Documentation

Mathlib.CategoryTheory.Category.TwoP

The category of two-pointed types #

This defines TwoP, the category of two-pointed types.

References #

source
structure TwoP :
Type (u + 1)

The category of two-pointed types.

Instances For
    source
    instance TwoP.instCoeSortTwoPType :
    CoeSort TwoP (Type u_3)
    source
    def TwoP.of {X : Type u_3} (toTwoPointing : TwoPointing X) :
    TwoP

    Turns a two-pointing into a two-pointed type.

    Instances For
      source
      @[simp]
      theorem TwoP.coe_of {X : Type u_3} (toTwoPointing : TwoPointing X) :
      (TwoP.of toTwoPointing).X = X
      source
      def TwoPointing.TwoP {X : Type u_3} (toTwoPointing : TwoPointing X) :
      TwoP

      Alias of TwoP.of.

      Turns a two-pointing into a two-pointed type.

      Instances For
        source
        instance TwoP.instInhabitedTwoP :
        Inhabited TwoP
        source
        noncomputable def TwoP.toBipointed (X : TwoP) :
        Bipointed

        Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are distinct.

        Instances For
          source
          @[simp]
          theorem TwoP.coe_toBipointed (X : TwoP) :
          (TwoP.toBipointed X).X = X.X
          source
          noncomputable instance TwoP.largeCategory :
          CategoryTheory.LargeCategory TwoP
          source
          noncomputable instance TwoP.concreteCategory :
          CategoryTheory.ConcreteCategory TwoP
          source
          noncomputable instance TwoP.hasForgetToBipointed :
          CategoryTheory.HasForget₂ TwoP Bipointed
          source
          @[simp]
          theorem TwoP.swap_obj_X (X : TwoP) :
          (TwoP.swap.obj X).X = X.X
          source
          @[simp]
          theorem TwoP.swap_map_toFun :
          ∀ {X Y : TwoP} (f : X ⟶ Y) (a : (TwoP.toBipointed X).X), Bipointed.Hom.toFun (TwoP.swap.map f) a = Bipointed.Hom.toFun f a
          source
          @[simp]
          theorem TwoP.swap_obj_toTwoPointing (X : TwoP) :
          (TwoP.swap.obj X).toTwoPointing = TwoPointing.swap X.toTwoPointing
          source
          noncomputable def TwoP.swap :
          CategoryTheory.Functor TwoP TwoP

          Swaps the pointed elements of a two-pointed type. TwoPointing.swap as a functor.

          Instances For
            source
            @[simp]
            theorem TwoP.swapEquiv_functor_obj_toTwoPointing_toProd (X : TwoP) :
            (TwoP.swapEquiv.functor.obj X).toTwoPointing.toProd = (X.toTwoPointing.snd, X.toTwoPointing.fst)
            source
            @[simp]
            theorem TwoP.swapEquiv_inverse_obj_toTwoPointing_toProd (X : TwoP) :
            (TwoP.swapEquiv.inverse.obj X).toTwoPointing.toProd = (X.toTwoPointing.snd, X.toTwoPointing.fst)
            source
            @[simp]
            theorem TwoP.swapEquiv_functor_obj_X (X : TwoP) :
            (TwoP.swapEquiv.functor.obj X).X = X.X
            source
            @[simp]
            theorem TwoP.swapEquiv_counitIso_hom_app_toFun (X : TwoP) (a : (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).X) :
            Bipointed.Hom.toFun (TwoP.swapEquiv.counitIso.hom.app X) a = a
            source
            @[simp]
            theorem TwoP.swapEquiv_inverse_obj_X (X : TwoP) :
            (TwoP.swapEquiv.inverse.obj X).X = X.X
            source
            @[simp]
            theorem TwoP.swapEquiv_inverse_map_toFun :
            ∀ {X Y : TwoP} (f : X ⟶ Y) (a : (TwoP.toBipointed X).X), Bipointed.Hom.toFun (TwoP.swapEquiv.inverse.map f) a = Bipointed.Hom.toFun f a
            source
            @[simp]
            theorem TwoP.swapEquiv_unitIso_inv_app_toFun (X : TwoP) :
            ∀ (a : (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).X), Bipointed.Hom.toFun (TwoP.swapEquiv.unitIso.inv.app X) a = Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.comp (TwoP.swap.map (TwoP.swap.map { toFun := id, map_fst := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.fst = id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.fst), map_snd := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.snd = id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.snd) })) (CategoryTheory.CategoryStruct.comp (TwoP.swap.map { toFun := id, map_fst := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj (TwoP.swap.obj X))).toProd.fst = id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj (TwoP.swap.obj X))).toProd.fst), map_snd := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj (TwoP.swap.obj X))).toProd.snd = id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj (TwoP.swap.obj X))).toProd.snd) }) { toFun := id, map_fst := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.fst = id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.fst), map_snd := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.snd = id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.snd) })) (Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.id (TwoP.swap.obj (TwoP.swap.obj X))) a)
            source
            @[simp]
            theorem TwoP.swapEquiv_counitIso_inv_app_toFun (X : TwoP) (a : (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).X) :
            Bipointed.Hom.toFun (TwoP.swapEquiv.counitIso.inv.app X) a = a
            source
            @[simp]
            theorem TwoP.swapEquiv_unitIso_hom_app_toFun (X : TwoP) :
            ∀ (a : (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).X), Bipointed.Hom.toFun (TwoP.swapEquiv.unitIso.hom.app X) a = Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.id (TwoP.swap.obj (TwoP.swap.obj X))) (Bipointed.Hom.toFun (CategoryTheory.CategoryStruct.comp { toFun := id, map_fst := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.fst = id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.fst), map_snd := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.snd = id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj X)).toProd.snd) } (CategoryTheory.CategoryStruct.comp (TwoP.swap.map { toFun := id, map_fst := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj (TwoP.swap.obj X))).toProd.fst = id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj (TwoP.swap.obj X))).toProd.fst), map_snd := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj (TwoP.swap.obj X))).toProd.snd = id (TwoP.toBipointed ((CategoryTheory.Functor.id TwoP).obj (TwoP.swap.obj X))).toProd.snd) }) (TwoP.swap.map (TwoP.swap.map { toFun := id, map_fst := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.fst = id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.fst), map_snd := (_ : id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.snd = id (TwoP.toBipointed ((CategoryTheory.Functor.comp TwoP.swap TwoP.swap).obj X)).toProd.snd) })))) a)
            source
            @[simp]
            theorem TwoP.swapEquiv_functor_map_toFun :
            ∀ {X Y : TwoP} (f : X ⟶ Y) (a : (TwoP.toBipointed X).X), Bipointed.Hom.toFun (TwoP.swapEquiv.functor.map f) a = Bipointed.Hom.toFun f a
            source
            noncomputable def TwoP.swapEquiv :
            TwoP ≌ TwoP

