The category of groups in a Cartesian monoidal category #

In fact, any monoid object whose associativity diagram is Cartesian can be made into a group object (we do not prove this in this file), so we should expect that many properties of group objects follow from this result.

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theorem CategoryTheory.AddGrpObj.isPullback {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : C) [AddGrpObj A] :

IsPullback (MonoidalCategoryStruct.whiskerRight AddMonObj.add A) (CategoryStruct.comp (MonoidalCategoryStruct.associator A A A).hom (MonoidalCategoryStruct.whiskerLeft A AddMonObj.add)) AddMonObj.add AddMonObj.add

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@[simp]

theorem CategoryTheory.GrpObj.inv_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {A B : C} [GrpObj A] [GrpObj B] (f : A ⟶ B) [IsMonHom f] :

CategoryStruct.comp inv f = CategoryStruct.comp f inv

Morphisms of group objects preserve inverses.

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@[simp]

theorem CategoryTheory.AddGrpObj.neg_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {A B : C} [AddGrpObj A] [AddGrpObj B] (f : A ⟶ B) [IsAddMonHom f] :

CategoryStruct.comp neg f = CategoryStruct.comp f neg

Morphisms of group objects preserve negations.

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@[simp]

theorem CategoryTheory.GrpObj.inv_hom_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {A B : C} [GrpObj A] [GrpObj B] (f : A ⟶ B) [IsMonHom f] {Z : C} (h : B ⟶ Z) :

CategoryStruct.comp inv (CategoryStruct.comp f h) = CategoryStruct.comp f (CategoryStruct.comp inv h)

Morphisms of group objects preserve inverses.

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@[simp]

theorem CategoryTheory.AddGrpObj.neg_hom_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {A B : C} [AddGrpObj A] [AddGrpObj B] (f : A ⟶ B) [IsAddMonHom f] {Z : C} (h : B ⟶ Z) :

CategoryStruct.comp neg (CategoryStruct.comp f h) = CategoryStruct.comp f (CategoryStruct.comp neg h)

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theorem CategoryTheory.GrpObj.toMonObj_injective {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X : C} :

Function.Injective (@toMonObj C inst✝ inst✝¹ X)

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theorem CategoryTheory.AddGrpObj.toAddMonObj_injective {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X : C} :

Function.Injective (@toAddMonObj C inst✝ inst✝¹ X)

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theorem CategoryTheory.GrpObj.ext {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X : C} (h₁ h₂ : GrpObj X) (H : h₁.toMonObj = h₂.toMonObj) :

h₁ = h₂

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theorem CategoryTheory.AddGrpObj.ext {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X : C} (h₁ h₂ : AddGrpObj X) (H : h₁.toAddMonObj = h₂.toAddMonObj) :

h₁ = h₂

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theorem CategoryTheory.AddGrpObj.ext_iff {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X : C} {h₁ h₂ : AddGrpObj X} :

h₁ = h₂ ↔ h₁.toAddMonObj = h₂.toAddMonObj

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theorem CategoryTheory.GrpObj.ext_iff {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X : C} {h₁ h₂ : GrpObj X} :

h₁ = h₂ ↔ h₁.toMonObj = h₂.toMonObj

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@[implicit_reducible]

def CategoryTheory.GrpObj.ofInvertible {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (G : C) [MonObj G] (h : (X : C) → (f : X ⟶ G) → Invertible f) :

GrpObj G

A monoid object with invertible homs is a group object.

Equations

One or more equations did not get rendered due to their size.

Instances For

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@[implicit_reducible]

instance CategoryTheory.GrpObj.tensorObj.instTensorObj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : C} [GrpObj G] [GrpObj H] :

GrpObj (MonoidalCategoryStruct.tensorObj G H)

Equations

One or more equations did not get rendered due to their size.

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@[implicit_reducible]

instance CategoryTheory.AddGrpObj.tensorObj.instTensorObj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : C} [AddGrpObj G] [AddGrpObj H] :

AddGrpObj (MonoidalCategoryStruct.tensorObj G H)

Equations

One or more equations did not get rendered due to their size.

