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0dd3af4 | 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 | 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
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