feat: Add monoidal functor for complex lorentz tensors
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jstoobysmith
parent
a39e7e5e65
commit
a39aeeed8b
4 changed files with 649 additions and 8 deletions
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@ -4,6 +4,7 @@ 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.ColorCat.Basic
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import HepLean.Mathematics.PiTensorProduct
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/-!
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## Monodial functor from color cat.
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@ -164,6 +165,18 @@ namespace colorFun
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open CategoryTheory
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open MonoidalCategory
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lemma map_tprod {X Y : OverColor Color} (p : (i : X.left) → (colorToRep (X.hom i)))
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(f : X ⟶ Y) : (colorFun.map f).hom (PiTensorProduct.tprod ℂ p) =
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PiTensorProduct.tprod ℂ fun (i : Y.left) => colorToRepCongr
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(OverColor.Hom.toEquiv_comp_inv_apply f i) (p ((OverColor.Hom.toEquiv f).symm i)) := by
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change (colorFun.map' f).hom ((PiTensorProduct.tprod ℂ) p) = _
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simp [colorFun.map', mapToLinearEquiv']
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erw [LinearEquiv.trans_apply]
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change (PiTensorProduct.congr fun i => colorToRepCongr _)
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((PiTensorProduct.reindex ℂ (fun x => _) (OverColor.Hom.toEquiv f))
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((PiTensorProduct.tprod ℂ) p)) = _
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rw [PiTensorProduct.reindex_tprod, PiTensorProduct.congr_tprod]
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@[simp]
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lemma obj_ρ_empty (g : SL(2, ℂ)) : (colorFun.obj (𝟙_ (OverColor Color))).ρ g = LinearMap.id := by
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erw [colorFun.obj'_ρ]
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@ -182,17 +195,286 @@ lemma obj_ρ_empty (g : SL(2, ℂ)) : (colorFun.obj (𝟙_ (OverColor Color))).
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exact Empty.elim i
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/-- The unit natural transformation. -/
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def ε : 𝟙_ (Rep ℂ SL(2, ℂ)) ⟶ colorFun.obj (𝟙_ (OverColor Color)) where
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hom := (PiTensorProduct.isEmptyEquiv Empty).symm.toLinearMap
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comm M := by
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refine LinearMap.ext (fun x => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorUnit_left,
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OverColor.instMonoidalCategoryStruct_tensorUnit_hom, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', Functor.id_obj, Category.id_comp, LinearEquiv.coe_coe]
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erw [obj_ρ_empty M]
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def ε : 𝟙_ (Rep ℂ SL(2, ℂ)) ≅ colorFun.obj (𝟙_ (OverColor Color)) where
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hom := {
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hom := (PiTensorProduct.isEmptyEquiv Empty).symm.toLinearMap
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comm := fun M => by
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refine LinearMap.ext (fun x => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorUnit_left,
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OverColor.instMonoidalCategoryStruct_tensorUnit_hom, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', Functor.id_obj, Category.id_comp, LinearEquiv.coe_coe]
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erw [obj_ρ_empty M]
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rfl}
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inv := {
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hom := (PiTensorProduct.isEmptyEquiv Empty).toLinearMap
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comm := fun M => by
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refine LinearMap.ext (fun x => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorUnit_left,
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OverColor.instMonoidalCategoryStruct_tensorUnit_hom, Action.instMonoidalCategory_tensorUnit_V,
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Action.tensorUnit_ρ', Functor.id_obj, Category.id_comp, LinearEquiv.coe_coe]
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erw [obj_ρ_empty M]
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rfl}
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hom_inv_id := by
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ext1
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simp [CategoryStruct.comp]
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rfl
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inv_hom_id := by
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ext1
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simp [CategoryStruct.comp]
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rfl