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@[simp]

theorem CategoryTheory.GrpObj.tensorObj.inv_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : C} [GrpObj G] [GrpObj H] :

inv = MonoidalCategoryStruct.tensorHom inv inv

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@[simp]

theorem CategoryTheory.AddGrpObj.tensorObj.neg_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : C} [AddGrpObj G] [AddGrpObj H] :

neg = MonoidalCategoryStruct.tensorHom neg neg

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def CategoryTheory.Grp.forget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Functor (Grp C) (Mon C)

The forgetful functor from group objects to monoid objects.

Equations

CategoryTheory.Grp.forget₂Mon C = CategoryTheory.inducedFunctor CategoryTheory.Grp.toMon

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def CategoryTheory.AddGrp.forget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Functor (AddGrp C) (AddMon C)

The forgetful functor from additive group objects to additive monoid objects.

Equations

CategoryTheory.AddGrp.forget₂Mon C = CategoryTheory.inducedFunctor CategoryTheory.AddGrp.toAddMon

Instances For

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@[simp]

theorem CategoryTheory.AddGrp.forget₂Mon_obj_X (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

((forget₂Mon C).obj A).X = A.X

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@[simp]

theorem CategoryTheory.Grp.forget₂Mon_obj_X (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

((forget₂Mon C).obj A).X = A.X

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def CategoryTheory.Grp.fullyFaithfulForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).FullyFaithful

The forgetful functor from group objects to monoid objects is fully faithful.

Equations

CategoryTheory.Grp.fullyFaithfulForget₂Mon C = CategoryTheory.fullyFaithfulInducedFunctor CategoryTheory.Grp.toMon

Instances For

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def CategoryTheory.AddGrp.fullyFaithfulForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).FullyFaithful

The forgetful functor from additive group objects to additive monoid objects is fully faithful.

Equations

CategoryTheory.AddGrp.fullyFaithfulForget₂Mon C = CategoryTheory.fullyFaithfulInducedFunctor CategoryTheory.AddGrp.toAddMon

Instances For

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instance CategoryTheory.Grp.instFullMonForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).Full

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instance CategoryTheory.AddGrp.instFullMonForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).Full

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instance CategoryTheory.Grp.instFaithfulMonForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).Faithful

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instance CategoryTheory.AddGrp.instFaithfulMonForget₂Mon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).Faithful

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@[simp]

theorem CategoryTheory.Grp.forget₂Mon_obj_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

MonObj.one = MonObj.one

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@[simp]

theorem CategoryTheory.AddGrp.forget₂AddMon_obj_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

AddMonObj.zero = AddMonObj.zero

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@[simp]

theorem CategoryTheory.Grp.forget₂Mon_obj_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

MonObj.mul = MonObj.mul

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@[simp]

theorem CategoryTheory.AddGrp.forget₂AddMon_obj_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

AddMonObj.add = AddMonObj.add

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@[simp]

theorem CategoryTheory.Grp.forget₂Mon_map_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {A B : Grp C} (f : A ⟶ B) :

((forget₂Mon C).map f).hom = f.hom.hom

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@[simp]

theorem CategoryTheory.AddGrp.forget₂AddMon_map_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {A B : AddGrp C} (f : A ⟶ B) :

((forget₂Mon C).map f).hom = f.hom.hom

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def CategoryTheory.Grp.forget (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Functor (Grp C) C

The forgetful functor from group objects to the ambient category.

Equations

CategoryTheory.Grp.forget C = (CategoryTheory.Grp.forget₂Mon C).comp (CategoryTheory.Mon.forget C)

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def CategoryTheory.AddGrp.forget (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Functor (AddGrp C) C

The forgetful functor from additive group objects to the ambient category.

Equations

CategoryTheory.AddGrp.forget C = (CategoryTheory.AddGrp.forget₂Mon C).comp (CategoryTheory.AddMon.forget C)

Instances For

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@[simp]

theorem CategoryTheory.AddGrp.forget_map (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X✝ Y✝ : AddGrp C} (f : X✝ ⟶ Y✝) :

(forget C).map f = f.hom.hom

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@[simp]

theorem CategoryTheory.AddGrp.forget_obj (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (X : AddGrp C) :

(forget C).obj X = X.X

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@[simp]

theorem CategoryTheory.Grp.forget_obj (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (X : Grp C) :

(forget C).obj X = X.X

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@[simp]

theorem CategoryTheory.Grp.forget_map (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {X✝ Y✝ : Grp C} (f : X✝ ⟶ Y✝) :

