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Mathlib.CategoryTheory.Preadditive.CommGrp_

Commutative group objects in additive categories. #

We construct an inverse of the forgetful functor CommGrp_ C ⥤ C if C is an additive category.

This looks slightly strange because the additive structure of C maps to the multiplicative structure of the commutative group objects.

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def CategoryTheory.Preadditive.toCommGrp (C : Type u) [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] :
Functor C (CommGrp_ C)

The canonical functor from an additive category into its commutative group objects. This is always an equivalence, see commGrpEquivalence.

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    @[simp]
    theorem CategoryTheory.Preadditive.toCommGrp_obj_X (C : Type u) [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
    ((toCommGrp C).obj X).X = X
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    @[simp]
    theorem CategoryTheory.Preadditive.toCommGrp_obj_inv (C : Type u) [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
    ((toCommGrp C).obj X).inv = -CategoryStruct.id X
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    @[simp]
    theorem CategoryTheory.Preadditive.toCommGrp_obj_mul (C : Type u) [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
    ((toCommGrp C).obj X).mul = ChosenFiniteProducts.fst X X + ChosenFiniteProducts.snd X X
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    @[simp]
    theorem CategoryTheory.Preadditive.toCommGrp_map_hom (C : Type u) [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] {X Y : C} (f : X ⟶ Y) :
    ((toCommGrp C).map f).hom = f
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    @[simp]
    theorem CategoryTheory.Preadditive.toCommGrp_obj_one (C : Type u) [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
    ((toCommGrp C).obj X).one = 0
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    def CategoryTheory.Preadditive.commGrpEquivalenceAux {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] :
    (CommGrp_.forget C).comp (toCommGrp C) ≅ Functor.id (CommGrp_ C)

    Auxiliary definition for commGrpEquivalence.

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      @[simp]
      theorem CategoryTheory.Preadditive.commGrpEquivalenceAux_hom_app_hom {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : CommGrp_ C) :
      (commGrpEquivalenceAux.hom.app X).hom = CategoryStruct.id ((Grp_.forget₂Mon_ C).obj ((CommGrp_.forget₂Grp_ C).obj X)).X
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      @[simp]
      theorem CategoryTheory.Preadditive.commGrpEquivalenceAux_inv_app_hom {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : CommGrp_ C) :
      (commGrpEquivalenceAux.inv.app X).hom = CategoryStruct.id ((Grp_.forget₂Mon_ C).obj ((CommGrp_.forget₂Grp_ C).obj X)).X
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      def CategoryTheory.Preadditive.commGrpEquivalence {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] :
      C ≌ CommGrp_ C

      An additive category is equivalent to its category of commutative group objects.

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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_functor_obj_mul {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
        (commGrpEquivalence.functor.obj X).mul = ChosenFiniteProducts.fst X X + ChosenFiniteProducts.snd X X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_inverse_obj {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : CommGrp_ C) :
        commGrpEquivalence.inverse.obj X = ((Grp_.forget₂Mon_ C).obj ((CommGrp_.forget₂Grp_ C).obj X)).X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_unitIso_hom_app {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
        commGrpEquivalence.unitIso.hom.app X = CategoryStruct.id X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_functor_obj_inv {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
        (commGrpEquivalence.functor.obj X).inv = -CategoryStruct.id X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_counitIso_inv_app_hom {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : CommGrp_ C) :
        (commGrpEquivalence.counitIso.inv.app X).hom = CategoryStruct.id ((Grp_.forget₂Mon_ C).obj ((CommGrp_.forget₂Grp_ C).obj X)).X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_functor_map_hom {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] {X Y : C} (f : X ⟶ Y) :
        (commGrpEquivalence.functor.map f).hom = f
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_functor_obj_one {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
        (commGrpEquivalence.functor.obj X).one = 0
        source
        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_unitIso_inv_app {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
        commGrpEquivalence.unitIso.inv.app X = CategoryStruct.id X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_counitIso_hom_app_hom {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : CommGrp_ C) :
        (commGrpEquivalence.counitIso.hom.app X).hom = CategoryStruct.id ((Grp_.forget₂Mon_ C).obj ((CommGrp_.forget₂Grp_ C).obj X)).X
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_inverse_map {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] {X✝ Y✝ : CommGrp_ C} (f : X✝ ⟶ Y✝) :
        commGrpEquivalence.inverse.map f = f.hom
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        @[simp]
        theorem CategoryTheory.Preadditive.commGrpEquivalence_functor_obj_X {C : Type u} [Category.{v, u} C] [Preadditive C] [ChosenFiniteProducts C] (X : C) :
        (commGrpEquivalence.functor.obj X).X = X