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ae00908 | 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 | 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
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