(forget C).map f = f.hom.hom

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instance CategoryTheory.Grp.instFaithfulForget (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget C).Faithful

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instance CategoryTheory.AddGrp.instFaithfulForget (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget C).Faithful

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@[simp]

theorem CategoryTheory.Grp.forget₂Mon_comp_forget (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).comp (Mon.forget C) = forget C

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@[simp]

theorem CategoryTheory.AddGrp.forget₂AddMon_comp_forget (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(forget₂Mon C).comp (AddMon.forget C) = forget C

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instance CategoryTheory.Grp.instIsIsoHomHomMon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : Grp C} {f : G ⟶ H} [IsIso f] :

IsIso f.hom.hom

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instance CategoryTheory.AddGrp.instIsIsoHomHomMon (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : AddGrp C} {f : G ⟶ H} [IsIso f] :

IsIso f.hom.hom

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def CategoryTheory.Grp.mkIso' {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : C} (e : G ≅ H) [GrpObj G] [GrpObj H] [IsMonHom e.hom] :

{ X := G, grp := inst✝ } ≅ { X := H, grp := inst✝¹ }

Construct an isomorphism of group objects by giving a monoid isomorphism between the underlying objects.

Equations

CategoryTheory.Grp.mkIso' e = (CategoryTheory.Grp.fullyFaithfulForget₂Mon C).preimageIso (CategoryTheory.Mon.mkIso' e)

Instances For

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def CategoryTheory.AddGrp.mkIso' {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : C} (e : G ≅ H) [AddGrpObj G] [AddGrpObj H] [IsAddMonHom e.hom] :

{ X := G, addGrp := inst✝ } ≅ { X := H, addGrp := inst✝¹ }

Construct an isomorphism of additive group objects by giving an additive monoid isomorphism between the underlying objects.

Equations

CategoryTheory.AddGrp.mkIso' e = (CategoryTheory.AddGrp.fullyFaithfulForget₂Mon C).preimageIso (CategoryTheory.AddMon.mkIso' e)

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@[simp]

theorem CategoryTheory.AddGrp.mkIso'_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : C} (e : G ≅ H) [AddGrpObj G] [AddGrpObj H] [IsAddMonHom e.hom] :

(mkIso' e).hom.hom.hom = e.hom

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@[simp]

theorem CategoryTheory.Grp.mkIso'_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : C} (e : G ≅ H) [GrpObj G] [GrpObj H] [IsMonHom e.hom] :

(mkIso' e).hom.hom.hom = e.hom

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@[simp]

theorem CategoryTheory.Grp.mkIso'_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : C} (e : G ≅ H) [GrpObj G] [GrpObj H] [IsMonHom e.hom] :

(mkIso' e).inv.hom.hom = e.inv

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@[simp]

theorem CategoryTheory.AddGrp.mkIso'_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : C} (e : G ≅ H) [AddGrpObj G] [AddGrpObj H] [IsAddMonHom e.hom] :

(mkIso' e).inv.hom.hom = e.inv

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@[reducible, inline]

abbrev CategoryTheory.Grp.mkIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : Grp C} (e : G.X ≅ H.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :

G ≅ H

Construct an isomorphism of group objects by giving an isomorphism between the underlying objects and checking compatibility with unit and multiplication only in the forward direction.

Equations

CategoryTheory.Grp.mkIso e one_f mul_f = CategoryTheory.Grp.mkIso' e

Instances For

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@[reducible, inline]

abbrev CategoryTheory.AddGrp.mkIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : AddGrp C} (e : G.X ≅ H.X) (zero_f : CategoryStruct.comp AddMonObj.zero e.hom = AddMonObj.zero := by cat_disch) (add_f : CategoryStruct.comp AddMonObj.add e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) AddMonObj.add := by cat_disch) :

G ≅ H

Construct an isomorphism of additive group objects by giving an isomorphism between the underlying objects and checking compatibility with zero and addition only in the forward direction.

