110 lines
4.5 KiB
Text
110 lines
4.5 KiB
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.Elab
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import HepLean.Tensors.Tree.NodeIdentities.Basic
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import HepLean.Tensors.Tree.NodeIdentities.Congr
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/-!
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## Units as tensors
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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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open TensorProduct
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noncomputable section
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namespace TensorSpecies
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/-- The unit of a tensor species in a `PiTensorProduct`. -/
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def unitTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![S.τ c, c]) :=
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(OverColor.Discrete.pairIsoSep S.FD).hom.hom ((S.unit.app (Discrete.mk c)).hom (1 : S.k))
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variable {S : TensorSpecies}
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open TensorTree
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/-- The relation between two units of colors which are equal. -/
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lemma unitTensor_congr {c c' : S.C} (h : c = c') : {S.unitTensor c | μ ν}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl))
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{S.unitTensor c' | μ ν}ᵀ).tensor := by
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subst h
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change _ = (S.F.map (𝟙 _)).hom (S.unitTensor c)
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simp
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lemma unitTensor_eq_dual_perm_eq (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
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![S.τ c, c] x = (![S.τ (S.τ c), S.τ c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := fun x => by
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fin_cases x
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· rfl
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· exact (S.τ_involution c).symm
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/-- The unit tensor is equal to a permutation of indices of the dual tensor. -/
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lemma unitTensor_eq_dual_perm (c : S.C) : {S.unitTensor c | μ ν}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1,0] ![1, 0]) (unitTensor_eq_dual_perm_eq c))
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{S.unitTensor (S.τ c) | ν μ }ᵀ).tensor := by
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simp [unitTensor, tensorNode_tensor, perm_tensor]
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have h1 := S.unit_symm c
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erw [h1]
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have hg : (Discrete.pairIsoSep S.FD).hom.hom ∘ₗ (S.FD.obj { as := S.τ c } ◁
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S.FD.map (Discrete.eqToHom (S.τ_involution c))).hom ∘ₗ
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(β_ (S.FD.obj { as := S.τ (S.τ c) }) (S.FD.obj { as := S.τ c })).hom.hom =
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(S.F.map (equivToHomEq (finMapToEquiv ![1, 0] ![1, 0]) (unitTensor_eq_dual_perm_eq c))).hom
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∘ₗ (Discrete.pairIsoSep S.FD).hom.hom := by
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apply TensorProduct.ext'
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intro x y
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_whiskerLeft_hom,
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LinearMap.coe_comp, Function.comp_apply, Fin.isValue]
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change (Discrete.pairIsoSep S.FD).hom.hom
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(((y ⊗ₜ[S.k] ((S.FD.map (Discrete.eqToHom _)).hom x)))) =
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((S.F.map (equivToHomEq (finMapToEquiv ![1, 0] ![1, 0]) _)).hom ∘ₗ
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(Discrete.pairIsoSep S.FD).hom.hom) (x ⊗ₜ[S.k] y)
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rw [Discrete.pairIsoSep_tmul]
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conv_rhs =>
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simp [Discrete.pairIsoSep_tmul]
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change _ =
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(S.F.map (equivToHomEq (finMapToEquiv ![1, 0] ![1, 0]) _)).hom
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((Discrete.pairIsoSep S.FD).hom.hom (x ⊗ₜ[S.k] y))
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rw [Discrete.pairIsoSep_tmul]
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simp only [F_def, Nat.succ_eq_add_one, Nat.reduceAdd, mk_hom, Functor.id_obj, Fin.isValue]
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erw [OverColor.lift.map_tprod]
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apply congrArg
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funext i
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fin_cases i
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· simp only [Fin.zero_eta, Fin.isValue, Matrix.cons_val_zero, Fin.cases_zero, mk_hom,
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Nat.succ_eq_add_one, Nat.reduceAdd, lift.discreteFunctorMapEqIso, eqToIso_refl,
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Functor.mapIso_refl, Iso.refl_hom, Action.id_hom, Iso.refl_inv, LinearEquiv.ofLinear_apply]
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rfl
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· simp only [Fin.mk_one, Fin.isValue, Matrix.cons_val_one, Matrix.head_cons, mk_hom,
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Nat.succ_eq_add_one, Nat.reduceAdd, lift.discreteFunctorMapEqIso, Functor.mapIso_hom,
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eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, LinearEquiv.ofLinear_apply]
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rfl
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exact congrFun (congrArg (fun f => f.toFun) hg) _
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lemma dual_unitTensor_eq_perm_eq (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
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![S.τ (S.τ c), S.τ c] x = (![S.τ c, c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := fun x => by
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fin_cases x
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· exact (S.τ_involution c)
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· rfl
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lemma dual_unitTensor_eq_perm (c : S.C) : {S.unitTensor (S.τ c) | ν μ}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0]) (dual_unitTensor_eq_perm_eq c))
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{S.unitTensor c | μ ν}ᵀ).tensor := by
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rw [unitTensor_eq_dual_perm]
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conv =>
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lhs
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rw [perm_tensor_eq <| unitTensor_congr (S.τ_involution c)]
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rw [perm_perm]
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refine perm_congr ?_ rfl
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
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rfl
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end TensorSpecies
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end
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