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Mathlib.Algebra.Category.Grp.Colimits

The category of additive commutative groups has all colimits. #

This file uses a "pre-automated" approach, just as for Algebra.Category.MonCat.Colimits. 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 AddCommGrp 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 AddCommGrp 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 AddCommGrp.Colimits.Prequotient {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
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 AddCommGrp.Colimits.instInhabitedPrequotient {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
    Inhabited (AddCommGrp.Colimits.Prequotient F)
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    inductive AddCommGrp.Colimits.Relation {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
    AddCommGrp.Colimits.Prequotient F → AddCommGrp.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 AddCommGrp.Colimits.colimitSetoid {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
      Setoid (AddCommGrp.Colimits.Prequotient F)

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

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

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

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

        The bundled abelian group giving the colimit of a diagram.

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

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

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

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

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              @[simp]
              theorem AddCommGrp.Colimits.cocone_naturality {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) {j : J} {j' : J} (f : j ⟶ j') :
              CategoryTheory.CategoryStruct.comp (F.map f) (AddCommGrp.Colimits.coconeMorphism F j') = AddCommGrp.Colimits.coconeMorphism F j
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              @[simp]
              theorem AddCommGrp.Colimits.cocone_naturality_components {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) (j : J) (j' : J) (f : j ⟶ j') (x : ↑(F.obj j)) :
              (AddCommGrp.Colimits.coconeMorphism F j') ((F.map f) x) = (AddCommGrp.Colimits.coconeMorphism F j) x
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              def AddCommGrp.Colimits.colimitCocone {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
              CategoryTheory.Limits.Cocone F

              The cocone over the proposed colimit abelian group.

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

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

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                    def AddCommGrp.Colimits.descMorphism {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) (s : CategoryTheory.Limits.Cocone F) :
                    AddCommGrp.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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                      def AddCommGrp.Colimits.colimitCoconeIsColimit {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
                      CategoryTheory.Limits.IsColimit (AddCommGrp.Colimits.colimitCocone F)

                      Evidence that the proposed colimit is the colimit.

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                        theorem AddCommGrp.hasColimit {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrp) :
                        CategoryTheory.Limits.HasColimit F
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                        theorem AddCommGrp.hasColimitsOfShape (J : Type u) [CategoryTheory.Category.{v, u} J] :
                        CategoryTheory.Limits.HasColimitsOfShape J AddCommGrp
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                        theorem AddCommGrp.hasColimitsOfSize :
                        CategoryTheory.Limits.HasColimitsOfSize.{v, u, max (max u v) w, max (max (u + 1) (v + 1)) (w + 1)} AddCommGrp
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                        instance AddCommGrp.hasColimits :
                        CategoryTheory.Limits.HasColimits AddCommGrp
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                        instance AddCommGrp.instHasColimitsOfSizeAddCommGrpMax :
                        CategoryTheory.Limits.HasColimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} AddCommGrpMax
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                        instance AddCommGrp.instHasColimitsOfSizeAddCommGrpMax_1 :
                        CategoryTheory.Limits.HasColimitsOfSize.{u, u, max u v, max (u + 1) (v + 1)} AddCommGrpMax
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                        instance AddCommGrp.instHasColimitsOfSizeAddCommGrpMax_2 :
                        CategoryTheory.Limits.HasColimitsOfSize.{u, v, max u v, max (u + 1) (v + 1)} AddCommGrpMax
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                        instance AddCommGrp.instHasColimitsOfSizeAddCommGrpMax_3 :
                        CategoryTheory.Limits.HasColimitsOfSize.{v, u, max u v, max (u + 1) (v + 1)} AddCommGrpMax
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                        instance AddCommGrp.instHasColimitsOfSize :
                        CategoryTheory.Limits.HasColimitsOfSize.{0, 0, u, u + 1} AddCommGrp
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                        noncomputable def AddCommGrp.cokernelIsoQuotient {G : AddCommGrp} {H : AddCommGrp} (f : G ⟶ H) :
                        CategoryTheory.Limits.cokernel f ≅ AddCommGrp.of (↑H ⧸ AddMonoidHom.range f)

                        The categorical cokernel of a morphism in AddCommGrp 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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