Equations

CategoryTheory.AddGrp.mkIso e one_f mul_f = CategoryTheory.AddGrp.mkIso' e

Instances For

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@[simp]

theorem CategoryTheory.AddGrp.mkIso_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : AddGrp C} (e : G.X ≅ H.X) (zero_f : CategoryStruct.comp AddMonObj.zero e.hom = AddMonObj.zero := by cat_disch) (add_f : CategoryStruct.comp AddMonObj.add e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) AddMonObj.add := by cat_disch) :

(mkIso e zero_f add_f).hom.hom.hom = e.hom

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@[simp]

theorem CategoryTheory.AddGrp.mkIso_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : AddGrp C} (e : G.X ≅ H.X) (zero_f : CategoryStruct.comp AddMonObj.zero e.hom = AddMonObj.zero := by cat_disch) (add_f : CategoryStruct.comp AddMonObj.add e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) AddMonObj.add := by cat_disch) :

(mkIso e zero_f add_f).inv.hom.hom = e.inv

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@[simp]

theorem CategoryTheory.Grp.mkIso_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : Grp C} (e : G.X ≅ H.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :

(mkIso e one_f mul_f).hom.hom.hom = e.hom

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@[simp]

theorem CategoryTheory.Grp.mkIso_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : Grp C} (e : G.X ≅ H.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :

(mkIso e one_f mul_f).inv.hom.hom = e.inv

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@[deprecated CategoryTheory.Grp.mkIso_hom_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.mkIso_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : Grp C} (e : G.X ≅ H.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :

(mkIso e one_f mul_f).hom.hom.hom = e.hom

Alias of CategoryTheory.Grp.mkIso_hom_hom_hom.

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@[deprecated CategoryTheory.Grp.mkIso_inv_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.mkIso_inv_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {G H : Grp C} (e : G.X ≅ H.X) (one_f : CategoryStruct.comp MonObj.one e.hom = MonObj.one := by cat_disch) (mul_f : CategoryStruct.comp MonObj.mul e.hom = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom e.hom e.hom) MonObj.mul := by cat_disch) :

(mkIso e one_f mul_f).inv.hom.hom = e.inv

Alias of CategoryTheory.Grp.mkIso_inv_hom_hom.

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@[implicit_reducible]

instance CategoryTheory.Grp.uniqueHomFromTrivial {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

Unique (trivial C ⟶ A)

Equations

A.uniqueHomFromTrivial = (have this := CategoryTheory.InducedCategory.homEquiv; this).unique

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@[implicit_reducible]

instance CategoryTheory.AddGrp.uniqueHomFromTrivial {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

Unique (trivial C ⟶ A)

Equations

A.uniqueHomFromTrivial = (have this := CategoryTheory.InducedCategory.homEquiv; this).unique

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@[implicit_reducible]

instance CategoryTheory.Grp.uniqueHomToTrivial {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

Unique (A ⟶ trivial C)

Equations

A.uniqueHomToTrivial = (have this := CategoryTheory.InducedCategory.homEquiv; this).unique

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@[implicit_reducible]

instance CategoryTheory.AddGrp.uniqueHomToTrivial {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

Unique (A ⟶ trivial C)

Equations

A.uniqueHomToTrivial = (have this := CategoryTheory.InducedCategory.homEquiv; this).unique

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theorem CategoryTheory.Grp.isZero_trivial (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Limits.IsZero (trivial C)

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theorem CategoryTheory.AddGrp.isZero_trivial (C : Type u₁) [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Limits.IsZero (trivial C)

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instance CategoryTheory.Grp.instHasZeroObject {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Limits.HasZeroObject (Grp C)

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instance CategoryTheory.AddGrp.instHasZeroObject {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Limits.HasZeroObject (AddGrp C)

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@[implicit_reducible]

noncomputable instance CategoryTheory.Grp.instZeroHom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (G H : Grp C) :

Zero (G ⟶ H)

Equations

One or more equations did not get rendered due to their size.

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@[implicit_reducible]

noncomputable instance CategoryTheory.AddGrp.instZeroHom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (G H : AddGrp C) :

Zero (G ⟶ H)

Equations

One or more equations did not get rendered due to their size.