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def colorToRepSumEquiv {X Y : OverColor Color} (i : X.left ⊕ Y.left) :
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Sum.elim (fun i => colorToRep (X.hom i)) (fun i => colorToRep (Y.hom i)) i ≃ₗ[ℂ]
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colorToRep (Sum.elim X.hom Y.hom i) :=
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match i with
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| Sum.inl _ => LinearEquiv.refl _ _
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| Sum.inr _ => LinearEquiv.refl _ _
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def μModEquiv (X Y : OverColor Color) : (colorFun.obj X ⊗ colorFun.obj Y).V ≃ₗ[ℂ] colorFun.obj (X ⊗ Y) :=
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HepLean.PiTensorProduct.tmulEquiv ≪≫ₗ PiTensorProduct.congr colorToRepSumEquiv
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lemma μModEquiv_tmul_tprod {X Y : OverColor Color}(p : (i : X.left) → (colorToRep (X.hom i)))
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(q : (i : Y.left) → (colorToRep (Y.hom i))) :
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(μModEquiv X Y) ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q) =
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(PiTensorProduct.tprod ℂ) fun i =>
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(colorToRepSumEquiv i) (HepLean.PiTensorProduct.elimPureTensor p q i) := by
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rw [μModEquiv]
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Action.instMonoidalCategory_tensorObj_V,
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Functor.id_obj, Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
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Action.FunctorCategoryEquivalence.functor_obj_obj]
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rw [LinearEquiv.trans_apply]
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erw [HepLean.PiTensorProduct.tmulEquiv_tmul_tprod]
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change (PiTensorProduct.congr colorToRepSumEquiv) ((PiTensorProduct.tprod ℂ) (HepLean.PiTensorProduct.elimPureTensor p q)) = _
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rw [PiTensorProduct.congr_tprod]
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rfl
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def μ (X Y : OverColor Color) : colorFun.obj X ⊗ colorFun.obj Y ≅ colorFun.obj (X ⊗ Y) where
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hom := {
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hom := (μModEquiv X Y).toLinearMap
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comm := fun M => by
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refine HepLean.PiTensorProduct.tensorProd_piTensorProd_ext (fun p q => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Functor.id_obj, CategoryStruct.comp,
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Action.instMonoidalCategory_tensorObj_V, Action.tensor_ρ', LinearMap.coe_comp,
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Function.comp_apply]
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change (μModEquiv X Y) (((((colorFun.obj X).ρ M) (PiTensorProduct.tprod ℂ p)) ⊗ₜ[ℂ]
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(((colorFun.obj Y).ρ M) (PiTensorProduct.tprod ℂ q))))
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= ((colorFun.obj (X ⊗ Y)).ρ M) ((μModEquiv X Y) ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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rw [μModEquiv_tmul_tprod]
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erw [obj'_ρ_tprod, obj'_ρ_tprod, obj'_ρ_tprod]
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rw [μModEquiv_tmul_tprod]
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apply congrArg
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funext i
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match i with
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| Sum.inl i =>
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rfl
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| Sum.inr i =>
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rfl
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}
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inv := {
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hom := (μModEquiv X Y).symm.toLinearMap
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comm := fun M => by
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simp [CategoryStruct.comp]
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erw [LinearEquiv.eq_comp_toLinearMap_symm,LinearMap.comp_assoc ,
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LinearEquiv.toLinearMap_symm_comp_eq ]
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refine HepLean.PiTensorProduct.tensorProd_piTensorProd_ext (fun p q => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Functor.id_obj, CategoryStruct.comp,
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Action.instMonoidalCategory_tensorObj_V, Action.tensor_ρ', LinearMap.coe_comp,
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Function.comp_apply]
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symm
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change (μModEquiv X Y) (((((colorFun.obj X).ρ M) (PiTensorProduct.tprod ℂ p)) ⊗ₜ[ℂ]
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(((colorFun.obj Y).ρ M) (PiTensorProduct.tprod ℂ q))))
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= ((colorFun.obj (X ⊗ Y)).ρ M) ((μModEquiv X Y) ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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rw [μModEquiv_tmul_tprod]
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erw [obj'_ρ_tprod, obj'_ρ_tprod, obj'_ρ_tprod]
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rw [μModEquiv_tmul_tprod]
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apply congrArg
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funext i
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match i with
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| Sum.inl i =>
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rfl
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| Sum.inr i =>
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rfl}
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hom_inv_id := by
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ext1
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simp only [Action.instMonoidalCategory_tensorObj_V, CategoryStruct.comp, Action.Hom.comp_hom,
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colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, LinearEquiv.comp_coe,
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LinearEquiv.self_trans_symm, LinearEquiv.refl_toLinearMap, Action.id_hom]
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rfl
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inv_hom_id := by
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ext1
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simp only [CategoryStruct.comp, Action.instMonoidalCategory_tensorObj_V, Action.Hom.comp_hom,
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colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, LinearEquiv.comp_coe,
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LinearEquiv.symm_trans_self, LinearEquiv.refl_toLinearMap, Action.id_hom]
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rfl
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@[simp]
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lemma μ_tmul_tprod {X Y : OverColor Color} (p : (i : X.left) → (colorToRep (X.hom i)))
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(q : (i : Y.left) → (colorToRep (Y.hom i))) :
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(μ X Y).hom.hom ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q) =
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(PiTensorProduct.tprod ℂ) fun i =>
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(colorToRepSumEquiv i) (HepLean.PiTensorProduct.elimPureTensor p q i) := by
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exact μModEquiv_tmul_tprod p q
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lemma μ_natural_left {X Y : OverColor Color} (f : X ⟶ Y) (Z : OverColor Color) :
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MonoidalCategory.whiskerRight (colorFun.map f) (colorFun.obj Z) ≫ (μ Y Z).hom =
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(μ X Z).hom ≫ colorFun.map (MonoidalCategory.whiskerRight f Z) := by
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ext1
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refine HepLean.PiTensorProduct.tensorProd_piTensorProd_ext (fun p q => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Functor.id_obj, CategoryStruct.comp,
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Action.Hom.comp_hom, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_whiskerRight_hom, LinearMap.coe_comp, Function.comp_apply]
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change _ = (colorFun.map (MonoidalCategory.whiskerRight f Z)).hom
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((μ X Z).hom.hom ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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rw [μ_tmul_tprod]
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change _ = (colorFun.map (f ▷ Z)).hom
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((PiTensorProduct.tprod ℂ) fun i => (colorToRepSumEquiv i)
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(HepLean.PiTensorProduct.elimPureTensor p q i))
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rw [colorFun.map_tprod]
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have h1 : (((colorFun.map f).hom ▷ (colorFun.obj Z).V) ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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= ((colorFun.map f).hom ((PiTensorProduct.tprod ℂ) p) ⊗ₜ[ℂ] ((PiTensorProduct.tprod ℂ) q)) := by rfl
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erw [h1]
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rw [colorFun.map_tprod]
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change (μ Y Z).hom.hom
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(((PiTensorProduct.tprod ℂ) fun i => (colorToRepCongr _) (p ((OverColor.Hom.toEquiv f).symm i))) ⊗ₜ[ℂ]
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(PiTensorProduct.tprod ℂ) q) = _
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rw [μ_tmul_tprod]
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apply congrArg
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funext i
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match i with
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| Sum.inl i => rfl
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| Sum.inr i => rfl
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lemma μ_natural_right {X Y : OverColor Color} (X' : OverColor Color) (f : X ⟶ Y) :
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MonoidalCategory.whiskerLeft (colorFun.obj X') (colorFun.map f) ≫ (μ X' Y).hom =
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(μ X' X).hom ≫ colorFun.map (MonoidalCategory.whiskerLeft X' f) := by
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ext1
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refine HepLean.PiTensorProduct.tensorProd_piTensorProd_ext (fun p q => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Functor.id_obj, CategoryStruct.comp,
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Action.Hom.comp_hom, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_whiskerLeft_hom, LinearMap.coe_comp, Function.comp_apply]
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change _ = (colorFun.map (X' ◁ f)).hom ((μ X' X).hom.hom ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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rw [μ_tmul_tprod]
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change _ = (colorFun.map (X' ◁ f)).hom
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((PiTensorProduct.tprod ℂ) fun i => (colorToRepSumEquiv i) (HepLean.PiTensorProduct.elimPureTensor p q i))
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rw [map_tprod]
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have h1 : (((colorFun.obj X').V ◁ (colorFun.map f).hom) ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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= ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (colorFun.map f).hom ((PiTensorProduct.tprod ℂ) q)) := by rfl
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erw [h1]
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rw [map_tprod]
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change (μ X' Y).hom.hom
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((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ]
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(PiTensorProduct.tprod ℂ) fun i => (colorToRepCongr _) (q ((OverColor.Hom.toEquiv f).symm i))) = _
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rw [μ_tmul_tprod]
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apply congrArg
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funext i
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match i with
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| Sum.inl i => rfl
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| Sum.inr i => rfl
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lemma associativity (X Y Z : OverColor Color) :
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whiskerRight (μ X Y).hom (colorFun.obj Z) ≫
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(μ (X ⊗ Y) Z).hom ≫ colorFun.map (associator X Y Z).hom =
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(associator (colorFun.obj X) (colorFun.obj Y) (colorFun.obj Z)).hom ≫
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whiskerLeft (colorFun.obj X) (μ Y Z).hom ≫ (μ X (Y ⊗ Z)).hom := by
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ext1
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refine HepLean.PiTensorProduct.induction_assoc' (fun p q m => ?_)
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simp only [colorFun_obj_V_carrier, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Functor.id_obj, CategoryStruct.comp,
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Action.Hom.comp_hom, Action.instMonoidalCategory_tensorObj_V,
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Action.instMonoidalCategory_whiskerRight_hom, LinearMap.coe_comp, Function.comp_apply,
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Action.instMonoidalCategory_whiskerLeft_hom, Action.instMonoidalCategory_associator_hom_hom]
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change (colorFun.map (α_ X Y Z).hom).hom ((μ (X ⊗ Y) Z).hom.hom
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((((μ X Y).hom.hom ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q)) ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) m))) =
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(μ X (Y ⊗ Z)).hom.hom
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(( ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] ((μ Y Z).hom.hom ((PiTensorProduct.tprod ℂ) q ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) m)))))
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rw [μ_tmul_tprod, μ_tmul_tprod]
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change (colorFun.map (α_ X Y Z).hom).hom
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((μ (X ⊗ Y) Z).hom.hom
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(((PiTensorProduct.tprod ℂ) fun i => (colorToRepSumEquiv i) (HepLean.PiTensorProduct.elimPureTensor p q i)) ⊗ₜ[ℂ]
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(PiTensorProduct.tprod ℂ) m)) =
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(μ X (Y ⊗ Z)).hom.hom
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((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ]
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(PiTensorProduct.tprod ℂ) fun i => (colorToRepSumEquiv i) (HepLean.PiTensorProduct.elimPureTensor q m i))
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rw [μ_tmul_tprod, μ_tmul_tprod]
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erw [map_tprod]
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apply congrArg
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funext i
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match i with
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| Sum.inl i => rfl
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| Sum.inr (Sum.inl i) => rfl
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| Sum.inr (Sum.inr i) => rfl
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lemma left_unitality (X : OverColor Color) : (leftUnitor (colorFun.obj X)).hom =
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whiskerRight colorFun.ε.hom (colorFun.obj X) ≫
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(μ (𝟙_ (OverColor Color)) X).hom ≫ colorFun.map (leftUnitor X).hom := by
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ext1
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apply HepLean.PiTensorProduct.induction_mod_tmul (fun x q => ?_)
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simp only [colorFun_obj_V_carrier, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Action.instMonoidalCategory_tensorUnit_V, Functor.id_obj,
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Action.instMonoidalCategory_leftUnitor_hom_hom, CategoryStruct.comp, Action.Hom.comp_hom,
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Action.instMonoidalCategory_tensorObj_V, OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorUnit_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom,
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Action.instMonoidalCategory_whiskerRight_hom, LinearMap.coe_comp, Function.comp_apply]
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change TensorProduct.lid ℂ (colorFun.obj X) (x ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q) = (colorFun.map (λ_ X).hom).hom
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((μ (𝟙_ (OverColor Color)) X).hom.hom ((((PiTensorProduct.isEmptyEquiv Empty).symm x) ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q)))
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simp [PiTensorProduct.isEmptyEquiv]
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rw [TensorProduct.smul_tmul, TensorProduct.tmul_smul]
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erw [LinearMap.map_smul, LinearMap.map_smul]
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apply congrArg
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change _ = (colorFun.map (λ_ X).hom).hom
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((μ (𝟙_ (OverColor Color)) X).hom.hom ((PiTensorProduct.tprod ℂ) _ ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) q))
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rw [μ_tmul_tprod]
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erw [map_tprod]
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rfl
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lemma right_unitality (X : OverColor Color) : (MonoidalCategory.rightUnitor (colorFun.obj X)).hom =
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whiskerLeft (colorFun.obj X) ε.hom ≫
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(μ X (𝟙_ (OverColor Color))).hom ≫ colorFun.map (MonoidalCategory.rightUnitor X).hom := by
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ext1
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apply HepLean.PiTensorProduct.induction_tmul_mod (fun p x => ?_)
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simp only [colorFun_obj_V_carrier, Functor.id_obj, Equivalence.symm_inverse,
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Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
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Action.instMonoidalCategory_tensorUnit_V, Action.instMonoidalCategory_rightUnitor_hom_hom,
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CategoryStruct.comp, Action.Hom.comp_hom, Action.instMonoidalCategory_tensorObj_V,
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OverColor.instMonoidalCategoryStruct_tensorObj_left,
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OverColor.instMonoidalCategoryStruct_tensorUnit_left,
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OverColor.instMonoidalCategoryStruct_tensorObj_hom, Action.instMonoidalCategory_whiskerLeft_hom,
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LinearMap.coe_comp, Function.comp_apply]
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change TensorProduct.rid ℂ (colorFun.obj X) ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] x ) =
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(colorFun.map (ρ_ X).hom).hom
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((μ X (𝟙_ (OverColor Color))).hom.hom ((((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] ((PiTensorProduct.isEmptyEquiv Empty).symm x)))))
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simp [PiTensorProduct.isEmptyEquiv]
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erw [LinearMap.map_smul, LinearMap.map_smul]
|
||||
apply congrArg
|
||||
change _ = (colorFun.map (ρ_ X).hom).hom
|
||||
((μ X (𝟙_ (OverColor Color))).hom.hom ((PiTensorProduct.tprod ℂ) p ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) _))
|
||||
rw [μ_tmul_tprod]
|
||||
erw [map_tprod]
|
||||
rfl
|
||||
|
||||
end colorFun
|
||||
|
||||
def colorFunMon : MonoidalFunctor (OverColor Color) (Rep ℂ SL(2, ℂ)) where
|
||||
toFunctor := colorFun
|
||||
ε := colorFun.ε.hom
|
||||
μ X Y := (colorFun.μ X Y).hom
|
||||
μ_natural_left := colorFun.μ_natural_left
|
||||
μ_natural_right := colorFun.μ_natural_right
|
||||
associativity := colorFun.associativity
|
||||
left_unitality := colorFun.left_unitality
|
||||
right_unitality := colorFun.right_unitality
|
||||
|
||||
|
||||
end
|
||||
end Fermion
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue