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