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[Merged by Bors] - feat(CategoryTheory/Monoidal): partially setting simp lemmas #10061

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[Merged by Bors] - feat(CategoryTheory/Monoidal): partially setting simp lemmas #10061

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update
yuma-mizuno Jan 28, 2024
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yuma-mizuno Jan 28, 2024
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clean up
yuma-mizuno Jan 28, 2024
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reassoc
yuma-mizuno Jan 30, 2024
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5 changes: 5 additions & 0 deletions Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean
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Expand Up @@ -82,6 +82,11 @@ noncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R
simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,
Iso.refl_hom, MonoidalCategory.tensor_id]
erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]
leftUnitor_eq := fun X => by
dsimp only [forget₂_module_obj, forget₂_module_map, Iso.refl_symm, Iso.trans_hom,
Iso.refl_hom, tensorIso_hom]
simp only [MonoidalCategory.leftUnitor_conjugation, Category.id_comp, Iso.hom_inv_id]
rfl
rightUnitor_eq := fun X => by
dsimp
erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]
Expand Down
23 changes: 5 additions & 18 deletions Mathlib/CategoryTheory/Bicategory/SingleObj.lean
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Expand Up @@ -53,18 +53,12 @@ instance : Bicategory (MonoidalSingleObj C) where
Hom _ _ := C
id _ := 𝟙_ C
comp X Y := tensorObj X Y
whiskerLeft X Y Z f := tensorHom (𝟙 X) f
whiskerRight f Z := tensorHom f (𝟙 Z)
whiskerLeft X Y Z f := X ◁ f
whiskerRight f Z := f ▷ Z
associator X Y Z := α_ X Y Z
leftUnitor X := λ_ X
rightUnitor X := ρ_ X
comp_whiskerLeft _ _ _ _ _ := by
simp_rw [associator_inv_naturality, Iso.hom_inv_id_assoc, tensor_id]
whisker_assoc _ _ _ _ _ := by simp_rw [associator_inv_naturality, Iso.hom_inv_id_assoc]
whiskerRight_comp _ _ _ := by simp_rw [← tensor_id, associator_naturality, Iso.inv_hom_id_assoc]
id_whiskerLeft _ := by simp_rw [leftUnitor_inv_naturality, Iso.hom_inv_id_assoc]
whiskerRight_id _ := by simp_rw [rightUnitor_inv_naturality, Iso.hom_inv_id_assoc]
pentagon _ _ _ _ := by simp_rw [pentagon]
whisker_exchange := whisker_exchange

namespace MonoidalSingleObj

Expand All @@ -74,6 +68,8 @@ protected def star : MonoidalSingleObj C :=
PUnit.unit
#align category_theory.monoidal_single_obj.star CategoryTheory.MonoidalSingleObj.star

attribute [local simp] id_tensorHom tensorHom_id in

/-- The monoidal functor from the endomorphisms of the single object
when we promote a monoidal category to a single object bicategory,
to the original monoidal category.
Expand All @@ -86,15 +82,6 @@ def endMonoidalStarFunctor : MonoidalFunctor (EndMonoidal (MonoidalSingleObj.sta
map f := f
ε := 𝟙 _
μ X Y := 𝟙 _
-- The proof will be automated after merging #6307.
μ_natural_left f g := by
simp_rw [Category.id_comp, Category.comp_id]
-- Should we provide further simp lemmas so this goal becomes visible?
exact (tensor_id_comp_id_tensor _ _).symm
μ_natural_right f g := by
simp_rw [Category.id_comp, Category.comp_id]
-- Should we provide further simp lemmas so this goal becomes visible?
exact (tensor_id_comp_id_tensor _ _).symm
#align category_theory.monoidal_single_obj.End_monoidal_star_functor CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor

/-- The equivalence between the endomorphisms of the single object
Expand Down
2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Closed/FunctorCategory.lean
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Expand Up @@ -56,7 +56,7 @@ def closedCounit (F : D ⥤ C) : closedIhom F ⋙ tensorLeft F ⟶ 𝟭 (D ⥤ C
dsimp
simp only [closedIhom_obj_map, pre_comm_ihom_map]
rw [← tensor_id_comp_id_tensor, id_tensor_comp]
simp }
simp [tensor_id_comp_id_tensor_assoc] }
#align category_theory.functor.closed_counit CategoryTheory.Functor.closedCounit

/-- If `C` is a monoidal closed category and `D` is a groupoid, then every functor `F : D ⥤ C` is
Expand Down
4 changes: 2 additions & 2 deletions Mathlib/CategoryTheory/Monoidal/Braided.lean
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Expand Up @@ -200,7 +200,7 @@ theorem braiding_leftUnitor_aux₁ (X : C) :
(α_ (𝟙_ C) (𝟙_ C) X).hom ≫
(𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ⊗ 𝟙 _) =
((λ_ _).hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv :=
by rw [← leftUnitor_tensor, leftUnitor_naturality]; simp
by rw [← leftUnitor_tensor, leftUnitor_naturality]; simp [id_tensorHom, tensorHom_id]
#align category_theory.braiding_left_unitor_aux₁ CategoryTheory.braiding_leftUnitor_aux₁

theorem braiding_leftUnitor_aux₂ (X : C) :
Expand Down Expand Up @@ -233,7 +233,7 @@ theorem braiding_rightUnitor_aux₁ (X : C) :
(α_ X (𝟙_ C) (𝟙_ C)).inv ≫
((β_ (𝟙_ C) X).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ X _).hom ≫ (𝟙 _ ⊗ (ρ_ X).hom) =
(𝟙 X ⊗ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv :=
by rw [← rightUnitor_tensor, rightUnitor_naturality]; simp
by rw [← rightUnitor_tensor, rightUnitor_naturality]; simp [id_tensorHom, tensorHom_id]
#align category_theory.braiding_right_unitor_aux₁ CategoryTheory.braiding_rightUnitor_aux₁

theorem braiding_rightUnitor_aux₂ (X : C) :
Expand Down