openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PolygonVerticalStrings.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonStringGroupoid | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonVerticalStrings | |
| open Classical CategoryTheory | |
| namespace PolygonObject | |
| variable {a n m : ℕ} | |
| def verticalStringProperty (a n : ℕ) : ObjectProperty (RawString a n) := | |
| fun F => ∀ (i j : Fin (n+1)) (f : i ⟶ j), Positional (F.map f) | |
| abbrev VerticalString (a n : ℕ) := (verticalStringProperty a n).FullSubcategory | |
| noncomputable def verticalEvaluation (a n : ℕ) : VerticalString a n ⥤ PolygonObject a where | |
| obj F := F.obj.obj 0 | |
| map f := f.hom.app 0 | |
| noncomputable def verticalConstant (a n : ℕ) : PolygonObject a ⥤ VerticalString a n where | |
| obj U := ⟨(Functor.const (Fin (n+1))).obj U,fun _ _ _ => identity_positional U⟩ | |
| map f := ObjectProperty.homMk ((Functor.const (Fin (n+1))).map f) | |
| noncomputable instance verticalEvaluation_full : (verticalEvaluation a n).Full where | |
| map_surjective h := ⟨ObjectProperty.homMk (extendLadder h),extendLadder_zero h⟩ | |
| instance verticalEvaluation_faithful : (verticalEvaluation a n).Faithful where | |
| map_injective h := by | |
| apply ObjectProperty.hom_ext | |
| exact ladder_ext h | |
| instance verticalEvaluation_essSurj : (verticalEvaluation a n).EssSurj where | |
| mem_essImage U := ⟨(verticalConstant a n).obj U,⟨Iso.refl U⟩⟩ | |
| instance verticalEvaluation_isEquivalence : (verticalEvaluation a n).IsEquivalence where | |
| noncomputable def verticalEquivalence (a n : ℕ) : VerticalString a n ≌ PolygonObject a := | |
| (verticalEvaluation a n).asEquivalence | |
| noncomputable def verticalReindex (u : Fin (n+1) ⥤ Fin (m+1)) : | |
| VerticalString a m ⥤ VerticalString a n where | |
| obj F := ⟨u⋙F.obj,fun i j f => F.property (u.obj i) (u.obj j) (u.map f)⟩ | |
| map f := ObjectProperty.homMk (Functor.whiskerLeft u f.hom) | |
| map_id F := by apply ObjectProperty.hom_ext; rfl | |
| map_comp f g := by apply ObjectProperty.hom_ext; rfl | |
| @[simp] theorem verticalConstant_evaluation (a n : ℕ) : | |
| verticalConstant a n ⋙ verticalEvaluation a n=𝟭 (PolygonObject a) := rfl | |
| instance verticalConstant_isEquivalence : (verticalConstant a n).IsEquivalence := by | |
| have : (verticalConstant a n ⋙ verticalEvaluation a n).IsEquivalence := by | |
| rw [verticalConstant_evaluation] | |
| infer_instance | |
| exact CategoryTheory.Functor.isEquivalence_of_comp_right | |
| (verticalConstant a n) (verticalEvaluation a n) | |
| theorem verticalConstant_reindex (u : Fin (n+1) ⥤ Fin (m+1)) : | |
| verticalConstant a m ⋙ verticalReindex u=verticalConstant a n := by | |
| apply CategoryTheory.Functor.ext | |
| · intro U V f; rfl | |
| · intro U; rfl | |
| end PolygonObject | |
| end PolygonVerticalStrings | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |