Documentation

Mathlib.CategoryTheory.Monoidal.CommMon_

The category of commutative monoids in a braided monoidal category. #

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structure CommMon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] extends Mon_ C :
Type (max u₁ v₁)

A commutative monoid object internal to a monoidal category.

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@[simp]
theorem CommMon_.mul_comm_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (self : CommMon_ C) {Z : C} (h : self.X ⟶ Z) :
CategoryTheory.CategoryStruct.comp (β_ self.X self.X).hom (CategoryTheory.CategoryStruct.comp self.mul h) = CategoryTheory.CategoryStruct.comp self.mul h
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def CommMon_.trivial (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CommMon_ C

The trivial commutative monoid object. We later show this is initial in CommMon_ C.

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theorem CommMon_.trivial_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(trivial C).one = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)
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@[simp]
theorem CommMon_.trivial_mul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(trivial C).mul = (CategoryTheory.MonoidalCategoryStruct.leftUnitor (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).hom
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@[simp]
theorem CommMon_.trivial_X (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(trivial C).X = CategoryTheory.MonoidalCategoryStruct.tensorUnit C
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instance CommMon_.instInhabited (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
Inhabited (CommMon_ C)
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instance CommMon_.instCategory {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Category.{v₁, max u₁ v₁} (CommMon_ C)
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theorem CommMon_.id_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
(CategoryTheory.CategoryStruct.id A).hom = CategoryTheory.CategoryStruct.id A.X
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@[simp]
theorem CommMon_.comp_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {R S T : CommMon_ C} (f : R ⟶ S) (g : S ⟶ T) :
(CategoryTheory.CategoryStruct.comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom
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theorem CommMon_.hom_ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {A B : CommMon_ C} (f g : A ⟶ B) (h : f.hom = g.hom) :
f = g
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theorem CommMon_.hom_ext_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {A B : CommMon_ C} {f g : A ⟶ B} :
f = g ↔ f.hom = g.hom
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@[simp]
theorem CommMon_.id' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
CategoryTheory.CategoryStruct.id A = CategoryTheory.CategoryStruct.id A.toMon_
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@[simp]
theorem CommMon_.comp' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {A₁ A₂ A₃ : CommMon_ C} (f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) :
CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f g
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def CommMon_.forget₂Mon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Functor (CommMon_ C) (Mon_ C)

The forgetful functor from commutative monoid objects to monoid objects.

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def CommMon_.fullyFaithfulForget₂Mon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(forget₂Mon_ C).FullyFaithful

The forgetful functor from commutative monoid objects to monoid objects is fully faithful.

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instance CommMon_.instFullMon_Forget₂Mon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(forget₂Mon_ C).Full
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instance CommMon_.instFaithfulMon_Forget₂Mon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(forget₂Mon_ C).Faithful
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@[simp]
theorem CommMon_.forget₂Mon_obj_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
((forget₂Mon_ C).obj A).one = A.one
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@[simp]
theorem CommMon_.forget₂Mon_obj_mul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
((forget₂Mon_ C).obj A).mul = A.mul
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theorem CommMon_.forget₂Mon_map_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {A B : CommMon_ C} (f : A ⟶ B) :
((forget₂Mon_ C).map f).hom = f.hom
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@[deprecated CommMon_.forget₂Mon_obj_one (since := "2025-02-07")]
theorem CommMon_.forget₂_Mon_obj_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
((forget₂Mon_ C).obj A).one = A.one

Alias of CommMon_.forget₂Mon_obj_one.

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@[deprecated CommMon_.forget₂Mon_obj_mul (since := "2025-02-07")]
theorem CommMon_.forget₂_Mon_obj_mul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
((forget₂Mon_ C).obj A).mul = A.mul

Alias of CommMon_.forget₂Mon_obj_mul.

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@[deprecated CommMon_.forget₂Mon_map_hom (since := "2025-02-07")]
theorem CommMon_.forget₂_Mon_map_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {A B : CommMon_ C} (f : A ⟶ B) :
((forget₂Mon_ C).map f).hom = f.hom

Alias of CommMon_.forget₂Mon_map_hom.

