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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 | import OAI.Algebra.Finitistic.Localization.EvaluationLinear
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.QuadraticEncoding
universe u v w z
section LinearFunctor
variable {k : Type u} [Field k]
variable {C : Type v} [Category.{w} C] [Preadditive C] [Linear k C]
variable {D : Type z} [Category D] [Preadditive D] [Linear k D]
variable (F : C ⥤ D) [F.Additive] [F.Linear k] (L : C) {T : D} (e : F.obj L ≅ T)
/-- Evaluation of endomorphisms, expressed in a specified identification of the target. -/
def identifiedEndHom : End L →ₐ[k] End T where
toFun a := e.inv ≫ F.map a ≫ e.hom
map_zero' := by simp
map_add' a b := by
change e.inv ≫ F.map ((a : L ⟶ L) + b) ≫ e.hom =
(e.inv ≫ F.map a ≫ e.hom) + (e.inv ≫ F.map b ≫ e.hom)
rw [F.map_add, Preadditive.add_comp, Preadditive.comp_add]
map_one' := by simp
map_mul' a b := by simp [End.mul_def, Functor.map_comp, Category.assoc]
commutes' c := by
change e.inv ≫ F.map (c • 𝟙 L) ≫ e.hom = c • 𝟙 T
simp [Linear.comp_smul, Linear.smul_comp]
end LinearFunctor
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)
/-- Canonical transport of every endomorphism of Y to its universe lift. -/
def liftedEndHom : Module.End k Y →ₐ[k] End (ModuleCat.of k (ULift.{u} Y)) where
toFun a := ModuleCat.ofHom (ULift.moduleEquiv.symm.toLinearMap.comp
(a.comp ULift.moduleEquiv.toLinearMap))
map_zero' := by apply ModuleCat.hom_ext; apply LinearMap.ext; intro y; rfl
map_add' a b := by apply ModuleCat.hom_ext; apply LinearMap.ext; intro y; rfl
map_one' := by apply ModuleCat.hom_ext; apply LinearMap.ext; intro y; rfl
map_mul' a b := by apply ModuleCat.hom_ext; apply LinearMap.ext; intro y; rfl
commutes' c := by apply ModuleCat.hom_ext; apply LinearMap.ext; intro y; rfl
/-- The presented algebra acts on Y by precisely its specified generator endomorphisms. -/
def givenAction : PresentedAlgebra P →ₐ[k] Module.End k Y :=
presentationLift P X.generator X.relation
def givenStalkAction : PresentedAlgebra P →ₐ[k]
End ((vectorStalk (k := k)).obj (ModuleCat.of k (ULift.{u} Y))) :=
(identifiedEndHom (vectorStalk (k := k)) _ (Iso.refl _)).comp
((liftedEndHom (k := k) (Y := Y)).comp (givenAction X))
/-- The quotient action, evaluated and identified with the stalk of Y. -/
def evaluatedAction : PresentedAlgebra P →ₐ[k]
End ((vectorStalk (k := k)).obj (ModuleCat.of k (ULift.{u} Y))) :=
(identifiedEndHom (boundedQuotientEvaluation X) (vertexQ P 0) (quotientVertexIso X 0)).comp
(quotientAction P)
lemma evaluatedAction_generator (h : Fin d) :
evaluatedAction X (presentedGenerator P h) =
givenStalkAction X (presentedGenerator P h) := by
change (quotientVertexIso X 0).inv ≫
(boundedQuotientEvaluation X).map (quotientAction P (presentedGenerator P h)) ≫
(quotientVertexIso X 0).hom = _
rw [quotientAction_generator, normalizedGenerator_evaluates]
simp only [← Category.assoc, Iso.inv_hom_id, Category.id_comp]
change (vectorStalk (k := k)).map ((liftedEndHom (k := k) (Y := Y)) (X.generator h)) =
𝟙 _ ≫ (vectorStalk (k := k)).map ((liftedEndHom (k := k) (Y := Y))
(givenAction X (presentedGenerator P h))) ≫ 𝟙 _
rw [Category.id_comp, Category.comp_id]
rw [show givenAction X (presentedGenerator P h) = X.generator h from
presentationLift_generator P X.generator X.relation h]
/-- Section 3, lines 167–168: the entire quotient action evaluates to the given action. -/
theorem quotientAction_evaluates : evaluatedAction X = givenStalkAction X := by
apply RingQuot.ringQuot_ext'
apply FreeAlgebra.hom_ext
funext h
exact evaluatedAction_generator X h
end LittleFinitistic.QuadraticEncoding
end
end OAI
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