| import OAI.Algebra.Finitistic.Localization.SourceCornerEvaluation |
| import OAI.Algebra.Finitistic.Selection.SelectionIteration |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.SelectedImageComparison |
| open QuadraticEncoding SourceImageEvaluation |
| universe v |
| variable {Y : Type v} [AddCommGroup Y] [Module ℂ Y] |
| [Module SelectionRing Y] [IsScalarTower ℂ SelectionRing Y] |
|
|
| /-- The evaluator's range is canonically the primary's actual H(Y), as a complex vector space. -/ |
| def equiv : imageSpace (k := ℂ) (Y := Y) (selectionIdempotent 0) ≃ₗ[ℂ] |
| complexObject (selectionStep (ModuleCat.of SelectionRing Y)) where |
| toFun x := ⟨x.val.down, by |
| obtain ⟨z,hz⟩ := x.property |
| have hz' : selectionIdempotent 0 • z.down = x.val.down := congrArg ULift.down hz |
| change selectionIdempotent 0 • x.val.down = x.val.down |
| rw [← hz',← mul_smul,selectionIdempotent_square]⟩ |
| invFun x := ⟨ULift.up x.val, ⟨ULift.up x.val, by |
| change ULift.up (selectionIdempotent 0 • x.val) = ULift.up x.val |
| exact congrArg ULift.up x.property⟩⟩ |
| left_inv x := by apply Subtype.ext; rfl |
| right_inv x := by rfl |
| map_add' x y := rfl |
| map_smul' c x := by |
| apply Subtype.ext |
| change c • x.val.down = selectionAutomorphism 1 (algebraMap ℂ SelectionRing c) • x.val.down |
| rw [AlgEquiv.commutes,algebraMap_smul] |
|
|
| abbrev moduleAction (r : SelectionRing) (x : selectionStep (ModuleCat.of SelectionRing Y)) : |
| selectionStep (ModuleCat.of SelectionRing Y) := r • x |
|
|
| /-- The evaluator's compressed alpha action is precisely the action on the primary's H(Y). -/ |
| theorem equiv_action (r : SelectionRing) |
| (x : imageSpace (k := ℂ) (Y := Y) (selectionIdempotent 0)) : |
| equiv (Y := Y) ((imageCornerEnd (action (k := ℂ) (Y := Y) (selectionIdempotent 0)).hom |
| (action (k := ℂ) (Y := Y) (selectionAutomorphism 1 r)).hom) x) = |
| moduleAction (Y := Y) r (equiv (Y := Y) x) := by |
| apply Subtype.ext |
| have h := imageCorner_restriction (k := ℂ) (Y := Y) (selectionIdempotent 0) |
| (selectionAutomorphism 1 r) (selectionIdempotent_square 0) |
| (selectionIdempotent_central 0 (selectionAutomorphism 1 r)).eq x |
| exact congrArg ULift.down h |
|
|
| theorem image_nontrivial_of (h : Nontrivial (selectionStep (ModuleCat.of SelectionRing Y))) : |
| Nontrivial (imageSpace (k := ℂ) (Y := Y) (selectionIdempotent 0)) := by |
| let i := equiv (Y := Y) |
| obtain ⟨x,y,hxy⟩ := h.exists_pair_ne |
| exact ⟨⟨i.symm x,i.symm y,fun h => hxy (i.symm.injective h)⟩⟩ |
|
|
| theorem image_subsingleton_of (h : Subsingleton (selectionStep (ModuleCat.of SelectionRing Y))) : |
| Subsingleton (imageSpace (k := ℂ) (Y := Y) (selectionIdempotent 0)) := |
| ⟨fun first second => (equiv (Y := Y)).injective |
| (h.elim (equiv (Y := Y) first) (equiv (Y := Y) second))⟩ |
| end LittleFinitistic.SelectedImageComparison |
|
|
| end |
|
|
| end OAI |
|
|
