feat: General properties of contractions
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jstoobysmith
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9b4cd5d0b8
10 changed files with 366 additions and 269 deletions
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@ -131,155 +131,6 @@ lemma perm_contr_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.s
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erw [Equiv.apply_eq_iff_eq]
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exact (Fin.succAbove_ne i j).symm
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/-- The isomorphism between the image of a map `Fin 1 ⊕ Fin 1 → S.C` contructed by `finExtractTwo`
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under `S.F.obj`, and an object in the image of `OverColor.Discrete.pairτ S.FD`. -/
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def contrFin1Fin1 {n : ℕ} (c : Fin n.succ.succ → S.C)
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(i : Fin n.succ.succ) (j : Fin n.succ) (h : c (i.succAbove j) = S.τ (c i)) :
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S.F.obj (OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)) ≅
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(OverColor.Discrete.pairτ S.FD S.τ).obj { as := c i } := by
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apply (S.F.mapIso
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(OverColor.mkSum (((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)))).trans
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apply (S.F.μIso _ _).symm.trans
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apply tensorIso ?_ ?_
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· symm
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apply (OverColor.forgetLiftApp S.FD (c i)).symm.trans
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apply S.F.mapIso
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apply OverColor.mkIso
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funext x
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fin_cases x
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rfl
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· symm
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apply (OverColor.forgetLiftApp S.FD (S.τ (c i))).symm.trans
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apply S.F.mapIso
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apply OverColor.mkIso
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funext x
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fin_cases x
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simp [h]
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lemma contrFin1Fin1_inv_tmul {n : ℕ} (c : Fin n.succ.succ → S.C)
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(i : Fin n.succ.succ) (j : Fin n.succ) (h : c (i.succAbove j) = S.τ (c i))
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(x : S.FD.obj { as := c i })
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(y : S.FD.obj { as := S.τ (c i) }) :
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(S.contrFin1Fin1 c i j h).inv.hom (x ⊗ₜ[S.k] y) =
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PiTensorProduct.tprod S.k (fun k =>
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match k with | Sum.inl 0 => x | Sum.inr 0 => (S.FD.map
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(eqToHom (by simp [h]))).hom y) := by
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simp only [Nat.succ_eq_add_one, contrFin1Fin1, Functor.comp_obj, Discrete.functor_obj_eq_as,
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Function.comp_apply, Iso.trans_symm, Iso.symm_symm_eq, Iso.trans_inv, tensorIso_inv,
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Iso.symm_inv, Functor.mapIso_hom, tensor_comp, MonoidalFunctor.μIso_hom, Category.assoc,
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LaxMonoidalFunctor.μ_natural, Functor.mapIso_inv, Action.comp_hom,
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Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorHom_hom,
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Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, ModuleCat.coe_comp, Functor.id_obj, mk_hom,
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Fin.isValue]
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change (S.F.map (OverColor.mkSum ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).inv).hom
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((S.F.map ((OverColor.mkIso _).hom ⊗ (OverColor.mkIso _).hom)).hom
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((S.F.μ (OverColor.mk fun _ => c i) (OverColor.mk fun _ => S.τ (c i))).hom
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((((OverColor.forgetLiftApp S.FD (c i)).inv.hom x) ⊗ₜ[S.k]
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((OverColor.forgetLiftApp S.FD (S.τ (c i))).inv.hom y))))) = _
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simp only [Nat.succ_eq_add_one, Action.instMonoidalCategory_tensorObj_V, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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forgetLiftApp, Action.mkIso_inv_hom, LinearEquiv.toModuleIso_inv, Fin.isValue]
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erw [OverColor.forgetLiftAppV_symm_apply,
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OverColor.forgetLiftAppV_symm_apply S.FD (S.τ (c i))]
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change ((OverColor.lift.obj S.FD).map (OverColor.mkSum
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((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).inv).hom
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(((OverColor.lift.obj S.FD).map ((OverColor.mkIso _).hom ⊗ (OverColor.mkIso _).hom)).hom
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(((OverColor.lift.obj S.FD).μ (OverColor.mk fun _ => c i)
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(OverColor.mk fun _ => S.τ (c i))).hom
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(((PiTensorProduct.tprod S.k) fun _ => x) ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) fun _ => y))) = _
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rw [OverColor.lift.obj_μ_tprod_tmul S.FD]
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change ((OverColor.lift.obj S.FD).map
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(OverColor.mkSum ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).inv).hom
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(((OverColor.lift.obj S.FD).map ((OverColor.mkIso _).hom ⊗ (OverColor.mkIso _).hom)).hom
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((PiTensorProduct.tprod S.k) _)) = _
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rw [OverColor.lift.map_tprod S.FD]
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change ((OverColor.lift.obj S.FD).map
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(OverColor.mkSum ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).inv).hom
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((PiTensorProduct.tprod S.k _)) = _
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rw [OverColor.lift.map_tprod S.FD]