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def CommMon_.forget (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Functor (CommMon_ C) C

The forgetful functor from commutative monoid objects to the ambient category.

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theorem CommMon_.forget_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {X✝ Y✝ : CommMon_ C} (f : X✝ ⟶ Y✝) :
(forget C).map f = f.hom
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@[simp]
theorem CommMon_.forget_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CommMon_ C) :
(forget C).obj X = ((forget₂Mon_ C).obj X).X
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instance CommMon_.instFaithfulForget (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(forget C).Faithful
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theorem CommMon_.forget₂Mon_comp_forget (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(forget₂Mon_ C).comp (Mon_.forget C) = forget C
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def CommMon_.mkIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {M N : CommMon_ C} (f : M.X ≅ N.X) (one_f : CategoryTheory.CategoryStruct.comp M.one f.hom = N.one := by aesop_cat) (mul_f : CategoryTheory.CategoryStruct.comp M.mul f.hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom f.hom f.hom) N.mul := by aesop_cat) :
M ≅ N

Constructor for isomorphisms in the category CommMon_ C.

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theorem CommMon_.mkIso_hom_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {M N : CommMon_ C} (f : M.X ≅ N.X) (one_f : CategoryTheory.CategoryStruct.comp M.one f.hom = N.one := by aesop_cat) (mul_f : CategoryTheory.CategoryStruct.comp M.mul f.hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom f.hom f.hom) N.mul := by aesop_cat) :
(mkIso f one_f mul_f).hom.hom = f.hom
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@[simp]
theorem CommMon_.mkIso_inv_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {M N : CommMon_ C} (f : M.X ≅ N.X) (one_f : CategoryTheory.CategoryStruct.comp M.one f.hom = N.one := by aesop_cat) (mul_f : CategoryTheory.CategoryStruct.comp M.mul f.hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom f.hom f.hom) N.mul := by aesop_cat) :
(mkIso f one_f mul_f).inv.hom = f.inv
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instance CommMon_.uniqueHomFromTrivial {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
Unique (trivial C ⟶ A)
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instance CommMon_.instHasInitial {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Limits.HasInitial (CommMon_ C)
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def CategoryTheory.Functor.mapCommMon {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (F : Functor C D) [F.LaxBraided] :
Functor (CommMon_ C) (CommMon_ D)

A lax braided functor takes commutative monoid objects to commutative monoid objects.

That is, a lax braided functor F : C ⥤ D induces a functor CommMon_ C ⥤ CommMon_ D.

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theorem CategoryTheory.Functor.mapCommMon_obj_mul {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (F : Functor C D) [F.LaxBraided] (A : CommMon_ C) :
(F.mapCommMon.obj A).mul = CategoryStruct.comp (LaxMonoidal.μ F A.X A.X) (F.map A.mul)
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theorem CategoryTheory.Functor.mapCommMon_map_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (F : Functor C D) [F.LaxBraided] {X✝ Y✝ : CommMon_ C} (f : X✝ ⟶ Y✝) :
(F.mapCommMon.map f).hom = F.map f.hom
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theorem CategoryTheory.Functor.mapCommMon_obj_one {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (F : Functor C D) [F.LaxBraided] (A : CommMon_ C) :
(F.mapCommMon.obj A).one = CategoryStruct.comp (LaxMonoidal.ε F) (F.map A.one)
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theorem CategoryTheory.Functor.mapCommMon_obj_X {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (F : Functor C D) [F.LaxBraided] (A : CommMon_ C) :
(F.mapCommMon.obj A).X = F.obj A.X
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noncomputable def CategoryTheory.Functor.mapCommMonIdIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
(Functor.id C).mapCommMon ≅ Functor.id (CommMon_ C)

The identity functor is also the identity on commutative monoid objects.

