Documentation

Mathlib.CategoryTheory.Monoidal.CommMon_

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

source
structure CategoryTheory.CommMon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
Type (max u₁ v₁)

A commutative monoid object internal to a monoidal category.

  • X : C

    The underlying object in the ambient monoidal category

  • mon : MonObj self.X
  • comm : IsCommMonObj self.X
Instances For
    source
    @[deprecated CategoryTheory.CommMon (since := "2025-09-15")]
    def CategoryTheory.CommMon_ (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
    Type (max u₁ v₁)

    Alias of CategoryTheory.CommMon.

    A commutative monoid object internal to a monoidal category.

    Equations
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      def CategoryTheory.CommMon.toMon {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
      Mon C

      A commutative monoid object is a monoid object.

      Equations
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        @[simp]
        theorem CategoryTheory.CommMon.toMon_X {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
        A.toMon.X = A.X
        source
        @[deprecated CategoryTheory.CommMon.toMon (since := "2025-09-15")]
        def CategoryTheory.CommMon.toMon_ {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
        Mon C

        Alias of CategoryTheory.CommMon.toMon.

        A commutative monoid object is a monoid object.

        Equations
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          def CategoryTheory.CommMon.trivial (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
          CommMon C

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

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            @[simp]
            theorem CategoryTheory.CommMon.trivial_X (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
            (trivial C).X = MonoidalCategoryStruct.tensorUnit C
            source
            @[simp]
            theorem CategoryTheory.CommMon.trivial_mon_mul (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
            MonObj.mul = (MonoidalCategoryStruct.leftUnitor (MonoidalCategoryStruct.tensorUnit C)).hom
            source
            @[simp]
            theorem CategoryTheory.CommMon.trivial_mon_one (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
            MonObj.one = CategoryStruct.id (MonoidalCategoryStruct.tensorUnit C)
            source
            instance CategoryTheory.CommMon.instInhabited {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
            Inhabited (CommMon C)
            Equations
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            instance CategoryTheory.CommMon.instCategory {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
            Category.{v₁, max u₁ v₁} (CommMon C)
            Equations
            • One or more equations did not get rendered due to their size.
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            @[simp]
            theorem CategoryTheory.CommMon.id_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
            (CategoryStruct.id A).hom.hom = CategoryStruct.id A.X
            source
            @[simp]
            theorem CategoryTheory.CommMon.comp_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {R S T : CommMon C} (f : R ⟶ S) (g : S ⟶ T) :
            (CategoryStruct.comp f g).hom.hom = CategoryStruct.comp f.hom.hom g.hom.hom
            source
            theorem CategoryTheory.CommMon.hom_ext {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {A B : CommMon C} (f g : A ⟶ B) (h : f.hom.hom = g.hom.hom) :
            f = g
            source
            theorem CategoryTheory.CommMon.hom_ext_iff {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {A B : CommMon C} {f g : A ⟶ B} :
            f = g ↔ f.hom.hom = g.hom.hom
            source
            def CategoryTheory.CommMon.homMk {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {A B : CommMon C} (f : A.toMon ⟶ B.toMon) :
            A ⟶ B

            Constructor for morphisms in CommMon C.

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              @[simp]
              theorem CategoryTheory.CommMon.homMk_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {A B : CommMon C} (f : A.toMon ⟶ B.toMon) :
              (homMk f).hom = f
              source
              def CategoryTheory.CommMon.forget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
              Functor (CommMon C) (Mon C)

              The forgetful functor from commutative monoid objects to monoid objects.

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                @[simp]
                theorem CategoryTheory.CommMon.forget₂Mon_obj_X (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                ((forget₂Mon C).obj A).X = A.X
                source
                @[deprecated CategoryTheory.CommMon.forget₂Mon (since := "2025-09-15")]
                def CategoryTheory.CommMon.forget₂Mon_ (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                Functor (CommMon C) (Mon C)

                Alias of CategoryTheory.CommMon.forget₂Mon.

                The forgetful functor from commutative monoid objects to monoid objects.

