36 lines
967 B
Text
36 lines
967 B
Text
|
/-
|
|||
|
|
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
|||
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|||
|
|
Authors: Joseph Tooby-Smith
|
|||
|
|
-/
|
|||
|
|
import HepLean.Tensors.Tree.Basic
|
|||
|
|
/-!
|
|||
|
|
|
|||
|
|
# Commuting products
|
|||
|
|
|
|||
|
|
The results here follow from the properties of Monoidal categories and monoidal functors.
|
|||
|
|
-/
|
|||
|
|
|
|||
|
|
open IndexNotation
|
|||
|
|
open CategoryTheory
|
|||
|
|
open MonoidalCategory
|
|||
|
|
open OverColor
|
|||
|
|
open HepLean.Fin
|
|||
|
|
|
|||
|
|
namespace TensorTree
|
|||
|
|
|
|||
|
|
variable {S : TensorStruct} {n n2 : ℕ}
|
|||
|
|
(c : Fin n → S.C) (c2 : Fin n2 → S.C)
|
|||
|
|
|
|||
|
|
def braidPerm : OverColor.mk (Sum.elim c2 c ∘ ⇑finSumFinEquiv.symm) ⟶
|
|||
|
|
OverColor.mk (Sum.elim c c2 ∘ ⇑finSumFinEquiv.symm) :=
|
|||
|
|
(equivToIso finSumFinEquiv).inv ≫
|
|||
|
|
(BraidedCategory.braiding (OverColor.mk c2) (OverColor.mk c)).hom
|
|||
|
|
≫ (equivToIso finSumFinEquiv).hom
|
|||
|
|
|
|||
|
|
theorem prod_comm (t : TensorTree S c) (t2 : TensorTree S c2) :
|
|||
|
|
(prod t t2).tensor = (perm (braidPerm c c2) (prod t2 t)).tensor := by
|
|||
|
|
|
|||
|
|
sorry
|
|||
|
|
end TensorTree
|