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apply congrArg
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funext r
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match r with
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| Sum.inl 0 =>
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simp only [Nat.succ_eq_add_one, mk_hom, Fin.isValue, Function.comp_apply,
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instMonoidalCategoryStruct_tensorObj_left, mkSum_inv_homToEquiv, Equiv.refl_symm,
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instMonoidalCategoryStruct_tensorObj_hom, Functor.id_obj, lift.discreteSumEquiv, Sum.elim_inl,
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Sum.elim_inr, HepLean.PiTensorProduct.elimPureTensor]
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simp only [Fin.isValue, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
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Iso.refl_hom, Action.id_hom, Iso.refl_inv, LinearEquiv.ofLinear_apply]
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rfl
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| Sum.inr 0 =>
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simp only [Nat.succ_eq_add_one, mk_hom, Fin.isValue, Function.comp_apply,
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instMonoidalCategoryStruct_tensorObj_left, mkSum_inv_homToEquiv, Equiv.refl_symm,
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instMonoidalCategoryStruct_tensorObj_hom, lift.discreteFunctorMapEqIso, eqToIso_refl,
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Functor.mapIso_refl, Iso.refl_hom, Action.id_hom, Iso.refl_inv, Functor.mapIso_hom,
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eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, Functor.id_obj, lift.discreteSumEquiv,
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Sum.elim_inl, Sum.elim_inr, HepLean.PiTensorProduct.elimPureTensor,
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LinearEquiv.ofLinear_apply]
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rfl
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lemma contrFin1Fin1_hom_hom_tprod {n : ℕ} (c : Fin n.succ.succ → S.C)
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(i : Fin n.succ.succ) (j : Fin n.succ) (h : c (i.succAbove j) = S.τ (c i))
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(x : (k : Fin 1 ⊕ Fin 1) → (S.FD.obj
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{ as := (OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl)).hom k })) :
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(S.contrFin1Fin1 c i j h).hom.hom (PiTensorProduct.tprod S.k x) =
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x (Sum.inl 0) ⊗ₜ[S.k] ((S.FD.map (eqToHom (by simp [h]))).hom (x (Sum.inr 0))) := by
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change ((Action.forget _ _).mapIso (S.contrFin1Fin1 c i j h)).hom _ = _
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trans ((Action.forget _ _).mapIso (S.contrFin1Fin1 c i j h)).toLinearEquiv
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(PiTensorProduct.tprod S.k x)
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· rfl
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erw [← LinearEquiv.eq_symm_apply]
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erw [contrFin1Fin1_inv_tmul]
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congr
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funext i
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match i with
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| Sum.inl 0 =>
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rfl
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| Sum.inr 0 =>
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simp only [Nat.succ_eq_add_one, Fin.isValue, mk_hom, Function.comp_apply,
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Discrete.functor_obj_eq_as]
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change _ = ((S.FD.map (eqToHom _)) ≫ (S.FD.map (eqToHom _))).hom (x (Sum.inr 0))
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rw [← Functor.map_comp]
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simp
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exact h
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/-- The isomorphism of objects in `Rep S.k S.G` given an `i` in `Fin n.succ.succ` and
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a `j` in `Fin n.succ` allowing us to undertake contraction. -/
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def contrIso {n : ℕ} (c : Fin n.succ.succ → S.C)
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(i : Fin n.succ.succ) (j : Fin n.succ) (h : c (i.succAbove j) = S.τ (c i)) :
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S.F.obj (OverColor.mk c) ≅ ((OverColor.Discrete.pairτ S.FD S.τ).obj
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(Discrete.mk (c i))) ⊗
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(OverColor.lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)) :=
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(S.F.mapIso (OverColor.equivToIso (HepLean.Fin.finExtractTwo i j))).trans <|
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(S.F.mapIso (OverColor.mkSum (c ∘ (HepLean.Fin.finExtractTwo i j).symm))).trans <|
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(S.F.μIso _ _).symm.trans <| by
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refine tensorIso (S.contrFin1Fin1 c i j h) (S.F.mapIso (OverColor.mkIso (by ext x; simp)))
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lemma contrIso_hom_hom {n : ℕ} {c1 : Fin n.succ.succ → S.C}
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{i : Fin n.succ.succ} {j : Fin n.succ} {h : c1 (i.succAbove j) = S.τ (c1 i)} :
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(S.contrIso c1 i j h).hom.hom =
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(S.F.map (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).hom ≫
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(S.F.map (mkSum (c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom ≫
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(S.F.μIso (OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c1 ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom ≫
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((S.contrFin1Fin1 c1 i j h).hom.hom ⊗
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(S.F.map (mkIso (contrIso.proof_1 S c1 i j)).hom).hom) := by
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rfl
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/-- `contrMap` is a function that takes a natural number `n`, a function `c` from
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`Fin n.succ.succ` to `S.C`, an index `i` of type `Fin n.succ.succ`, an index `j` of type
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`Fin n.succ`, and a proof `h` that `c (i.succAbove j) = S.τ (c i)`. It returns a morphism
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corresponding to the contraction of the `i`th index with the `i.succAbove j` index.