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theorem CategoryTheory.Functor.mapCommMonIdIso_inv_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon_ C) :
(mapCommMonIdIso.inv.app X).hom = CategoryStruct.id X.X
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theorem CategoryTheory.Functor.mapCommMonIdIso_hom_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon_ C) :
(mapCommMonIdIso.hom.app X).hom = CategoryStruct.id X.X
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noncomputable def CategoryTheory.Functor.mapCommMonCompIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [MonoidalCategory E] [BraidedCategory E] {F : Functor C D} [F.LaxBraided] {G : Functor D E} [G.LaxBraided] :
(F.comp G).mapCommMon ≅ F.mapCommMon.comp G.mapCommMon

The composition functor is also the composition on commutative monoid objects.

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theorem CategoryTheory.Functor.mapCommMonCompIso_hom_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [MonoidalCategory E] [BraidedCategory E] {F : Functor C D} [F.LaxBraided] {G : Functor D E} [G.LaxBraided] (X : CommMon_ C) :
(mapCommMonCompIso.hom.app X).hom = CategoryStruct.id (G.obj (F.obj X.X))
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@[simp]
theorem CategoryTheory.Functor.mapCommMonCompIso_inv_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [MonoidalCategory E] [BraidedCategory E] {F : Functor C D} [F.LaxBraided] {G : Functor D E} [G.LaxBraided] (X : CommMon_ C) :
(mapCommMonCompIso.inv.app X).hom = CategoryStruct.id (G.obj (F.obj X.X))
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def CategoryTheory.Functor.mapCommMonFunctor (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (D : Type u₂) [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] :
Functor (LaxBraidedFunctor C D) (Functor (CommMon_ C) (CommMon_ D))

mapCommMon is functorial in the lax braided functor.

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theorem CategoryTheory.Functor.mapCommMonFunctor_obj (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (D : Type u₂) [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (F : LaxBraidedFunctor C D) :
(mapCommMonFunctor C D).obj F = F.mapCommMon
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@[simp]
theorem CategoryTheory.Functor.mapCommMonFunctor_map_app_hom (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (D : Type u₂) [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {X✝ Y✝ : LaxBraidedFunctor C D} (α : X✝ ⟶ Y✝) (A : CommMon_ C) :
(((mapCommMonFunctor C D).map α).app A).hom = α.hom.app A.X
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noncomputable def CategoryTheory.Functor.mapCommMonNatTrans {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F F' : Functor C D} [F.LaxBraided] [F'.LaxBraided] (f : F ⟶ F') [NatTrans.IsMonoidal f] :
F.mapCommMon ⟶ F'.mapCommMon

Natural transformations between functors lift to monoid objects.

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theorem CategoryTheory.Functor.mapCommMonNatTrans_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F F' : Functor C D} [F.LaxBraided] [F'.LaxBraided] (f : F ⟶ F') [NatTrans.IsMonoidal f] (X : CommMon_ C) :
((mapCommMonNatTrans f).app X).hom = f.app X.X
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noncomputable def CategoryTheory.Functor.mapCommMonNatIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F F' : Functor C D} [F.LaxBraided] [F'.LaxBraided] (e : F ≅ F') [NatTrans.IsMonoidal e.hom] :
F.mapCommMon ≅ F'.mapCommMon

Natural isomorphisms between functors lift to monoid objects.

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@[simp]
theorem CategoryTheory.Functor.mapCommMonNatIso_inv_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F F' : Functor C D} [F.LaxBraided] [F'.LaxBraided] (e : F ≅ F') [NatTrans.IsMonoidal e.hom] (X : CommMon_ C) :
((mapCommMonNatIso e).inv.app X).hom = e.inv.app X.X
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@[simp]
theorem CategoryTheory.Functor.mapCommMonNatIso_hom_app_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F F' : Functor C D} [F.LaxBraided] [F'.LaxBraided] (e : F ≅ F') [NatTrans.IsMonoidal e.hom] (X : CommMon_ C) :
((mapCommMonNatIso e).hom.app X).hom = e.hom.app X.X
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noncomputable def CategoryTheory.Adjunction.mapCommMon {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Braided] [G.LaxBraided] [a.IsMonoidal] :
F.mapCommMon ⊣ G.mapCommMon

An adjunction of braided functors lifts to an adjunction of their lifts to commutative monoid objects.

