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.

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

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

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theorem CommMon_.trivial_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :

(CommMon_.trivial C).one = CategoryTheory.CategoryStruct.id (𝟙_ C)

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

(CommMon_.trivial C).X = 𝟙_ C

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theorem CommMon_.trivial_mul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :

(CommMon_.trivial C).mul = (CategoryTheory.MonoidalCategoryStruct.leftUnitor (𝟙_ C)).hom

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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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CommMon_.instInhabited C = { default := CommMon_.trivial 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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CommMon_.instCategory = CategoryTheory.InducedCategory.category CommMon_.toMon_

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

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

(CommMon_.forget₂Mon_ C).FullyFaithful

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

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CommMon_.fullyFaithfulForget₂Mon_ C = CategoryTheory.fullyFaithfulInducedFunctor CommMon_.toMon_

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

(CommMon_.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] :

(CommMon_.forget₂Mon_ C).Faithful

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theorem CommMon_.forget₂_Mon_obj_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :

((CommMon_.forget₂Mon_ C).obj A).one = A.one

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theorem CommMon_.forget₂_Mon_obj_mul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : CommMon_ C) :

((CommMon_.forget₂Mon_ C).obj A).mul = A.mul

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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) :

((CommMon_.forget₂Mon_ C).map f).hom = f.hom

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

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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) :

(CommMon_.mkIso f one_f mul_f).hom.hom = f.hom

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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) :

(CommMon_.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 (CommMon_.trivial C ⟶ A)

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A.uniqueHomFromTrivial = A.uniqueHomFromTrivial

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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} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : CategoryTheory.Functor C D) [F.LaxBraided] :

CategoryTheory.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} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : CategoryTheory.Functor C D) [F.LaxBraided] (A : CommMon_ C) :

(F.mapCommMon.obj A).mul = CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.μ F A.X A.X) (F.map A.mul)

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theorem CategoryTheory.Functor.mapCommMon_obj_X {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : CategoryTheory.Functor C D) [F.LaxBraided] (A : CommMon_ C) :

(F.mapCommMon.obj A).X = F.obj A.X

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theorem CategoryTheory.Functor.mapCommMon_map_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : CategoryTheory.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} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : CategoryTheory.Functor C D) [F.LaxBraided] (A : CommMon_ C) :

(F.mapCommMon.obj A).one = CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.ε F) (F.map A.one)

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def CategoryTheory.Functor.mapCommMonFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] :

CategoryTheory.Functor (CategoryTheory.LaxBraidedFunctor C D) (CategoryTheory.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) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : CategoryTheory.LaxBraidedFunctor C D) :

(CategoryTheory.Functor.mapCommMonFunctor C D).obj F = F.mapCommMon

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theorem CategoryTheory.Functor.mapCommMonFunctor_map_app_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] {X✝ Y✝ : CategoryTheory.LaxBraidedFunctor C D} (α : X✝ ⟶ Y✝) (A : CommMon_ C) :

(((CategoryTheory.Functor.mapCommMonFunctor C D).map α).app A).hom = α.hom.app A.X

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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✝) :

(CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C).map α = ((CategoryTheory.Functor.mapCommMonFunctor (CategoryTheory.Discrete PUnit.{u + 1}) C).map α).app (CommMon_.trivial (CategoryTheory.Discrete PUnit.{u + 1}))

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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) :

(CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C).obj F = F.mapCommMon.obj (CommMon_.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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CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj A = (CategoryTheory.Functor.const (CategoryTheory.Discrete PUnit.{?u.40 + 1})).obj A.X

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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}) :

(CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj A).obj x✝ = A.X

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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✝) :

(CommMon_.EquivLaxBraidedFunctorPUnit.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) :

(CommMon_.EquivLaxBraidedFunctorPUnit.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 (CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj A) = A.one

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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 (CommMon_.EquivLaxBraidedFunctorPUnit.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) :

(CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj A).LaxBraided

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

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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) :

(CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided C).obj A = CategoryTheory.LaxBraidedFunctor.of (CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj A)

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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}) :

((CommMon_.EquivLaxBraidedFunctorPUnit.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) ≅ (CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C).comp (CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided C)

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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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}) :

((CommMon_.EquivLaxBraidedFunctorPUnit.unitIso C).inv.app X).hom.app X✝ = CategoryTheory.CategoryStruct.id (X.obj (𝟙_ (CategoryTheory.Discrete PUnit.{u + 1})))

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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}) :

((CommMon_.EquivLaxBraidedFunctorPUnit.unitIso C).hom.app X).hom.app X✝ = CategoryTheory.CategoryStruct.id (X.obj X✝)

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

(CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided C).comp (CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C) ≅ CategoryTheory.Functor.id (CommMon_ C)

Implementation of CommMon_.equivLaxBraidedFunctorPUnit.

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theorem CommMon_.EquivLaxBraidedFunctorPUnit.counitIso_inv_app_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (X : CommMon_ C) :

((CommMon_.EquivLaxBraidedFunctorPUnit.counitIso C).inv.app X).hom = CategoryTheory.CategoryStruct.id X.X

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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) :

((CommMon_.EquivLaxBraidedFunctorPUnit.counitIso C).hom.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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Instances For

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@[simp]

theorem CommMon_.equivLaxBraidedFunctorPUnit_counitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :

(CommMon_.equivLaxBraidedFunctorPUnit C).counitIso = CommMon_.EquivLaxBraidedFunctorPUnit.counitIso C

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@[simp]

theorem CommMon_.equivLaxBraidedFunctorPUnit_inverse (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :

(CommMon_.equivLaxBraidedFunctorPUnit C).inverse = CommMon_.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided C

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@[simp]

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

(CommMon_.equivLaxBraidedFunctorPUnit C).unitIso = CommMon_.EquivLaxBraidedFunctorPUnit.unitIso C

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@[simp]

theorem CommMon_.equivLaxBraidedFunctorPUnit_functor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :

(CommMon_.equivLaxBraidedFunctorPUnit C).functor = CommMon_.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C

Documentation

Mathlib.CategoryTheory.Monoidal.CommMon_

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