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--/
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def contrMap {n : ℕ} (c : Fin n.succ.succ → S.C)
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(i : Fin n.succ.succ) (j : Fin n.succ) (h : c (i.succAbove j) = S.τ (c i)) :
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S.F.obj (OverColor.mk c) ⟶
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S.F.obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)) :=
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(S.contrIso c i j h).hom ≫
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(tensorHom (S.contr.app (Discrete.mk (c i))) (𝟙 _)) ≫
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(MonoidalCategory.leftUnitor _).hom
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/-- Casts an element of the monoidal unit of `Rep S.k S.G` to the field `S.k`. -/
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def castToField (v : (↑((𝟙_ (Discrete S.C ⥤ Rep S.k S.G)).obj { as := c }).V)) : S.k := v
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@ -294,120 +145,6 @@ lemma castFin0ToField_tprod {c : Fin 0 → S.C}
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simp only [castFin0ToField, mk_hom, Functor.id_obj, LinearEquiv.coe_coe]
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erw [PiTensorProduct.isEmptyEquiv_apply_tprod]
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lemma contrMap_tprod {n : ℕ} (c : Fin n.succ.succ → S.C)
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(i : Fin n.succ.succ) (j : Fin n.succ) (h : c (i.succAbove j) = S.τ (c i))
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(x : (i : Fin n.succ.succ) → S.FD.obj (Discrete.mk (c i))) :
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(S.contrMap c i j h).hom (PiTensorProduct.tprod S.k x) =
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(S.castToField ((S.contr.app (Discrete.mk (c i))).hom ((x i) ⊗ₜ[S.k]
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(S.FD.map (Discrete.eqToHom h)).hom (x (i.succAbove j)))) : S.k)
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• (PiTensorProduct.tprod S.k (fun k => x (i.succAbove (j.succAbove k))) :
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S.F.obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))) := by
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rw [contrMap, contrIso]
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simp only [Nat.succ_eq_add_one, S.F_def, Iso.trans_hom, Functor.mapIso_hom, Iso.symm_hom,
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tensorIso_hom, Monoidal.tensorUnit_obj, tensorHom_id,
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Category.assoc, Action.comp_hom, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_tensorHom_hom, Action.instMonoidalCategory_tensorUnit_V,
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Action.instMonoidalCategory_whiskerRight_hom, Functor.id_obj, mk_hom, ModuleCat.coe_comp,
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Function.comp_apply, Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj, Functor.comp_obj, Discrete.functor_obj_eq_as]
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change (λ_ ((lift.obj S.FD).obj _)).hom.hom
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(((S.contr.app { as := c i }).hom ▷ ((lift.obj S.FD).obj
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(OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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(((S.contrFin1Fin1 c i j h).hom.hom ⊗ ((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso (OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)
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∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom
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(((lift.obj S.FD).map (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom
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(((lift.obj S.FD).map (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).hom
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((PiTensorProduct.tprod S.k) x)))))) = _
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rw [lift.map_tprod]
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change (λ_ ((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)))).hom.hom
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(((S.contr.app { as := c i }).hom ▷
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((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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(((S.contrFin1Fin1 c i j h).hom.hom ⊗ ((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso (OverColor.mk
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((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom
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(((lift.obj S.FD).map (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).hom
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((PiTensorProduct.tprod S.k) fun i_1 =>
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(lift.discreteFunctorMapEqIso S.FD _)
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(x ((Hom.toEquiv (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).symm i_1))))))) = _
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rw [lift.map_tprod]
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change (λ_ ((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)))).hom.hom
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(((S.contr.app { as := c i }).hom ▷ ((lift.obj S.FD).obj
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(OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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(((S.contrFin1Fin1 c i j h).hom.hom ⊗ ((lift.obj S.FD).map (mkIso _).hom).hom)
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(((lift.obj S.FD).μIso
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inl))
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(OverColor.mk ((c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm) ∘ Sum.inr))).inv.hom
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((PiTensorProduct.tprod S.k) fun i_1 =>
