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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.Algebra.Finitistic.Localization.CompletedImageEvaluation
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Idempotents
attribute [local instance] HasDerivedCategory.standard
namespace LittleFinitistic.QuadraticEncoding
universe u v w z
section General
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable {D : Type w} [Category.{z} D] [Preadditive D] [IsIdempotentComplete D]
variable (F : C ⥤ D) [F.Additive] (U : Karoubi C) (I : D)
variable (π : F.obj U.X ⟶ I) (ι : I ⟶ F.obj U.X)
variable (hπι : π ≫ ι = F.map U.p) (hιπ : ι ≫ π = 𝟙 I)
omit [Preadditive C] [Preadditive D] [F.Additive] in
theorem completedExtensionImageSquare (a : U ⟶ U) (b : I ⟶ I)
(h : F.map a.f ≫ π = π ≫ b) :
((functorExtension C D).obj F).map a ≫
(completedExtensionImageIso F U I π ι hπι hιπ).hom =
(completedExtensionImageIso F U I π ι hπι hιπ).hom ≫ b := by
let e := splitKaroubiIso (F.map U.p) (by rw [← F.map_comp, U.idem]) π ι hπι hιπ
have he : ((functorExtension₂ C D).obj F).map a ≫ e.hom =
e.hom ≫ (toKaroubi D).map b := by
apply Karoubi.Hom.ext
exact h
change (toKaroubiEquivalence D).inverse.map (((functorExtension₂ C D).obj F).map a) ≫
((toKaroubiEquivalence D).inverse.map e.hom ≫ (toKaroubiEquivalence D).unitIso.inv.app I) =
((toKaroubiEquivalence D).inverse.map e.hom ≫ (toKaroubiEquivalence D).unitIso.inv.app I) ≫ b
erw [← Category.assoc, ← Functor.map_comp, he, Functor.map_comp, Category.assoc]
erw [(toKaroubiEquivalence D).unitIso.inv.naturality b]
exact (Category.assoc _ _ _).symm
end General
section Image
variable {k : Type u} [Field k] {V : Type v} [AddCommGroup V] [Module k V]
variable (p a : Module.End k V) (hp : p*p=p)
def imageCornerEnd : ModuleCat.of k (LinearMap.range p) ⟶ ModuleCat.of k (LinearMap.range p) :=
imageInclusion p ≫ ModuleCat.ofHom a ≫ imageProjection p
include hp in
lemma image_corner_square :
ModuleCat.ofHom (p*a*p) ≫ imageProjection p = imageProjection p ≫ imageCornerEnd p a := by
change ((ModuleCat.ofHom p ≫ ModuleCat.ofHom a) ≫ ModuleCat.ofHom p) ≫ imageProjection p = _
rw [← image_projection_inclusion p]
simp only [imageCornerEnd, Category.assoc, image_inclusion_projection p hp, Category.comp_id]
include hp in
/-- When a commutes with p, the corner operator is the restriction of a to the image. -/
lemma imageCornerEnd_apply (ha : p*a=a*p) (x : LinearMap.range p) :
((imageCornerEnd p a) x).val = a x.val := by
change p (a x.val) = a x.val
have h := LinearMap.congr_fun ha x.val
change p (a x.val) = a (p x.val) at h
rw [h]
obtain ⟨y,hy⟩ := x.property
have hp' : p x.val = x.val := by rw [← hy]; exact LinearMap.congr_fun hp y
rw [hp']
end Image
variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t}
variable {Y : Type v} [AddCommGroup Y] [Module k Y] [FiniteDimensional k Y]
variable (X : Representation P Y) (e a : PresentedAlgebra P) (he : e*e=e)
def quotientCornerEnd : quotientSummand e he ⟶ quotientSummand e he :=
⟨quotientAction P (e*a*e), by
change (quotientAction P e * quotientAction P (e*a*e)) * quotientAction P e = _
simp only [← map_mul]
congr 1
simp only [← mul_assoc, he]
rw [mul_assoc (e*a) e e, he]⟩
def evaluatedCornerEnd : evaluatedImageDerived X e ⟶ evaluatedImageDerived X e :=
(derivedTargetInclusion (k := k)).preimage (DerivedCategory.Qh.map
((vectorStalk (k := k)).map (imageCornerEnd
(liftedEndHom (givenAction X e)).hom (liftedEndHom (givenAction X a)).hom)))
include he in
lemma derived_corner_projection_square :
(derivedQuotientEvaluation X ⋙ derivedTargetInclusion (k := k)).map
(quotientAction P (e*a*e)) ≫ derivedImageProjection X e =
derivedImageProjection X e ≫ (derivedTargetInclusion (k := k)).map (evaluatedCornerEnd X e a) := by
unfold evaluatedCornerEnd
erw [Functor.map_preimage]
unfold derivedImageProjection
erw [← Category.assoc, derived_quotientAction_square, Category.assoc]
erw [← Functor.map_comp, ← Functor.map_comp]
have h := image_corner_square (liftedEndHom (givenAction X e)).hom
(liftedEndHom (givenAction X a)).hom (lifted_action_idempotent X e he)
have hc : liftedEndHom (givenAction X (e*a*e)) = ModuleCat.ofHom
((liftedEndHom (givenAction X e)).hom * (liftedEndHom (givenAction X a)).hom *
(liftedEndHom (givenAction X e)).hom) := by
simp only [map_mul]
rfl
erw [hc, h, Functor.map_comp, Functor.map_comp, ← Category.assoc]
rfl
/-- The complete derived evaluation intertwines every corner endomorphism with
its concrete action on eY; this includes e alpha(r) e from Section 3. -/
theorem completed_corner_square :
(completedDerivedEvaluation X).map (quotientCornerEnd e a he) ≫
(completedSummandImageIso X e he).hom =
(completedSummandImageIso X e he).hom ≫ evaluatedCornerEnd X e a := by
apply completedExtensionImageSquare
apply (derivedTargetInclusion (k := k)).map_injective
rw [Functor.map_comp, Functor.map_comp]
erw [Functor.map_preimage]
exact derived_corner_projection_square X e a he
omit [FiniteDimensional k Y] in
include he in
/-- For a central e and any algebra endomorphism alpha, the concrete map on eY
is the restriction of alpha(r), exactly as in the manuscript. -/
theorem central_alpha_restriction (hec : ∀ b : PresentedAlgebra P, e*b=b*e)
(α : PresentedAlgebra P →ₐ[k] PresentedAlgebra P) (b : PresentedAlgebra P)
(y : evaluatedImage X e) :
((imageCornerEnd (liftedEndHom (givenAction X e)).hom
(liftedEndHom (givenAction X (α b))).hom) y).val =
(liftedEndHom (givenAction X (α b))).hom y.val := by
apply imageCornerEnd_apply _ _ (lifted_action_idempotent X e he)
have h := congrArg (fun c => (liftedEndHom (givenAction X c)).hom) (hec (α b))
simp only [map_mul] at h
exact h
end LittleFinitistic.QuadraticEncoding
end
end OAI
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