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Mathlib.Algebra.Category.GroupCat.Colimits

The category of additive commutative groups has all colimits. #

This file uses a "pre-automated" approach, just as for Mon/Colimits.lean. It is a very uniform approach, that conceivably could be synthesised directly by a tactic that analyses the shape of AddCommGroup and MonoidHom.

TODO: In fact, in AddCommGroupCat there is a much nicer model of colimits as quotients of finitely supported functions, and we really should implement this as well (or instead).

We build the colimit of a diagram in AddCommGroupCat by constructing the free group on the disjoint union of all the abelian groups in the diagram, then taking the quotient by the abelian group laws within each abelian group, and the identifications given by the morphisms in the diagram.

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inductive AddCommGroupCat.Colimits.Prequotient {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
Type (max (max u v) w)

An inductive type representing all group expressions (without relations) on a collection of types indexed by the objects of J.

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    instance AddCommGroupCat.Colimits.instInhabitedPrequotient {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
    Inhabited (AddCommGroupCat.Colimits.Prequotient F)
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    inductive AddCommGroupCat.Colimits.Relation {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
    AddCommGroupCat.Colimits.Prequotient F → AddCommGroupCat.Colimits.Prequotient F → Prop

    The relation on prequotient saying when two expressions are equal because of the abelian group laws, or because one element is mapped to another by a morphism in the diagram.

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      instance AddCommGroupCat.Colimits.colimitSetoid {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
      Setoid (AddCommGroupCat.Colimits.Prequotient F)

      The setoid corresponding to group expressions modulo abelian group relations and identifications.

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      def AddCommGroupCat.Colimits.ColimitType {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
      Type (max u v w)

      The underlying type of the colimit of a diagram in AddCommGroupCat.

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        instance AddCommGroupCat.Colimits.instAddCommGroupColimitType {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
        AddCommGroup (AddCommGroupCat.Colimits.ColimitType F)
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        instance AddCommGroupCat.Colimits.ColimitTypeInhabited {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
        Inhabited (AddCommGroupCat.Colimits.ColimitType F)
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        @[simp]
        theorem AddCommGroupCat.Colimits.quot_zero {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
        Quot.mk Setoid.r AddCommGroupCat.Colimits.Prequotient.zero = 0
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        @[simp]
        theorem AddCommGroupCat.Colimits.quot_neg {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (x : AddCommGroupCat.Colimits.Prequotient F) :
        Quot.mk Setoid.r (AddCommGroupCat.Colimits.Prequotient.neg x) = -Quot.mk Setoid.r x
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        @[simp]
        theorem AddCommGroupCat.Colimits.quot_add {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (x : AddCommGroupCat.Colimits.Prequotient F) (y : AddCommGroupCat.Colimits.Prequotient F) :
        Quot.mk Setoid.r (AddCommGroupCat.Colimits.Prequotient.add x y) = Quot.mk Setoid.r x + Quot.mk Setoid.r y
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        def AddCommGroupCat.Colimits.colimit {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
        AddCommGroupCat

        The bundled abelian group giving the colimit of a diagram.

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          def AddCommGroupCat.Colimits.coconeFun {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (j : J) (x : ↑(F.toPrefunctor.obj j)) :
          AddCommGroupCat.Colimits.ColimitType F

          The function from a given abelian group in the diagram to the colimit abelian group.

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            def AddCommGroupCat.Colimits.coconeMorphism {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (j : J) :
            F.toPrefunctor.obj j ⟶ AddCommGroupCat.Colimits.colimit F

            The group homomorphism from a given abelian group in the diagram to the colimit abelian group.

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            • One or more equations did not get rendered due to their size.
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              @[simp]
              theorem AddCommGroupCat.Colimits.cocone_naturality {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) {j : J} {j' : J} (f : j ⟶ j') :
              CategoryTheory.CategoryStruct.comp (F.toPrefunctor.map f) (AddCommGroupCat.Colimits.coconeMorphism F j') = AddCommGroupCat.Colimits.coconeMorphism F j
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              @[simp]
              theorem AddCommGroupCat.Colimits.cocone_naturality_components {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (j : J) (j' : J) (f : j ⟶ j') (x : ↑(F.toPrefunctor.obj j)) :
              (AddCommGroupCat.Colimits.coconeMorphism F j') ((F.toPrefunctor.map f) x) = (AddCommGroupCat.Colimits.coconeMorphism F j) x
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              def AddCommGroupCat.Colimits.colimitCocone {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
              CategoryTheory.Limits.Cocone F

              The cocone over the proposed colimit abelian group.

