The category of commutative monoids in a braided monoidal category.

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.

• X : C
• one : 𝟙_ C ⟶ self.X
• mul : CategoryTheory.MonoidalCategoryStruct.tensorObj self.X self.X ⟶ self.X
• one_mul : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight self.one self.X) self.mul = (CategoryTheory.MonoidalCategoryStruct.leftUnitor self.X).hom
• mul_one : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft self.X self.one) self.mul = (CategoryTheory.MonoidalCategoryStruct.rightUnitor self.X).hom
• mul_assoc : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight self.mul self.X) self.mul = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator self.X self.X self.X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft self.X self.mul) self.mul)
• mul_comm : CategoryTheory.CategoryStruct.comp (β_ self.X self.X).hom self.mul = self.mul

@[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

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.

Equations

• CommMon_.trivial C = { toMon_ := Mon_.trivial C, mul_comm := ⋯ }

@[simp]

theorem CommMon_.trivial_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (trivial C).one = CategoryTheory.CategoryStruct.id (𝟙_ C)

@[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 (𝟙_ C)).hom

@[simp]

theorem CommMon_.trivial_X (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (trivial C).X = 𝟙_ C

instance CommMon_.instInhabited (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : Inhabited (CommMon_ C)

Equations

• CommMon_.instInhabited C = { default := CommMon_.trivial C }

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)

Equations

• CommMon_.instCategory = CategoryTheory.InducedCategory.category CommMon_.toMon_

@[simp]

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

@[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

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

@[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_

@[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

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.

Equations

• CommMon_.forget₂Mon_ C = CategoryTheory.inducedFunctor CommMon_.toMon_

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.

Equations

• CommMon_.fullyFaithfulForget₂Mon_ C = CategoryTheory.fullyFaithfulInducedFunctor CommMon_.toMon_

instance CommMon_.instFullMon_Forget₂Mon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (forget₂Mon_ C).Full

instance CommMon_.instFaithfulMon_Forget₂Mon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (forget₂Mon_ C).Faithful

@[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

@[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

@[simp]

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

@[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.

@[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.

@[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.

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.

Equations

• CommMon_.forget C = (CommMon_.forget₂Mon_ C).comp (Mon_.forget C)

@[simp]

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

@[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

instance CommMon_.instFaithfulForget (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (forget C).Faithful

@[simp]

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

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.

Equations

• CommMon_.mkIso f one_f mul_f = (CommMon_.fullyFaithfulForget₂Mon_ C).preimageIso (Mon_.mkIso f one_f mul_f)

@[simp]

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

@[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

instance CommMon_.uniqueHomFromTrivial {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) : Unique (trivial C ⟶ A)

Equations

• A.uniqueHomFromTrivial = A.uniqueHomFromTrivial

instance CommMon_.instHasInitial {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : CategoryTheory.Limits.HasInitial (CommMon_ C)

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.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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)

@[simp]

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

@[simp]

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

@[simp]

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)

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.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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

@[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

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.

Equations

• One or more equations did not get rendered due to their size.

@[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}))

@[simp]

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}))

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.

Equations

• CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj A = (CategoryTheory.Functor.const (CategoryTheory.Discrete PUnit.{?u.45 + 1})).obj A.X

@[simp]

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

@[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

instance CommMon_.EquivLaxBraidedFunctorPUnit.instLaxMonoidalDiscretePUnitCommMonToLaxBraidedObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) : (commMonToLaxBraidedObj A).LaxMonoidal

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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

@[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

instance CommMon_.EquivLaxBraidedFunctorPUnit.instLaxBraidedDiscretePUnitCommMonToLaxBraidedObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) : (commMonToLaxBraidedObj A).LaxBraided

Equations

• CommMon_.EquivLaxBraidedFunctorPUnit.instLaxBraidedDiscretePUnitCommMonToLaxBraidedObj A = CategoryTheory.Functor.LaxBraided.mk ⋯

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.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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)

@[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

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.

Equations

• One or more equations did not get rendered due to their size.

@[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 (𝟙_ (CategoryTheory.Discrete PUnit.{u + 1})))

@[simp]

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✝)

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.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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

@[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

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.

Equations

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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

@[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

@[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

@[simp]

theorem CommMon_.equivLaxBraidedFunctorPUnit_unitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] : (equivLaxBraidedFunctorPUnit C).unitIso = EquivLaxBraidedFunctorPUnit.unitIso C