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@[simp]
theorem CategoryTheory.Adjunction.mapCommMon_counit {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Braided] [G.LaxBraided] [a.IsMonoidal] :
a.mapCommMon.counit = CategoryStruct.comp Functor.mapCommMonCompIso.inv (CategoryStruct.comp (Functor.mapCommMonNatTrans a.counit) Functor.mapCommMonIdIso.hom)
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@[simp]
theorem CategoryTheory.Adjunction.mapCommMon_unit {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Braided] [G.LaxBraided] [a.IsMonoidal] :
a.mapCommMon.unit = CategoryStruct.comp Functor.mapCommMonIdIso.inv (CategoryStruct.comp (Functor.mapCommMonNatTrans a.unit) Functor.mapCommMonCompIso.hom)
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noncomputable def CategoryTheory.Equivalence.mapCommMon {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (e : C ≌ D) [e.functor.Braided] [e.inverse.Braided] [e.IsMonoidal] :
CommMon_ C ≌ CommMon_ D

An equivalence of categories lifts to an equivalence of their commutative monoid objects.

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@[simp]
theorem CategoryTheory.Equivalence.mapCommMon_unitIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (e : C ≌ D) [e.functor.Braided] [e.inverse.Braided] [e.IsMonoidal] :
e.mapCommMon.unitIso = Functor.mapCommMonIdIso.symm ≪≫ Functor.mapCommMonNatIso e.unitIso ≪≫ Functor.mapCommMonCompIso
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@[simp]
theorem CategoryTheory.Equivalence.mapCommMon_counitIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (e : C ≌ D) [e.functor.Braided] [e.inverse.Braided] [e.IsMonoidal] :
e.mapCommMon.counitIso = Functor.mapCommMonCompIso.symm ≪≫ Functor.mapCommMonNatIso e.counitIso ≪≫ Functor.mapCommMonIdIso
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@[simp]
theorem CategoryTheory.Equivalence.mapCommMon_inverse {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (e : C ≌ D) [e.functor.Braided] [e.inverse.Braided] [e.IsMonoidal] :
e.mapCommMon.inverse = e.inverse.mapCommMon
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@[simp]
theorem CategoryTheory.Equivalence.mapCommMon_functor {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [MonoidalCategory D] [BraidedCategory D] (e : C ≌ D) [e.functor.Braided] [e.inverse.Braided] [e.IsMonoidal] :
e.mapCommMon.functor = e.functor.mapCommMon
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def CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Functor (CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C) (CommMon_ C)

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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theorem CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {X✝ Y✝ : CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C} (α : X✝ ⟶ Y✝) :
(laxBraidedToCommMon C).map α = ((CategoryTheory.Functor.mapCommMonFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C).map α).app (trivial (CategoryTheory.Discrete PUnit.{u + 1}))
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@[simp]
theorem CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (F : CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C) :
(laxBraidedToCommMon C).obj F = F.mapCommMon.obj (trivial (CategoryTheory.Discrete PUnit.{u + 1}))
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def CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{u + 1}) C

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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theorem CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) (x✝ : CategoryTheory.Discrete PUnit.{u + 1}) :
(commMonToLaxBraidedObj A).obj x✝ = A.X
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@[simp]
theorem CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) {X✝ Y✝ : CategoryTheory.Discrete PUnit.{u + 1}} (x✝ : X✝ ⟶ Y✝) :
(commMonToLaxBraidedObj A).map x✝ = CategoryTheory.CategoryStruct.id A.X
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instance CommMon_.EquivLaxBraidedFunctorPUnit.instLaxMonoidalDiscretePUnitCommMonToLaxBraidedObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
(commMonToLaxBraidedObj A).LaxMonoidal
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theorem CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_ε {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
CategoryTheory.Functor.LaxMonoidal.ε (commMonToLaxBraidedObj A) = A.one
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@[simp]
theorem CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_μ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) (X Y : CategoryTheory.Discrete PUnit.{u_1 + 1}) :
CategoryTheory.Functor.LaxMonoidal.μ (commMonToLaxBraidedObj A) X Y = A.mul
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instance CommMon_.EquivLaxBraidedFunctorPUnit.instLaxBraidedDiscretePUnitCommMonToLaxBraidedObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
(commMonToLaxBraidedObj A).LaxBraided
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def CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Functor (CommMon_ C) (CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C)

