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feat: Writing toDualRep in terms of contrOneTwoLeft

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jstoobysmith 2024-11-18 06:07:03 +00:00
parent 5d21f74062
commit 2c26584cfc
5 changed files with 49 additions and 13 deletions
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21
HepLean/Tensors/TensorSpecies/ContractLemmas.lean
View file

@ -29,13 +29,11 @@ variable {S : TensorSpecies}
def contrOneTwoLeft {c1 c2 : S.C}
(x : S.F.obj (OverColor.mk ![c1])) (y : S.F.obj (OverColor.mk ![S.τ c1, c2])) :
S.F.obj (OverColor.mk ![c2]) :=
(S.F.map (OverColor.mkIso (by funext x; fin_cases x; rfl)).hom).hom <|
(OverColor.forgetLiftApp S.FD c2).inv.hom <|
(OverColor.forgetLiftAppCon S.FD c2).inv.hom <|
(λ_ (S.FD.obj (Discrete.mk c2))).hom.hom <|
((S.contr.app (Discrete.mk c1)) ▷ (S.FD.obj (Discrete.mk (c2 )))).hom <|
(α_ _ _ (S.FD.obj (Discrete.mk (c2 )))).inv.hom <|
(OverColor.forgetLiftApp S.FD c1).hom.hom ((S.F.map (OverColor.mkIso (by
funext x; fin_cases x; rfl)).hom).hom x) ⊗ₜ
(OverColor.forgetLiftAppCon S.FD c1).hom.hom (x) ⊗ₜ
(OverColor.Discrete.pairIsoSep S.FD).inv.hom y
@[simp]
@ -72,19 +70,20 @@ lemma contrOneTwoLeft_tprod_eq {c1 c2 : S.C}
(fy : (i : (𝟭 Type).obj (OverColor.mk ![S.τ c1, c2]).left)
→ CoeSort.coe (S.FD.obj { as := (OverColor.mk ![S.τ c1, c2]).hom i })) :
contrOneTwoLeft (PiTensorProduct.tprod S.k fx) (PiTensorProduct.tprod S.k fy) =
(S.F.map (OverColor.mkIso (by funext x; fin_cases x; rfl)).hom).hom
((OverColor.forgetLiftApp S.FD c2).inv.hom (
((OverColor.forgetLiftAppCon S.FD c2).inv.hom (
((S.contr.app (Discrete.mk c1)).hom (fx (0 : Fin 1) ⊗ₜ fy (0 : Fin 2)) •
fy (1 : Fin 2)))) := by
rw [contrOneTwoLeft]
apply congrArg
apply congrArg
rw [Discrete.pairIsoSep_inv_tprod S.FD fy]
rw [Discrete.pairIsoSep_inv_tprod S.FD fy, OverColor.forgetLiftAppCon]
change (S.contr.app { as := c1 }).hom (_ ⊗ₜ[S.k] fy (0 : Fin 2)) • fy (1 : Fin 2) = _
congr
simp
simp only [Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, Nat.succ_eq_add_one, Nat.reduceAdd,
Iso.trans_hom, Functor.mapIso_hom, Action.comp_hom, mk_left, Functor.id_obj, mk_hom,
ModuleCat.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq, Fin.isValue]
rw [forgetLiftApp_hom_hom_apply_eq]
simp [S.F_def]
simp only [mk_left, Functor.id_obj, Fin.isValue]
erw [OverColor.lift.map_tprod]
congr
funext x
@ -107,6 +106,7 @@ lemma contr_one_two_left_eq_contrOneTwoLeft_tprod {c1 c2 : S.C} (x : S.F.obj (Ov
subst hy
conv_rhs =>
rw [contrOneTwoLeft_tprod_eq]
rw [OverColor.forgetLiftAppCon_inv_apply_expand]
simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Fin.succAbove_zero, mk_left,
Functor.id_obj, mk_hom, contr_tensor, prod_tensor, Action.instMonoidalCategory_tensorObj_V,
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
@ -184,7 +184,6 @@ lemma contr_one_two_left_eq_contrOneTwoLeft {c1 c2 : S.C} (x : S.F.obj (OverColo
simpa using contr_one_two_left_eq_contrOneTwoLeft_tprod (PiTensorProduct.tprod S.k fx)
(PiTensorProduct.tprod S.k fy) fx fy
/-- Expands the inner contraction of two 2-tensors which are
tprods in terms of basic categorical
constructions and fields of the tensor species. -/

19
HepLean/Tensors/TensorSpecies/RepIso.lean
View file

@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.Tensors.TensorSpecies.Basic
import HepLean.Tensors.TensorSpecies.MetricTensor
/-!
# Isomorphism between rep of color `c` and rep of dual color.
@ -40,6 +41,24 @@ lemma toDualRep_congr {c c' : S.C} (h : c = c') : S.toDualRep c = S.FD.map (Disc
def fromDualRep (c : S.C) : S.FD.obj (Discrete.mk (S.τ c)) ⟶ S.FD.obj (Discrete.mk c) :=
S.toDualRep (S.τ c) ≫ S.FD.map (Discrete.eqToHom (S.τ_involution c))
/-- The rewriting of `toDualRep` in terms of `contrOneTwoLeft`. -/
lemma toDualRep_apply_eq_contrOneTwoLeft (c : S.C) (x : S.FD.obj (Discrete.mk c)) :
(S.toDualRep c).hom x = (OverColor.forgetLiftAppCon S.FD (S.τ c)).hom.hom
(contrOneTwoLeft (((OverColor.forgetLiftAppCon S.FD c).inv.hom x))
(S.metricTensor (S.τ c))) := by
simp only [toDualRep, Monoidal.tensorUnit_obj, Action.comp_hom,
Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
Action.instMonoidalCategory_rightUnitor_inv_hom, Action.instMonoidalCategory_whiskerLeft_hom,
Action.instMonoidalCategory_associator_inv_hom, Action.instMonoidalCategory_whiskerRight_hom,
Action.instMonoidalCategory_leftUnitor_hom_hom, ModuleCat.coe_comp, Function.comp_apply,
ModuleCat.MonoidalCategory.rightUnitor_inv_apply, ModuleCat.MonoidalCategory.whiskerLeft_apply,
Nat.succ_eq_add_one, Nat.reduceAdd, contrOneTwoLeft, Functor.comp_obj,
Discrete.functor_obj_eq_as, Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, OverColor.Discrete.rep_iso_hom_inv_apply]
repeat apply congrArg
erw [pairIsoSep_inv_metricTensor]
rfl
end TensorSpecies
end

3
HepLean/Tensors/TensorSpecies/UnitTensor.lean
View file

@ -120,7 +120,8 @@ lemma contrOneTwoLeft_unitTensor {c1 : S.C} (x : S.F.obj (OverColor.mk ![c1]))
Action.instMonoidalCategory_leftUnitor_hom_hom, Monoidal.tensorUnit_obj,
Action.instMonoidalCategory_whiskerRight_hom, Functor.comp_obj, Discrete.functor_obj_eq_as,
Function.comp_apply, Action.instMonoidalCategory_associator_inv_hom, Equivalence.symm_inverse,
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj]
Action.functorCategoryEquivalence_functor, Action.FunctorCategoryEquivalence.functor_obj_obj,
forgetLiftAppCon_inv_apply_expand]
erw [pairIsoSep_inv_unitTensor (S := S) (c := c1)]
change (S.F.mapIso (mkIso _)).hom.hom _ = _
rw [Discrete.rep_iso_apply_iff, Discrete.rep_iso_inv_apply_iff]