Documentation

Mathlib.Algebra.Category.GrpWithZero

The category of groups with zero #

This file defines GrpWithZero, the category of groups with zero.

source
structure GrpWithZero :
Type (u_1 + 1)

The category of groups with zero.

Instances For
    source
    instance GrpWithZero.instCoeSortType :
    CoeSort GrpWithZero (Type u_1)
    Equations
      source
      @[reducible, inline]
      abbrev GrpWithZero.of (α : Type u_1) [GroupWithZero α] :
      GrpWithZero

      Construct a bundled GrpWithZero from a GroupWithZero.

      Equations
        Instances For
          source
          instance GrpWithZero.instInhabited :
          Inhabited GrpWithZero
          Equations
            source
            instance GrpWithZero.instLargeCategory :
            CategoryTheory.LargeCategory GrpWithZero
            Equations
              source
              instance GrpWithZero.groupWithZeroConcreteCategory :
              CategoryTheory.ConcreteCategory GrpWithZero fun (x1 x2 : GrpWithZero) => x1.carrier →*₀ x2.carrier
              Equations
                source
                @[reducible, inline]
                abbrev GrpWithZero.ofHom {X Y : Type u} [GroupWithZero X] [GroupWithZero Y] (f : X →*₀ Y) :
                of X ⟶ of Y

                Typecheck a MonoidWithZeroHom as a morphism in GrpWithZero.

                Equations
                  Instances For
                    source
                    @[simp]
                    theorem GrpWithZero.hom_id {X : GrpWithZero} :
                    CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X) = MonoidWithZeroHom.id X.carrier
                    source
                    @[simp]
                    theorem GrpWithZero.hom_comp {X Y Z : GrpWithZero} {f : X ⟶ Y} {g : Y ⟶ Z} :
                    CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g) = MonoidWithZeroHom.comp g f
                    source
                    theorem GrpWithZero.coe_id {X : GrpWithZero} :
                    ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) = id
                    source
                    theorem GrpWithZero.coe_comp {X Y Z : GrpWithZero} {f : X ⟶ Y} {g : Y ⟶ Z} :
                    ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g)) = ⇑(CategoryTheory.ConcreteCategory.hom g) ∘ ⇑(CategoryTheory.ConcreteCategory.hom f)
                    source
                    @[simp]
                    theorem GrpWithZero.forget_map {X Y : GrpWithZero} (f : X ⟶ Y) :
                    (CategoryTheory.forget GrpWithZero).map f = ⇑(CategoryTheory.ConcreteCategory.hom f)
                    source
                    instance GrpWithZero.hasForgetToBipointed :
                    CategoryTheory.HasForget₂ GrpWithZero Bipointed
                    Equations
                      source
                      instance GrpWithZero.hasForgetToMon :
                      CategoryTheory.HasForget₂ GrpWithZero MonCat
                      Equations
                        source
                        def GrpWithZero.Iso.mk {α β : GrpWithZero} (e : α.carrier ≃* β.carrier) :
                        α ≅ β

                        Constructs an isomorphism of groups with zero from a group isomorphism between them.

                        Equations
                          Instances For
                            source
                            @[simp]
                            theorem GrpWithZero.Iso.mk_inv {α β : GrpWithZero} (e : α.carrier ≃* β.carrier) :
                            (mk e).inv = ofHom ↑e.symm
                            source
                            @[simp]
                            theorem GrpWithZero.Iso.mk_hom {α β : GrpWithZero} (e : α.carrier ≃* β.carrier) :
                            (mk e).hom = ofHom ↑e