File size: 4,069 Bytes
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 | import OAI.Algebra.Finitistic.Localization.CompletedDerivedEvaluation
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Idempotents
attribute [local instance] HasDerivedCategory.standard
namespace LittleFinitistic.QuadraticEncoding
open FiniteProjectiveHomotopy
universe u v
private theorem sandwich_cancel {C : Type*} [Category C] {A B I : C}
(e : A ≅ B) (p : B ⟶ I) (i : I ⟶ B) :
((e.hom ≫ p) ≫ (i ≫ e.inv)) ≫ e.hom = e.hom ≫ (p ≫ i) := by
simp only [Category.assoc, e.inv_hom_id, Category.comp_id]
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) (r : PresentedAlgebra P) (hr : r*r=r)
/-- The image vector space, as an actual bounded finite-dimensional stalk complex. -/
def evaluatedImageK : K.{u,max u v} k :=
⟨(vectorStalk (k := k)).obj (ModuleCat.of k (evaluatedImage X r)),
single_property (ModuleCat.of k (evaluatedImage X r)) 0⟩
def evaluatedImageDerived : FiniteVectorDerived.{u,v} (k := k) :=
(derivedEmbedding k).toEssImage.obj (evaluatedImageK X r)
abbrev derivedTargetInclusion := (derivedEmbedding.{u,max u v} k).essImage.ι
def derivedImageProjection :
(derivedTargetInclusion (k := k)).obj ((derivedQuotientEvaluation X).obj (vertexQ P 0)) ⟶
(derivedTargetInclusion (k := k)).obj (evaluatedImageDerived X r) :=
(derivedVertexIso X 0).hom ≫ DerivedCategory.Qh.map
((vectorStalk (k := k)).map (imageProjection (liftedEndHom (givenAction X r)).hom))
def derivedImageInclusion :
(derivedTargetInclusion (k := k)).obj (evaluatedImageDerived X r) ⟶
(derivedTargetInclusion (k := k)).obj ((derivedQuotientEvaluation X).obj (vertexQ P 0)) :=
DerivedCategory.Qh.map
((vectorStalk (k := k)).map (imageInclusion (liftedEndHom (givenAction X r)).hom)) ≫
(derivedVertexIso X 0).inv
include hr in
lemma derived_image_inclusion_projection :
derivedImageInclusion X r ≫ derivedImageProjection X r = 𝟙 _ := by
dsimp [derivedImageInclusion, derivedImageProjection]
erw [Category.assoc, Iso.inv_hom_id_assoc]
erw [← Functor.map_comp, ← Functor.map_comp,
image_inclusion_projection _ (lifted_action_idempotent X r hr)]
erw [CategoryTheory.Functor.map_id, CategoryTheory.Functor.map_id]
rfl
lemma derived_image_projection_inclusion :
derivedImageProjection X r ≫ derivedImageInclusion X r =
(derivedQuotientEvaluation X ⋙ derivedTargetInclusion (k := k)).map (quotientAction P r) := by
apply (cancel_mono (derivedVertexIso X 0).hom).mp
dsimp [derivedImageProjection, derivedImageInclusion]
erw [sandwich_cancel (derivedVertexIso X 0)
(DerivedCategory.Qh.map ((vectorStalk (k := k)).map
(imageProjection (liftedEndHom (givenAction X r)).hom)))
(DerivedCategory.Qh.map ((vectorStalk (k := k)).map
(imageInclusion (liftedEndHom (givenAction X r)).hom)))]
erw [← Functor.map_comp, ← Functor.map_comp, image_projection_inclusion]
exact (derived_quotientAction_square X r).symm
/-- The full completed derived evaluation takes U to the actual finite-dimensional eY stalk. -/
def completedSummandImageIso :
(completedDerivedEvaluation X).obj (quotientSummand r hr) ≅ evaluatedImageDerived X r :=
completedExtensionImageIso (derivedQuotientEvaluation X) (quotientSummand r hr)
(evaluatedImageDerived X r)
((derivedTargetInclusion (k := k)).preimage (derivedImageProjection X r))
((derivedTargetInclusion (k := k)).preimage (derivedImageInclusion X r))
(by
apply (derivedTargetInclusion (k := k)).map_injective
rw [Functor.map_comp]
erw [Functor.map_preimage, Functor.map_preimage]
exact derived_image_projection_inclusion X r)
(by
apply (derivedTargetInclusion (k := k)).map_injective
erw [Functor.map_comp, CategoryTheory.Functor.map_id]
erw [Functor.map_preimage, Functor.map_preimage]
exact derived_image_inclusion_projection X r hr)
end LittleFinitistic.QuadraticEncoding
end
end OAI
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