singleMapHomologicalComplex |
| (tensorFunctor k (B P) X.Evaluation) (.up ℤ) (0 : ℤ)).hom.app (vertexModule P 1) ≫ f) hm) |
| simpa only [Functor.map_comp, Category.assoc, Functor.comp_map, degreeZero] using |
| congrArg (fun f => (HomotopyCategory.quotient (ModuleCat.{max u v} k) (.up ℤ)).map f) hc |
| end LittleFinitistic.QuadraticEncoding |
|
|
| end |
|
|
| end OAI |
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