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Mathlib.Algebra.Category.Grp.ForgetCorepresentable

The forget functor is corepresentable #

It is shown that the forget functor AddCommGrp.{u} ⥤ Type u is corepresentable by ULift ℤ. Similar results are obtained for the variants CommGrp, AddGrp and Grp.

(ULift ℤ →+ G) ≃ G #

These universe-monomorphic variants of zmultiplesHom/zpowersHom are put here since they shouldn't be useful outside of category theory.

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def uliftZMultiplesHom (G : Type u) [AddGroup G] :
G ≃ (ULift.{u, 0} ℤ →+ G)

The equivalence (ULift ℤ →+ G) ≃ G for any additive group G.

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    @[simp]
    theorem uliftZMultiplesHom_symm_apply (G : Type u) [AddGroup G] (a✝ : ULift.{u, 0} ℤ →+ G) :
    (uliftZMultiplesHom G).symm a✝ = a✝ (AddEquiv.ulift.symm 1)
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    @[simp]
    theorem uliftZMultiplesHom_apply_apply (G : Type u) [AddGroup G] (a✝ : G) (a✝¹ : ULift.{u, 0} ℤ) :
    ((uliftZMultiplesHom G) a✝) a✝¹ = AddEquiv.ulift a✝¹ • a✝
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    def uliftZPowersHom (G : Type u) [Group G] :
    G ≃ (ULift.{u, 0} (Multiplicative ℤ) →* G)

    The equivalence (ULift (Multiplicative ℤ) →* G) ≃ G for any group G.

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      @[simp]
      theorem uliftZPowersHom_symm_apply (G : Type u) [Group G] (a✝ : ULift.{u, 0} (Multiplicative ℤ) →* G) :
      (uliftZPowersHom G).symm a✝ = a✝ (MulEquiv.ulift.symm (Multiplicative.ofAdd 1))
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      @[simp]
      theorem uliftZPowersHom_apply_apply (G : Type u) [Group G] (a✝ : G) (a✝¹ : ULift.{u, 0} (Multiplicative ℤ)) :
      ((uliftZPowersHom G) a✝) a✝¹ = a✝ ^ Multiplicative.toAdd (MulEquiv.ulift a✝¹)
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      @[deprecated zpowersHom (since := "2025-05-11")]
      def MonoidHom.fromMultiplicativeIntEquiv (α : Type u) [Group α] :
      (Multiplicative ℤ →* α) ≃ α

      The equivalence (Multiplicative ℤ →* α) ≃ α for any group α.

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        @[deprecated uliftZPowersHom (since := "2025-05-11")]
        def MonoidHom.fromULiftMultiplicativeIntEquiv (α : Type u) [Group α] :
        (ULift.{u, 0} (Multiplicative ℤ) →* α) ≃ α

        The equivalence (ULift (Multiplicative ℤ) →* α) ≃ α for any group α.

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          @[deprecated zmultiplesHom (since := "2025-05-11")]
          def AddMonoidHom.fromIntEquiv (α : Type u) [AddGroup α] :
          (ℤ →+ α) ≃ α

          The equivalence (ℤ →+ α) ≃ α for any additive group α.

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            @[deprecated uliftZMultiplesHom (since := "2025-05-11")]
            def AddMonoidHom.fromULiftIntEquiv (α : Type u) [AddGroup α] :
            (ULift.{u, 0} ℤ →+ α) ≃ α

            The equivalence (ULift ℤ →+ α) ≃ α for any additive group α.

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              def Grp.coyonedaObjIsoForget :
              CategoryTheory.coyoneda.obj (Opposite.op (of (ULift.{u, 0} (Multiplicative ℤ)))) ≅ CategoryTheory.forget Grp

              The forget functor Grp.{u} ⥤ Type u is corepresentable.

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              • One or more equations did not get rendered due to their size.
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                def CommGrp.coyonedaObjIsoForget :
                CategoryTheory.coyoneda.obj (Opposite.op (of (ULift.{u, 0} (Multiplicative ℤ)))) ≅ CategoryTheory.forget CommGrp

                The forget functor CommGrp.{u} ⥤ Type u is corepresentable.

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                • One or more equations did not get rendered due to their size.
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                  def AddGrp.coyonedaObjIsoForget :
                  CategoryTheory.coyoneda.obj (Opposite.op (of (ULift.{u, 0} ℤ))) ≅ CategoryTheory.forget AddGrp

                  The forget functor AddGrp.{u} ⥤ Type u is corepresentable.

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                    def AddCommGrp.coyonedaObjIsoForget :
                    CategoryTheory.coyoneda.obj (Opposite.op (of (ULift.{u, 0} ℤ))) ≅ CategoryTheory.forget AddCommGrp

                    The forget functor AddCommGrp.{u} ⥤ Type u is corepresentable.

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                      instance Grp.forget_isCorepresentable :
                      (CategoryTheory.forget Grp).IsCorepresentable
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                      instance CommGrp.forget_isCorepresentable :
                      (CategoryTheory.forget CommGrp).IsCorepresentable
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                      instance AddGrp.forget_isCorepresentable :
                      (CategoryTheory.forget AddGrp).IsCorepresentable
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                      instance AddCommGrp.forget_isCorepresentable :
                      (CategoryTheory.forget AddCommGrp).IsCorepresentable