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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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | 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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