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                def AddCommGroupCat.Colimits.descFunLift {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (s : CategoryTheory.Limits.Cocone F) :
                AddCommGroupCat.Colimits.Prequotient F → ↑s.pt

                The function from the free abelian group on the diagram to the cone point of any other cocone.

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                  def AddCommGroupCat.Colimits.descFun {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (s : CategoryTheory.Limits.Cocone F) :
                  AddCommGroupCat.Colimits.ColimitType F → ↑s.pt

                  The function from the colimit abelian group to the cone point of any other cocone.

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                  • One or more equations did not get rendered due to their size.
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                    def AddCommGroupCat.Colimits.descMorphism {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) (s : CategoryTheory.Limits.Cocone F) :
                    AddCommGroupCat.Colimits.colimit F ⟶ s.pt

                    The group homomorphism from the colimit abelian group to the cone point of any other cocone.

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                    • One or more equations did not get rendered due to their size.
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                      def AddCommGroupCat.Colimits.colimitCoconeIsColimit {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
                      CategoryTheory.Limits.IsColimit (AddCommGroupCat.Colimits.colimitCocone F)

                      Evidence that the proposed colimit is the colimit.

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                        theorem AddCommGroupCat.hasColimit {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGroupCat) :
                        CategoryTheory.Limits.HasColimit F
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                        theorem AddCommGroupCat.hasColimitsOfShape (J : Type u) [CategoryTheory.Category.{v, u} J] :
                        CategoryTheory.Limits.HasColimitsOfShape J AddCommGroupCat
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                        theorem AddCommGroupCat.hasColimitsOfSize :
                        CategoryTheory.Limits.HasColimitsOfSize.{v, u, max (max u v) w, max (max (u + 1) (v + 1)) (w + 1)} AddCommGroupCat
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                        instance AddCommGroupCat.hasColimits :
                        CategoryTheory.Limits.HasColimits AddCommGroupCat
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                        instance AddCommGroupCat.instHasColimitsOfSizeAddCommGroupCatMaxInstAddCommGroupCatLargeCategory :
                        CategoryTheory.Limits.HasColimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} AddCommGroupCatMax
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                        instance AddCommGroupCat.instHasColimitsOfSizeAddCommGroupCatMaxInstAddCommGroupCatLargeCategory_1 :
                        CategoryTheory.Limits.HasColimitsOfSize.{u, u, max u v, max (u + 1) (v + 1)} AddCommGroupCatMax
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                        instance AddCommGroupCat.instHasColimitsOfSizeAddCommGroupCatMaxInstAddCommGroupCatLargeCategory_2 :
                        CategoryTheory.Limits.HasColimitsOfSize.{u, v, max u v, max (u + 1) (v + 1)} AddCommGroupCatMax
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                        instance AddCommGroupCat.instHasColimitsOfSizeAddCommGroupCatMaxInstAddCommGroupCatLargeCategory_3 :
                        CategoryTheory.Limits.HasColimitsOfSize.{v, u, max u v, max (u + 1) (v + 1)} AddCommGroupCatMax
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                        instance AddCommGroupCat.instHasColimitsOfSizeAddCommGroupCatInstAddCommGroupCatLargeCategory :
                        CategoryTheory.Limits.HasColimitsOfSize.{0, 0, u, u + 1} AddCommGroupCat
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                        noncomputable def AddCommGroupCat.cokernelIsoQuotient {G : AddCommGroupCat} {H : AddCommGroupCat} (f : G ⟶ H) :
                        CategoryTheory.Limits.cokernel f ≅ AddCommGroupCat.of (↑H ⧸ AddMonoidHom.range f)

                        The categorical cokernel of a morphism in AddCommGroupCat agrees with the usual group-theoretical quotient.

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                        • One or more equations did not get rendered due to their size.
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