Documentation

Mathlib.Algebra.Category.GroupCat.Biproducts

The category of abelian groups has finite biproducts #

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instance AddCommGroupCat.instHasBinaryBiproductsAddCommGroupCatInstAddCommGroupCatLargeCategoryPreadditiveHasZeroMorphismsInstPreadditiveAddCommGroupCatInstAddCommGroupCatLargeCategory :
CategoryTheory.Limits.HasBinaryBiproducts AddCommGroupCat
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instance AddCommGroupCat.instHasFiniteBiproductsAddCommGroupCatInstAddCommGroupCatLargeCategoryPreadditiveHasZeroMorphismsInstPreadditiveAddCommGroupCatInstAddCommGroupCatLargeCategory :
CategoryTheory.Limits.HasFiniteBiproducts AddCommGroupCat
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@[simp]
theorem AddCommGroupCat.binaryProductLimitCone_cone_pt (G : AddCommGroupCat) (H : AddCommGroupCat) :
(AddCommGroupCat.binaryProductLimitCone G H).cone.pt = AddCommGroupCat.of (↑G × ↑H)
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@[simp]
theorem AddCommGroupCat.binaryProductLimitCone_isLimit_lift (G : AddCommGroupCat) (H : AddCommGroupCat) (s : CategoryTheory.Limits.Cone (CategoryTheory.Limits.pair G H)) :
(AddCommGroupCat.binaryProductLimitCone G H).isLimit.lift s = AddMonoidHom.prod (s.π.app { as := CategoryTheory.Limits.WalkingPair.left }) (s.π.app { as := CategoryTheory.Limits.WalkingPair.right })
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def AddCommGroupCat.binaryProductLimitCone (G : AddCommGroupCat) (H : AddCommGroupCat) :
CategoryTheory.Limits.LimitCone (CategoryTheory.Limits.pair G H)

Construct limit data for a binary product in AddCommGroupCat, using AddCommGroupCat.of (G × H).

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  • One or more equations did not get rendered due to their size.
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    @[simp]
    theorem AddCommGroupCat.binaryProductLimitCone_cone_π_app_left (G : AddCommGroupCat) (H : AddCommGroupCat) :
    (AddCommGroupCat.binaryProductLimitCone G H).cone.π.app { as := CategoryTheory.Limits.WalkingPair.left } = AddMonoidHom.fst ↑G ↑H
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    @[simp]
    theorem AddCommGroupCat.binaryProductLimitCone_cone_π_app_right (G : AddCommGroupCat) (H : AddCommGroupCat) :
    (AddCommGroupCat.binaryProductLimitCone G H).cone.π.app { as := CategoryTheory.Limits.WalkingPair.right } = AddMonoidHom.snd ↑G ↑H
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    @[simp]
    theorem AddCommGroupCat.biprodIsoProd_hom_apply (G : AddCommGroupCat) (H : AddCommGroupCat) (i : ↑(CategoryTheory.Limits.BinaryBicone.toCone (CategoryTheory.Limits.BinaryBiproduct.bicone G H)).pt) :
    (AddCommGroupCat.biprodIsoProd G H).hom i = (CategoryTheory.Limits.biprod.fst i, CategoryTheory.Limits.biprod.snd i)
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    noncomputable def AddCommGroupCat.biprodIsoProd (G : AddCommGroupCat) (H : AddCommGroupCat) :
    G ⊞ H ≅ AddCommGroupCat.of (↑G × ↑H)

    We verify that the biproduct in AddCommGroupCat is isomorphic to the cartesian product of the underlying types:

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      theorem AddCommGroupCat.biprodIsoProd_inv_comp_fst_apply (G : AddCommGroupCat) (H : AddCommGroupCat) (x : (CategoryTheory.forget AddCommGroupCat).obj (AddCommGroupCat.binaryProductLimitCone G H).cone.pt) :
      CategoryTheory.Limits.biprod.fst ((CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso (CategoryTheory.Limits.BinaryBiproduct.isLimit G H) (AddCommGroupCat.binaryProductLimitCone G H).isLimit).inv x) = (AddMonoidHom.fst ↑G ↑H) x
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      @[simp]
      theorem AddCommGroupCat.biprodIsoProd_inv_comp_fst (G : AddCommGroupCat) (H : AddCommGroupCat) :
      CategoryTheory.CategoryStruct.comp (AddCommGroupCat.biprodIsoProd G H).inv CategoryTheory.Limits.biprod.fst = AddMonoidHom.fst ↑G ↑H
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      theorem AddCommGroupCat.biprodIsoProd_inv_comp_snd_apply (G : AddCommGroupCat) (H : AddCommGroupCat) (x : (CategoryTheory.forget AddCommGroupCat).obj (AddCommGroupCat.binaryProductLimitCone G H).cone.pt) :
      CategoryTheory.Limits.biprod.snd ((CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso (CategoryTheory.Limits.BinaryBiproduct.isLimit G H) (AddCommGroupCat.binaryProductLimitCone G H).isLimit).inv x) = (AddMonoidHom.snd ↑G ↑H) x
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      @[simp]
      theorem AddCommGroupCat.biprodIsoProd_inv_comp_snd (G : AddCommGroupCat) (H : AddCommGroupCat) :
      CategoryTheory.CategoryStruct.comp (AddCommGroupCat.biprodIsoProd G H).inv CategoryTheory.Limits.biprod.snd = AddMonoidHom.snd ↑G ↑H
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      @[simp]
      theorem AddCommGroupCat.HasLimit.lift_apply {J : Type w} (f : J → AddCommGroupCat) (s : CategoryTheory.Limits.Fan f) (x : ↑s.pt) (j : J) :
      (AddCommGroupCat.HasLimit.lift f s) x j = (s.π.app { as := j }) x
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      def AddCommGroupCat.HasLimit.lift {J : Type w} (f : J → AddCommGroupCat) (s : CategoryTheory.Limits.Fan f) :
      s.pt ⟶ AddCommGroupCat.of ((j : J) → ↑(f j))