                Equations
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                  def CategoryTheory.CommMon.fullyFaithfulForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                  (forget₂Mon C).FullyFaithful

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

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                    @[deprecated CategoryTheory.CommMon.fullyFaithfulForget₂Mon (since := "2025-09-15")]
                    def CategoryTheory.CommMon.fullyFaithfulForget₂Mon_ (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                    (forget₂Mon C).FullyFaithful

                    Alias of CategoryTheory.CommMon.fullyFaithfulForget₂Mon.

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

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                      instance CategoryTheory.CommMon.instFullMonForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                      (forget₂Mon C).Full
                      source
                      instance CategoryTheory.CommMon.instFaithfulMonForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                      (forget₂Mon C).Faithful
                      source
                      @[simp]
                      theorem CategoryTheory.CommMon.forget₂Mon_obj_one (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                      MonObj.one = MonObj.one
                      source
                      @[simp]
                      theorem CategoryTheory.CommMon.forget₂Mon_obj_mul (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                      MonObj.mul = MonObj.mul
                      source
                      @[simp]
                      theorem CategoryTheory.CommMon.forget₂Mon_map_hom (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {A B : CommMon C} (f : A ⟶ B) :
                      ((forget₂Mon C).map f).hom = f.hom.hom
                      source
                      def CategoryTheory.CommMon.forget (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                      Functor (CommMon C) C

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

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                        @[simp]
                        theorem CategoryTheory.CommMon.forget_obj (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon C) :
                        (forget C).obj X = X.X
                        source
                        @[simp]
                        theorem CategoryTheory.CommMon.forget_map (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {X✝ Y✝ : CommMon C} (f : X✝ ⟶ Y✝) :
                        (forget C).map f = f.hom.hom
                        source
                        instance CategoryTheory.CommMon.instFaithfulForget (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                        (forget C).Faithful
                        source
                        @[simp]
                        theorem CategoryTheory.CommMon.forget₂Mon_comp_forget (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                        (forget₂Mon C).comp (Mon.forget C) = forget C
                        source
                        instance CategoryTheory.CommMon.instIsIsoHomHomMon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : CommMon C} {f : M ⟶ N} [IsIso f] :
                        IsIso f.hom.hom
                        source
                        def CategoryTheory.CommMon.mkIso' {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : C} (e : M ≅ N) [MonObj M] [IsCommMonObj M] [MonObj N] [IsCommMonObj N] [IsMonHom e.hom] :
                        { X := M, mon := inst✝, comm := inst✝¹ } ≅ { X := N, mon := inst✝², comm := inst✝³ }

                        Construct an isomorphism of commutative monoid objects by giving a monoid isomorphism between the underlying objects.

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                          @[simp]
                          theorem CategoryTheory.CommMon.mkIso'_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : C} (e : M ≅ N) [MonObj M] [IsCommMonObj M] [MonObj N] [IsCommMonObj N] [IsMonHom e.hom] :
                          (mkIso' e).inv.hom.hom = e.inv
                          source
                          @[simp]
                          theorem CategoryTheory.CommMon.mkIso'_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : C} (e : M ≅ N) [MonObj M] [IsCommMonObj M] [MonObj N] [IsCommMonObj N] [IsMonHom e.hom] :
                          (mkIso' e).hom.hom.hom = e.hom
                          source
                          @[reducible, inline]
                          abbrev CategoryTheory.CommMon.mkIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : CommMon C} (e : M.X ≅ N.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :
                          M ≅ N

                          Construct an isomorphism of commutative monoid objects by giving an isomorphism between the underlying objects and checking compatibility with unit and multiplication only in the forward direction.

