| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Homology.LabelRelabel |
| import OAI.GroupTheory.SimpleAmenable.Configurations.CellFunctor |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.CellMonoidal |
|
|
| namespace OAI |
|
|
| section |
| open CategoryTheory MonoidalCategory |
| namespace SimpleAmenable.PolygonObject.Labelled.Cell |
|
|
| variable {a n m : ℕ} |
| variable (f : (Fin n → CutRing × CutRing) → (Fin m → CutRing × CutRing)) |
| (hf : ∀g, Reduced g → Reduced (f g)) |
| variable (P : polygonAlgebra a) (l : LabelledStage.ReducedLabel n) |
| noncomputable def relabelIso : |
| functor P l ⋙ relabel f hf ≅ functor P ⟨f l.val,hf l.val l.property⟩ := |
| NatIso.ofComponents (fun U => asIso (X := (relabel f hf).obj (object P l U)) |
| (Y := object P ⟨f l.val,hf l.val l.property⟩ U) |
| { arrow := 𝟙 _ |
| positional _ := rfl |
| labelled _ := rfl }) (by |
| intro U V g |
| apply Hom.ext |
| change _ ≫ 𝟙 _ = 𝟙 _ ≫ _ |
| simp only [Category.comp_id] |
| rfl) |
| noncomputable instance relabelIsoMonoidal : NatTrans.IsMonoidal (relabelIso f hf P l).hom where |
| unit := by |
| apply Hom.ext |
| change (𝟙 _ ≫ (unitArrow P l).arrow) ≫ 𝟙 _ = (unitArrow P ⟨f l.val,hf l.val l.property⟩).arrow |
| simp only [Category.id_comp,Category.comp_id] |
| rfl |
| tensor U V := by |
| apply Hom.ext |
| change (𝟙 _ ≫ (tensorArrow P l U V).arrow) ≫ 𝟙 _ = |
| sumArrow (𝟙 _) (𝟙 _) ≫ (tensorArrow P ⟨f l.val,hf l.val l.property⟩ U V).arrow |
| simp only [Category.id_comp,Category.comp_id,sumArrow_id] |
| rfl |
| end SimpleAmenable.PolygonObject.Labelled.Cell |
|
|
| end |
|
|
| end OAI |
|
|