      The map from an arbitrary cone over an indexed family of abelian groups to the cartesian product of those groups.

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      • AddCommGroupCat.HasLimit.lift f s = { toZeroHom := { toFun := fun (x : ↑s.pt) (j : J) => (s.π.app { as := j }) x, map_zero' := ⋯ }, map_add' := ⋯ }
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        @[simp]
        theorem AddCommGroupCat.HasLimit.productLimitCone_cone_π {J : Type w} (f : J → AddCommGroupCat) :
        (AddCommGroupCat.HasLimit.productLimitCone f).cone.π = CategoryTheory.Discrete.natTrans fun (j : CategoryTheory.Discrete J) => Pi.evalAddMonoidHom (fun (j : J) => ↑(f j)) j.as
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        @[simp]
        theorem AddCommGroupCat.HasLimit.productLimitCone_cone_pt {J : Type w} (f : J → AddCommGroupCat) :
        (AddCommGroupCat.HasLimit.productLimitCone f).cone.pt = AddCommGroupCat.of ((j : J) → ↑(f j))
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        @[simp]
        theorem AddCommGroupCat.HasLimit.productLimitCone_isLimit_lift {J : Type w} (f : J → AddCommGroupCat) (s : CategoryTheory.Limits.Fan f) :
        (AddCommGroupCat.HasLimit.productLimitCone f).isLimit.lift s = AddCommGroupCat.HasLimit.lift f s
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        def AddCommGroupCat.HasLimit.productLimitCone {J : Type w} (f : J → AddCommGroupCat) :
        CategoryTheory.Limits.LimitCone (CategoryTheory.Discrete.functor f)

        Construct limit data for a product in AddCommGroupCat, using AddCommGroupCat.of (∀ j, F.obj j).

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        • One or more equations did not get rendered due to their size.
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          @[simp]
          theorem AddCommGroupCat.biproductIsoPi_hom_apply {J : Type} [Finite J] (f : J → AddCommGroupCat) (x : ↑(CategoryTheory.Limits.Bicone.toCone (CategoryTheory.Limits.biproduct.bicone f)).pt) (j : J) :
          (AddCommGroupCat.biproductIsoPi f).hom x j = (CategoryTheory.Limits.biproduct.π f j) x
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          noncomputable def AddCommGroupCat.biproductIsoPi {J : Type} [Finite J] (f : J → AddCommGroupCat) :
          ⨁ f ≅ AddCommGroupCat.of ((j : J) → ↑(f j))

          We verify that the biproduct we've just defined is isomorphic to the AddCommGroupCat structure on the dependent function type.

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            theorem AddCommGroupCat.biproductIsoPi_inv_comp_π_apply {J : Type} [Finite J] (f : J → AddCommGroupCat) (j : J) (x : (CategoryTheory.forget AddCommGroupCat).obj (AddCommGroupCat.HasLimit.productLimitCone f).cone.pt) :
            (CategoryTheory.Limits.biproduct.π f j) ((CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso (CategoryTheory.Limits.biproduct.isLimit f) (AddCommGroupCat.HasLimit.productLimitCone f).isLimit).inv x) = (Pi.evalAddMonoidHom (fun (j : J) => ↑(f j)) j) x
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            @[simp]
            theorem AddCommGroupCat.biproductIsoPi_inv_comp_π {J : Type} [Finite J] (f : J → AddCommGroupCat) (j : J) :
            CategoryTheory.CategoryStruct.comp (AddCommGroupCat.biproductIsoPi f).inv (CategoryTheory.Limits.biproduct.π f j) = Pi.evalAddMonoidHom (fun (j : J) => ↑(f j)) j