            The equivalence between TwoP and itself induced by Prod.swap both ways.

            Instances For
              source
              @[simp]
              theorem TwoP.swapEquiv_symm :
              TwoP.swapEquiv.symm = TwoP.swapEquiv
              source
              @[simp]
              theorem TwoP_swap_comp_forget_to_Bipointed :
              CategoryTheory.Functor.comp TwoP.swap (CategoryTheory.forget₂ TwoP Bipointed) = CategoryTheory.Functor.comp (CategoryTheory.forget₂ TwoP Bipointed) Bipointed.swap
              source
              @[simp]
              theorem pointedToTwoPFst_obj_X (X : Pointed) :
              (pointedToTwoPFst.obj X).X = Option X.X
              source
              @[simp]
              theorem pointedToTwoPFst_obj_toTwoPointing_toProd (X : Pointed) :
              (pointedToTwoPFst.obj X).toTwoPointing.toProd = (some X.point, none)
              source
              @[simp]
              theorem pointedToTwoPFst_map_toFun :
              ∀ {X Y : Pointed} (f : X ⟶ Y) (a : Option X.X), Bipointed.Hom.toFun (pointedToTwoPFst.map f) a = Option.map f.toFun a
              source
              noncomputable def pointedToTwoPFst :
              CategoryTheory.Functor Pointed TwoP

              The functor from Pointed to TwoP which adds a second point.

              Instances For
                source
                @[simp]
                theorem pointedToTwoPSnd_obj_X (X : Pointed) :
                (pointedToTwoPSnd.obj X).X = Option X.X
                source
                @[simp]
                theorem pointedToTwoPSnd_map_toFun :
                ∀ {X Y : Pointed} (f : X ⟶ Y) (a : Option X.X), Bipointed.Hom.toFun (pointedToTwoPSnd.map f) a = Option.map f.toFun a
                source
                @[simp]
                theorem pointedToTwoPSnd_obj_toTwoPointing_toProd (X : Pointed) :
                (pointedToTwoPSnd.obj X).toTwoPointing.toProd = (none, some X.point)
                source
                noncomputable def pointedToTwoPSnd :
                CategoryTheory.Functor Pointed TwoP

                The functor from Pointed to TwoP which adds a first point.

                Instances For
                  source
                  @[simp]
                  theorem pointedToTwoPFst_comp_swap :
                  CategoryTheory.Functor.comp pointedToTwoPFst TwoP.swap = pointedToTwoPSnd
                  source
                  @[simp]
                  theorem pointedToTwoPSnd_comp_swap :
                  CategoryTheory.Functor.comp pointedToTwoPSnd TwoP.swap = pointedToTwoPFst
                  source
                  @[simp]
                  theorem pointedToTwoPFst_comp_forget_to_bipointed :
                  CategoryTheory.Functor.comp pointedToTwoPFst (CategoryTheory.forget₂ TwoP Bipointed) = pointedToBipointedFst
                  source
                  @[simp]
                  theorem pointedToTwoPSnd_comp_forget_to_bipointed :
                  CategoryTheory.Functor.comp pointedToTwoPSnd (CategoryTheory.forget₂ TwoP Bipointed) = pointedToBipointedSnd
                  source
                  noncomputable def pointedToTwoPFstForgetCompBipointedToPointedFstAdjunction :
                  pointedToTwoPFst ⊣ CategoryTheory.Functor.comp (CategoryTheory.forget₂ TwoP Bipointed) bipointedToPointedFst

                  Adding a second point is left adjoint to forgetting the second point.

                  Instances For
                    source
                    noncomputable def pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction :
                    pointedToTwoPSnd ⊣ CategoryTheory.Functor.comp (CategoryTheory.forget₂ TwoP Bipointed) bipointedToPointedSnd

                    Adding a first point is left adjoint to forgetting the first point.

                    Instances For