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Mathlib.Algebra.Category.Ring.Colimits

The category of commutative rings has all colimits. #

This file uses a "pre-automated" approach, just as for Mathlib/Algebra/Category/MonCat/Colimits.lean. It is a very uniform approach, that conceivably could be synthesised directly by a tactic that analyses the shape of CommRing and RingHom.

We build the colimit of a diagram in CommRingCat by constructing the free commutative ring on the disjoint union of all the commutative rings in the diagram, then taking the quotient by the commutative ring laws within each commutative ring, and the identifications given by the morphisms in the diagram.

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inductive CommRingCat.Colimits.Prequotient {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
Type v

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

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    instance CommRingCat.Colimits.instInhabitedPrequotient {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
    Inhabited (CommRingCat.Colimits.Prequotient F)
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    inductive CommRingCat.Colimits.Relation {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
    CommRingCat.Colimits.Prequotient F → CommRingCat.Colimits.Prequotient F → Prop

    The Relation on Prequotient saying when two expressions are equal because of the commutative ring laws, or because one element is mapped to another by a morphism in the diagram.

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      instance CommRingCat.Colimits.colimitSetoid {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
      Setoid (CommRingCat.Colimits.Prequotient F)

      The setoid corresponding to commutative expressions modulo monoid Relations and identifications.

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      def CommRingCat.Colimits.ColimitType {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
      Type v

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

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        instance CommRingCat.Colimits.ColimitType.instZero {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        Zero (CommRingCat.Colimits.ColimitType F)
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        instance CommRingCat.Colimits.ColimitType.instAdd {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        Add (CommRingCat.Colimits.ColimitType F)
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        instance CommRingCat.Colimits.ColimitType.instNeg {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        Neg (CommRingCat.Colimits.ColimitType F)
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        instance CommRingCat.Colimits.ColimitType.AddGroup {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        AddGroup (CommRingCat.Colimits.ColimitType F)
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        instance CommRingCat.Colimits.InhabitedColimitType {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        Inhabited (CommRingCat.Colimits.ColimitType F)
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        instance CommRingCat.Colimits.ColimitType.AddGroupWithOne {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        AddGroupWithOne (CommRingCat.Colimits.ColimitType F)
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        instance CommRingCat.Colimits.instCommRingColimitType {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        CommRing (CommRingCat.Colimits.ColimitType F)
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        @[simp]
        theorem CommRingCat.Colimits.quot_zero {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        Quot.mk Setoid.r CommRingCat.Colimits.Prequotient.zero = 0
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        @[simp]
        theorem CommRingCat.Colimits.quot_one {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        Quot.mk Setoid.r CommRingCat.Colimits.Prequotient.one = 1
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        @[simp]
        theorem CommRingCat.Colimits.quot_neg {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) (x : CommRingCat.Colimits.Prequotient F) :
        Quot.mk Setoid.r (CommRingCat.Colimits.Prequotient.neg x) = -Quot.mk Setoid.r x
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        @[simp]
        theorem CommRingCat.Colimits.quot_add {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) (x : CommRingCat.Colimits.Prequotient F) (y : CommRingCat.Colimits.Prequotient F) :
        Quot.mk Setoid.r (CommRingCat.Colimits.Prequotient.add x y) = Quot.mk Setoid.r x + Quot.mk Setoid.r y
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        @[simp]
        theorem CommRingCat.Colimits.quot_mul {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) (x : CommRingCat.Colimits.Prequotient F) (y : CommRingCat.Colimits.Prequotient F) :
        Quot.mk Setoid.r (CommRingCat.Colimits.Prequotient.mul x y) = Quot.mk Setoid.r x * Quot.mk Setoid.r y
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        def CommRingCat.Colimits.colimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
        CommRingCat

        The bundled commutative ring giving the colimit of a diagram.

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          def CommRingCat.Colimits.coconeFun {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) (j : J) (x : ↑(F.obj j)) :
          CommRingCat.Colimits.ColimitType F

          The function from a given commutative ring in the diagram to the colimit commutative ring.

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            def CommRingCat.Colimits.coconeMorphism {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) (j : J) :
            F.obj j ⟶ CommRingCat.Colimits.colimit F

            The ring homomorphism from a given commutative ring in the diagram to the colimit commutative ring.

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

              The cocone over the proposed colimit commutative ring.

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

                The function from the free commutative ring on the diagram to the cone point of any other cocone.

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

                  The function from the colimit commutative ring to the cone point of any other cocone.

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                    def CommRingCat.Colimits.descMorphism {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) (s : CategoryTheory.Limits.Cocone F) :
                    CommRingCat.Colimits.colimit F ⟶ s.pt

                    The ring homomorphism from the colimit commutative ring to the cone point of any other cocone.

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                      def CommRingCat.Colimits.colimitIsColimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat) :
                      CategoryTheory.Limits.IsColimit (CommRingCat.Colimits.colimitCocone F)

                      Evidence that the proposed colimit is the colimit.

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                        instance CommRingCat.Colimits.hasColimits_commRingCat :
                        CategoryTheory.Limits.HasColimits CommRingCat
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