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Mathlib.CategoryTheory.Monoidal.Comon_

The category of comonoids in a monoidal category. #

We define comonoids in a monoidal category C, and show that they are equivalently monoid objects in the opposite category.

We construct the monoidal structure on Comon_ C, when C is braided.

An oplax monoidal functor takes comonoid objects to comonoid objects. That is, a oplax monoidal functor F : C ⥤ D induces a functor Comon_ C ⥤ Comon_ D.

TODO #

A comonoid object internal to a monoidal category.

When the monoidal category is preadditive, this is also sometimes called a "coalgebra object".

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    The trivial comonoid object. We later show this is terminal in Comon_ C.

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      theorem Comon_.Hom.ext_iff {C : Type u₁} :
      ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {M N : Comon_ C} (x y : M.Hom N), x = y x.hom = y.hom
      theorem Comon_.Hom.ext {C : Type u₁} :
      ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {M N : Comon_ C} (x y : M.Hom N), x.hom = y.homx = y

      A morphism of comonoid objects.

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        theorem Comon_.Hom.hom_counit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (self : M.Hom N) :
        CategoryTheory.CategoryStruct.comp self.hom N.counit = M.counit

        The identity morphism on a comonoid object.

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          • M.homInhabited = { default := M.id }
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          theorem Comon_.comp_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} {O : Comon_ C} (f : M.Hom N) (g : N.Hom O) :
          def Comon_.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} {O : Comon_ C} (f : M.Hom N) (g : N.Hom O) :
          M.Hom O

          Composition of morphisms of monoid objects.

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            theorem Comon_.ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X : Comon_ C} {Y : Comon_ C} {f : X Y} {g : X Y} (w : f.hom = g.hom) :
            f = g
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            theorem Comon_.forget_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :
            ∀ {X Y : Comon_ C} (f : X Y), (Comon_.forget C).map f = f.hom

            The forgetful functor from comonoid objects to the ambient category.

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              The forgetful functor from comonoid objects to the ambient category reflects isomorphisms.

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              theorem Comon_.mkIso_hom_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (f : M.X N.X) (f_counit : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.counit = M.counit) _auto✝) (f_comul : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom f.hom f.hom)) _auto✝) :
              (Comon_.mkIso f f_counit f_comul).hom.hom = f.hom
              @[simp]
              theorem Comon_.mkIso_inv_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (f : M.X N.X) (f_counit : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.counit = M.counit) _auto✝) (f_comul : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom f.hom f.hom)) _auto✝) :
              (Comon_.mkIso f f_counit f_comul).inv.hom = f.inv

              Construct an isomorphism of comonoids by giving an isomorphism between the underlying objects and checking compatibility with counit and comultiplication only in the forward direction.

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                • A.uniqueHomToTrivial = { default := { hom := A.counit, hom_counit := , hom_comul := }, uniq := }

                Turn a comonoid object into a monoid object in the opposite category.

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                  theorem Comon_.Comon_ToMon_OpOp_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :
                  ∀ {X Y : Comon_ C} (f : X Y), (Comon_.Comon_ToMon_OpOp C).map f = Opposite.op { hom := f.hom.op, one_hom := , mul_hom := }

                  The contravariant functor turning comonoid objects into monoid objects in the opposite category.

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                    Turn a monoid object in the opposite category into a comonoid object.

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                      @[simp]

                      The contravariant functor turning monoid objects in the opposite category into comonoid objects.

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                        Comonoid objects are contravariantly equivalent to monoid objects in the opposite category.

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                          Comonoid objects in a braided category form a monoidal category.

                          This definition is via transporting back and forth to monoids in the opposite category,

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                          Preliminary statement of the comultiplication for a tensor product of comonoids. This version is the definitional equality provided by transport, and not quite as good as the version provided in tensorObj_comul below.

                          The comultiplication on the tensor product of two comonoids is the tensor product of the comultiplications followed by the tensor strength (to shuffle the factors back into order).

                          The forgetful functor from Comon_ C to C is monoidal when C is braided monoidal.

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                            A oplax monoidal functor takes comonoid objects to comonoid objects.

                            That is, a oplax monoidal functor F : C ⥤ D induces a functor Comon_ C ⥤ Comon_ D.

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