[Merged by Bors] - refactor(CategoryTheory/Monoidal/Braided): use monoidalComp in the proofs #10078
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yuma-mizuno
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Jan 28, 2024
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yuma-mizuno
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- depends on: [Merged by Bors] - feat(CategoryTheory/Monoidal): partially setting simp lemmas #10061
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joelriou
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| #align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂ | ||
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| @[simp] | ||
| theorem braiding_leftUnitor (X : C) : (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom = (ρ_ X).hom := by |
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We could add the reassoc attribute here.
joelriou
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Feb 1, 2024
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| theorem braiding_rightUnitor (X : C) : (β_ (𝟙_ C) X).hom ≫ (ρ_ X).hom = (λ_ X).hom := by | ||
| rw [← tensor_left_iff, id_tensor_comp, braiding_rightUnitor_aux₂] | ||
| rw [← whiskerLeft_iff, MonoidalCategory.whiskerLeft_comp, braiding_rightUnitor_aux₂] | ||
| #align category_theory.braiding_right_unitor CategoryTheory.braiding_rightUnitor | ||
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| @[simp] | ||
| theorem braiding_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).hom = (λ_ X).hom ≫ (ρ_ X).inv := by | ||
| simp [← braiding_rightUnitor] | ||
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| theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv := by | ||
| apply (cancel_mono (ρ_ X).hom).1 | ||
| simp only [assoc, braiding_rightUnitor, Iso.inv_hom_id] | ||
| simp | ||
| #align category_theory.left_unitor_inv_braiding CategoryTheory.leftUnitor_inv_braiding | ||
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| @[simp] | ||
| theorem rightUnitor_inv_braiding (X : C) : (ρ_ X).inv ≫ (β_ X (𝟙_ C)).hom = (λ_ X).inv := by |
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These lemmas braiding_rightUnitor, braiding_tensorUnit_left, leftUnitor_inv_braiding and rightUnitor_inv_braiding should have the reassoc attr, as well as braiding_tensorUnit_right below.
joelriou
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Feb 1, 2024
| (𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv := | ||
| by dsimp [tensor_μ]; simp | ||
| #align category_theory.tensor_μ_def₂ CategoryTheory.tensor_μ_def₂ | ||
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| theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : U₁ ⟶ V₁) |
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reassoc could be useful here also.
joelriou
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Feb 1, 2024
| convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1; simp | ||
| (Z₁ ⊗ Z₂) ◁ (f₁ ⊗ f₂) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) = | ||
| tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂) := by | ||
| convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1 <;> simp | ||
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| theorem tensor_left_unitality (X₁ X₂ : C) : |
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reassoc should also be added to lemmas tensor_left_unitality, tensor_right_unitality, tensor_associativity, leftUnitor_monoidal, rightUnitor_monoidal, associator_monoidal.
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Adding reassoc to tensor_associativity and associator_monoidal seems to give errors unless we increase the maxHeartbeats. Do you think it's OK to do this, or should we postpone adding reassoc until some optimization is done?
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Adding the line attribute [reassoc] tensor_associativity associator_monoidal after the declarations seem to work without increasing maxHeartbeats.
joelriou
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Feb 1, 2024
| @[reassoc] | ||
| theorem leftUnitor_inv_naturality' {X Y : C} (f : X ⟶ Y) : | ||
| f ≫ (λ_ Y).inv = (λ_ X).inv ≫ (_ ◁ f) := by simp | ||
| f ≫ (λ_ Y).inv = (λ_ X).inv ≫ _ ◁ f := by simp | ||
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| theorem id_whiskerLeft_symm {X X' : C} (f : X ⟶ X') : |
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reassoc can be added to id_whiskerLeft_symm and whiskerRight_id_symm
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Feb 1, 2024
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Thanks! bors merge |
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