| 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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|