                          Equations
                          Instances For
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                            @[simp]
                            theorem CategoryTheory.CommMon.mkIso_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : CommMon C} (e : M.X ≅ N.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :
                            (mkIso e one_f mul_f).hom.hom.hom = e.hom
                            source
                            @[simp]
                            theorem CategoryTheory.CommMon.mkIso_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] {M N : CommMon C} (e : M.X ≅ N.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :
                            (mkIso e one_f mul_f).inv.hom.hom = e.inv
                            source
                            instance CategoryTheory.CommMon.uniqueHomFromTrivial {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                            Unique (trivial C ⟶ A)
                            Equations
                            source
                            instance CategoryTheory.CommMon.instHasInitial {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                            Limits.HasInitial (CommMon C)
                            source
                            instance CategoryTheory.Functor.isCommMonObj_obj {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] {M : C} [MonObj M] [IsCommMonObj M] :
                            IsCommMonObj (F.obj M)
                            source
                            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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                            • One or more equations did not get rendered due to their size.
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                              @[simp]
                              theorem CategoryTheory.Functor.mapCommMon_map_hom_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.hom = F.map f.hom.hom
                              source
                              @[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
                              source
                              @[simp]
                              theorem CategoryTheory.Functor.mapCommMon_obj_mon_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) :
                              MonObj.one = CategoryStruct.comp (LaxMonoidal.ε F) (F.map MonObj.one)
                              source
                              @[simp]
                              theorem CategoryTheory.Functor.mapCommMon_obj_mon_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) :
                              MonObj.mul = CategoryStruct.comp (LaxMonoidal.μ F A.X A.X) (F.map MonObj.mul)
                              source
                              @[simp]
                              theorem CategoryTheory.Functor.mapCommMon_id_one {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                              MonObj.one = CategoryStruct.comp (CategoryStruct.id (MonoidalCategoryStruct.tensorUnit C)) MonObj.one
                              source
                              @[simp]
                              theorem CategoryTheory.Functor.mapCommMon_id_mul {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                              MonObj.mul = CategoryStruct.comp (CategoryStruct.id (MonoidalCategoryStruct.tensorObj A.X A.X)) MonObj.mul
                              source
                              @[simp]
                              theorem CategoryTheory.Functor.comp_mapCommMon_one {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} {G : Functor D E} [F.LaxBraided] [G.LaxBraided] (A : CommMon C) :
                              MonObj.one = CategoryStruct.comp (LaxMonoidal.ε (F.comp G)) ((F.comp G).map MonObj.one)
                              source
                              @[simp]
                              theorem CategoryTheory.Functor.comp_mapCommMon_mul {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} {G : Functor D E} [F.LaxBraided] [G.LaxBraided] (A : CommMon C) :
                              MonObj.mul = CategoryStruct.comp (LaxMonoidal.μ (F.comp G) A.X A.X) ((F.comp G).map MonObj.mul)
                              source
                              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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                              • One or more equations did not get rendered due to their size.
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                                @[simp]
                                theorem CategoryTheory.Functor.mapCommMonIdIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon C) :
                                (mapCommMonIdIso.inv.app X).hom.hom = CategoryStruct.id X.X
                                source
                                @[simp]
                                theorem CategoryTheory.Functor.mapCommMonIdIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon C) :
                                (mapCommMonIdIso.hom.app X).hom.hom = CategoryStruct.id X.X
                                source
                                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} {G : Functor D E} [F.LaxBraided] [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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                                • One or more equations did not get rendered due to their size.
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                                  @[simp]
                                  theorem CategoryTheory.Functor.mapCommMonCompIso_hom_app_hom_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} {G : Functor D E} [F.LaxBraided] [G.LaxBraided] (X : CommMon C) :
                                  (mapCommMonCompIso.hom.app X).hom.hom = CategoryStruct.id (G.obj (F.obj X.X))
                                  source
                                  @[simp]
                                  theorem CategoryTheory.Functor.mapCommMonCompIso_inv_app_hom_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} {G : Functor D E} [F.LaxBraided] [G.LaxBraided] (X : CommMon C) :
                                  (mapCommMonCompIso.inv.app X).hom.hom = CategoryStruct.id (G.obj (F.obj X.X))
                                  source
                                  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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                                  • One or more equations did not get rendered due to their size.
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                                    @[simp]
                                    theorem CategoryTheory.Functor.mapCommMonFunctor_map_app (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 = CommMon.homMk (Mon.Hom.mk' (α.hom.hom.app A.X) ⋯ ⋯)
                                    source
                                    @[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
                                    source
                                    instance CategoryTheory.Functor.Faithful.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] [F.Faithful] :
                                    F.mapCommMon.Faithful
                                    source
                                    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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                                    • One or more equations did not get rendered due to their size.
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                                      @[simp]
                                      theorem CategoryTheory.Functor.mapCommMonNatTrans_app_hom_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.hom = f.app X.X
                                      source
                                      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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                                        theorem CategoryTheory.Functor.mapCommMonNatIso_hom_app_hom_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.hom = e.hom.app X.X
                                        source
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                                        theorem CategoryTheory.Functor.mapCommMonNatIso_inv_app_hom_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.hom = e.inv.app X.X
                                        source
                                        def CategoryTheory.Functor.FullyFaithful.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.Braided] (hF : F.FullyFaithful) :
                                        F.mapCommMon.FullyFaithful

                                        If F : C ⥤ D is a fully faithful monoidal functor, then CommMonCat(F) : CommMonCat C ⥤ CommMonCat D is fully faithful too.