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(lift.discreteFunctorMapEqIso S.FD _)
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((lift.discreteFunctorMapEqIso S.FD _)
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(x ((Hom.toEquiv (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).symm
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((Hom.toEquiv (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).symm i_1)))))))) = _
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rw [lift.μIso_inv_tprod]
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change (λ_ ((lift.obj S.FD).obj (OverColor.mk (c ∘ i.succAbove ∘ j.succAbove)))).hom.hom
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(((S.contr.app { as := c i }).hom ▷ ((lift.obj S.FD).obj
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(OverColor.mk (c ∘ i.succAbove ∘ j.succAbove))).V)
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((TensorProduct.map (S.contrFin1Fin1 c i j h).hom.hom
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((lift.obj S.FD).map (mkIso _).hom).hom)
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(((PiTensorProduct.tprod S.k) fun i_1 =>
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(lift.discreteFunctorMapEqIso S.FD _)
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((lift.discreteFunctorMapEqIso S.FD _) (x
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((Hom.toEquiv (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).symm
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((Hom.toEquiv (mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).symm
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(Sum.inl i_1)))))) ⊗ₜ[S.k] (PiTensorProduct.tprod S.k) fun i_1 =>
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(lift.discreteFunctorMapEqIso S.FD _) ((lift.discreteFunctorMapEqIso S.FD _)
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(x ((Hom.toEquiv (equivToIso (HepLean.Fin.finExtractTwo i j)).hom).symm
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((Hom.toEquiv
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(mkSum (c ∘ ⇑(HepLean.Fin.finExtractTwo i j).symm)).hom).symm (Sum.inr i_1)))))))) = _
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rw [TensorProduct.map_tmul]
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rw [contrFin1Fin1_hom_hom_tprod]
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simp only [Nat.succ_eq_add_one, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_tensorUnit_V, Fin.isValue, mk_hom, Function.comp_apply,
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Discrete.functor_obj_eq_as, instMonoidalCategoryStruct_tensorObj_left, mkSum_homToEquiv,
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Equiv.refl_symm, Functor.id_obj, ModuleCat.MonoidalCategory.whiskerRight_apply]
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rw [Action.instMonoidalCategory_leftUnitor_hom_hom]
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simp only [Monoidal.tensorUnit_obj, Action.instMonoidalCategory_tensorUnit_V, Fin.isValue,
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ModuleCat.MonoidalCategory.leftUnitor_hom_apply]
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congr 1
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/- The contraction. -/
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· simp only [Fin.isValue, castToField]
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congr 2
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· simp only [Fin.isValue, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
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Iso.refl_hom, Action.id_hom, Iso.refl_inv, LinearEquiv.ofLinear_apply]
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rfl
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· simp only [Fin.isValue, lift.discreteFunctorMapEqIso, eqToIso_refl, Functor.mapIso_refl,
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Iso.refl_hom, Action.id_hom, Iso.refl_inv, LinearEquiv.ofLinear_apply]
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change (S.FD.map (eqToHom _)).hom
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(x (((HepLean.Fin.finExtractTwo i j)).symm ((Sum.inl (Sum.inr 0))))) = _
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simp only [Nat.succ_eq_add_one, Fin.isValue]
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have h1' {a b d: Fin n.succ.succ} (hbd : b =d) (h : c d = S.τ (c a)) (h' : c b = S.τ (c a)) :
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(S.FD.map (Discrete.eqToHom (h))).hom (x d) =
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(S.FD.map (Discrete.eqToHom h')).hom (x b) := by
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subst hbd
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rfl
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refine h1' ?_ ?_ ?_
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simp only [Nat.succ_eq_add_one, Fin.isValue, HepLean.Fin.finExtractTwo_symm_inl_inr_apply]
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simp [h]
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/- The tensor. -/
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· erw [lift.map_tprod]
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apply congrArg
|
||||
funext d
|
||||
simp only [mk_hom, Function.comp_apply, lift.discreteFunctorMapEqIso, Functor.mapIso_hom,
|
||||
eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, eqToIso_refl, Functor.mapIso_refl, Iso.refl_hom,
|
||||
Action.id_hom, Iso.refl_inv, LinearEquiv.ofLinear_apply]
|
||||
change (S.FD.map (eqToHom _)).hom
|
||||
((x ((HepLean.Fin.finExtractTwo i j).symm (Sum.inr (d))))) = _
|
||||
simp only [Nat.succ_eq_add_one]
|
||||
have h1 : ((HepLean.Fin.finExtractTwo i j).symm (Sum.inr d))
|
||||
= (i.succAbove (j.succAbove d)) := HepLean.Fin.finExtractTwo_symm_inr_apply i j d
|
||||
have h1' {a b : Fin n.succ.succ} (h : a = b) :
|
||||
(S.FD.map (eqToHom (by rw [h]))).hom (x a) = x b := by
|
||||
subst h
|
||||
simp
|
||||
exact h1' h1
|
||||
|
||||
/-!
|
||||
|
||||
## Evalutation of indices.
|
||||
|
|
|
|||
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Add table
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Reference in a new issue