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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theorem CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :
(commMonToLaxBraided C).obj A = CategoryTheory.LaxBraidedFunctor.of (commMonToLaxBraidedObj A)
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@[simp]
theorem CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided_map_hom_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {X✝ Y✝ : CommMon_ C} (f : X✝ ⟶ Y✝) (x✝ : CategoryTheory.Discrete PUnit.{u + 1}) :
((commMonToLaxBraided C).map f).hom.app x✝ = f.hom
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def CommMon_.EquivLaxBraidedFunctorPUnit.unitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.Functor.id (CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C) ≅ (laxBraidedToCommMon C).comp (commMonToLaxBraided C)

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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theorem CommMon_.EquivLaxBraidedFunctorPUnit.unitIso_hom_app_hom_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C) (X✝ : CategoryTheory.Discrete PUnit.{u + 1}) :
((unitIso C).hom.app X).hom.app X✝ = CategoryTheory.CategoryStruct.id (X.obj X✝)
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@[simp]
theorem CommMon_.EquivLaxBraidedFunctorPUnit.unitIso_inv_app_hom_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C) (X✝ : CategoryTheory.Discrete PUnit.{u + 1}) :
((unitIso C).inv.app X).hom.app X✝ = CategoryTheory.CategoryStruct.id (X.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Discrete PUnit.{u + 1})))
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def CommMon_.EquivLaxBraidedFunctorPUnit.counitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(commMonToLaxBraided C).comp (laxBraidedToCommMon C) ≅ CategoryTheory.Functor.id (CommMon_ C)

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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theorem CommMon_.EquivLaxBraidedFunctorPUnit.counitIso_hom_app_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CommMon_ C) :
((counitIso C).hom.app X).hom = CategoryTheory.CategoryStruct.id X.X
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@[simp]
theorem CommMon_.EquivLaxBraidedFunctorPUnit.counitIso_inv_app_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CommMon_ C) :
((counitIso C).inv.app X).hom = CategoryTheory.CategoryStruct.id X.X
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def CommMon_.equivLaxBraidedFunctorPUnit (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
CategoryTheory.LaxBraidedFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C ≌ CommMon_ C

Commutative monoid objects in C are "just" braided lax monoidal functors from the trivial braided monoidal category to C.

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theorem CommMon_.equivLaxBraidedFunctorPUnit_unitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(equivLaxBraidedFunctorPUnit C).unitIso = EquivLaxBraidedFunctorPUnit.unitIso C
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@[simp]
theorem CommMon_.equivLaxBraidedFunctorPUnit_inverse (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(equivLaxBraidedFunctorPUnit C).inverse = EquivLaxBraidedFunctorPUnit.commMonToLaxBraided C
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@[simp]
theorem CommMon_.equivLaxBraidedFunctorPUnit_functor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(equivLaxBraidedFunctorPUnit C).functor = EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C
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@[simp]
theorem CommMon_.equivLaxBraidedFunctorPUnit_counitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :
(equivLaxBraidedFunctorPUnit C).counitIso = EquivLaxBraidedFunctorPUnit.counitIso C
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def CommMon_.mk' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : C) [Mon_Class X] [IsCommMon X] :
CommMon_ C

Construct an object of CommMon_ C from an object X : C a Mon_Class X instance and a IsCommMon X instance.

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instance CommMon_.instIsCommMonX {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CommMon_ C) :
IsCommMon X.X