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                                          theorem CategoryTheory.Functor.FullyFaithful.mapCommMon_preimage {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.Braided] (hF : F.FullyFaithful) {X✝ Y✝ : CommMon C} (f : F.mapCommMon.obj X✝ ⟶ F.mapCommMon.obj Y✝) :
                                          hF.mapCommMon.preimage f = CommMon.homMk (hF.mapMon.preimage f.hom)
                                          source
                                          instance CategoryTheory.Functor.Full.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.Braided] [F.Full] [F.Faithful] :
                                          F.mapCommMon.Full
                                          source
                                          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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                                            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)
                                            source
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                                            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)
                                            source
                                            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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                                              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
                                              source
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                                              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
                                              source
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                                              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
                                              source
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                                              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
                                              source
                                              def CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                              Functor (LaxBraidedFunctor (Discrete PUnit.{u + 1}) C) (CommMon C)

                                              Implementation of CommMon.equivLaxBraidedFunctorPUnit.

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

                                                Implementation of CommMon.equivLaxBraidedFunctorPUnit.

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

                                                  Implementation of CommMon.equivLaxBraidedFunctorPUnit.

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

                                                    Implementation of CommMon.equivLaxBraidedFunctorPUnit.

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                                                      theorem CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.unitIso_inv_app_hom_hom_app (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : LaxBraidedFunctor (Discrete PUnit.{u + 1}) C) (X✝ : Discrete PUnit.{u + 1}) :
                                                      ((unitIso C).inv.app X).hom.hom.app X✝ = CategoryStruct.id (X.obj (MonoidalCategoryStruct.tensorUnit (Discrete PUnit.{u + 1})))
                                                      source
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                                                      theorem CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.unitIso_hom_app_hom_hom_app (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : LaxBraidedFunctor (Discrete PUnit.{u + 1}) C) (X✝ : Discrete PUnit.{u + 1}) :
                                                      ((unitIso C).hom.app X).hom.hom.app X✝ = CategoryStruct.id (X.obj X✝)
                                                      source
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                                                      theorem CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_aux_one (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                                                      MonObj.one = CategoryStruct.comp MonObj.one (CategoryStruct.id A.X)
                                                      source
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                                                      theorem CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_aux_mul (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (A : CommMon C) :
                                                      MonObj.mul = CategoryStruct.comp MonObj.mul (CategoryStruct.id A.X)
                                                      source
                                                      def CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                                      (commMonToLaxBraided C).comp (laxBraidedToCommMon C) ≅ Functor.id (CommMon C)

                                                      Implementation of CommMon.equivLaxBraidedFunctorPUnit.

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                                                        theorem CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_inv_app_hom_hom (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon C) :
                                                        ((counitIso C).inv.app X).hom.hom = CategoryStruct.id X.X
                                                        source
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                                                        theorem CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_hom_app_hom_hom (C : Type u₁) [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] (X : CommMon C) :
                                                        ((counitIso C).hom.app X).hom.hom = CategoryStruct.id X.X
                                                        source
                                                        def CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                                        LaxBraidedFunctor (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 CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_unitIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                                          equivLaxBraidedFunctorPUnit.unitIso = EquivLaxBraidedFunctorPUnit.unitIso C
                                                          source
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                                                          theorem CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_counitIso {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                                          equivLaxBraidedFunctorPUnit.counitIso = EquivLaxBraidedFunctorPUnit.counitIso C
                                                          source
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                                                          theorem CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_functor {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                                          equivLaxBraidedFunctorPUnit.functor = EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon C
                                                          source
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                                                          theorem CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_inverse {C : Type u₁} [Category.{v₁, u₁} C] [MonoidalCategory C] [BraidedCategory C] :
                                                          equivLaxBraidedFunctorPUnit.inverse = EquivLaxBraidedFunctorPUnit.commMonToLaxBraided C