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@[simp]

theorem CategoryTheory.Grp.zero_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (G H : Grp C) :

InducedCategory.Hom.hom 0 = 0

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@[simp]

theorem CategoryTheory.AddGrp.zero_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (G H : AddGrp C) :

InducedCategory.Hom.hom 0 = 0

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@[implicit_reducible]

noncomputable instance CategoryTheory.Grp.instHasZeroMorphisms {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Limits.HasZeroMorphisms (Grp C)

Equations

CategoryTheory.Grp.instHasZeroMorphisms = { zero := CategoryTheory.Grp.instZeroHom, comp_zero := ⋯, zero_comp := ⋯ }

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@[implicit_reducible]

noncomputable instance CategoryTheory.AddGrp.instHasZeroMorphisms {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

Limits.HasZeroMorphisms (AddGrp C)

Equations

CategoryTheory.AddGrp.instHasZeroMorphisms = { zero := CategoryTheory.AddGrp.instZeroHom, comp_zero := ⋯, zero_comp := ⋯ }

`Grp C` is cartesian-monoidal #

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@[implicit_reducible]

instance CategoryTheory.Grp.instMonoidalCategoryStruct {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

MonoidalCategoryStruct (Grp C)

Equations

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@[implicit_reducible]

instance CategoryTheory.AddGrp.instMonoidalCategoryStruct {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

MonoidalCategoryStruct (AddGrp C)

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@[simp]

theorem CategoryTheory.AddGrp.tensorHom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {X₁✝ Y₁✝ X₂✝ Y₂✝ : AddGrp C} (f : X₁✝ ⟶ Y₁✝) (g : X₂✝ ⟶ Y₂✝) :

(MonoidalCategoryStruct.tensorHom f g).hom = MonoidalCategoryStruct.tensorHom f.hom g.hom

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@[simp]

theorem CategoryTheory.Grp.tensorHom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {X₁✝ Y₁✝ X₂✝ Y₂✝ : Grp C} (f : X₁✝ ⟶ Y₁✝) (g : X₂✝ ⟶ Y₂✝) :

(MonoidalCategoryStruct.tensorHom f g).hom = MonoidalCategoryStruct.tensorHom f.hom g.hom

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@[simp]

theorem CategoryTheory.AddGrp.tensorObj_X {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

(MonoidalCategoryStruct.tensorObj G H).X = MonoidalCategoryStruct.tensorObj G.X H.X

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@[simp]

theorem CategoryTheory.Grp.tensorObj_X {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(MonoidalCategoryStruct.tensorObj G H).X = MonoidalCategoryStruct.tensorObj G.X H.X

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@[simp]

theorem CategoryTheory.Grp.tensorUnit_X {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

(MonoidalCategoryStruct.tensorUnit (Grp C)).X = MonoidalCategoryStruct.tensorUnit C

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@[simp]

theorem CategoryTheory.AddGrp.tensorAddUnit_X {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

(MonoidalCategoryStruct.tensorUnit (AddGrp C)).X = MonoidalCategoryStruct.tensorUnit C

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@[simp]

theorem CategoryTheory.Grp.tensorUnit_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

MonObj.one = MonObj.one

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@[simp]

theorem CategoryTheory.AddGrp.tensorAddUnit_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

AddMonObj.zero = AddMonObj.zero

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@[simp]

theorem CategoryTheory.Grp.tensorUnit_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

MonObj.mul = MonObj.mul

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@[simp]

theorem CategoryTheory.AddGrp.tensorAddUnit_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

AddMonObj.add = AddMonObj.add

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@[simp]

theorem CategoryTheory.Grp.tensorObj_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

MonObj.one = MonObj.one

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@[simp]

theorem CategoryTheory.AddGrp.tensorObj_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

AddMonObj.zero = AddMonObj.zero

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@[simp]

theorem CategoryTheory.Grp.tensorObj_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

MonObj.mul = MonObj.mul

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@[simp]

theorem CategoryTheory.AddGrp.tensorObj_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

AddMonObj.add = AddMonObj.add

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@[simp]

theorem CategoryTheory.Grp.whiskerLeft_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : Grp C} (f : G ⟶ H) (I : Grp C) :

(MonoidalCategoryStruct.whiskerRight f I).hom.hom = MonoidalCategoryStruct.whiskerRight f.hom.hom I.X

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@[simp]

theorem CategoryTheory.AddGrp.whiskerLeft_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : AddGrp C} (f : G ⟶ H) (I : AddGrp C) :

(MonoidalCategoryStruct.whiskerRight f I).hom.hom = MonoidalCategoryStruct.whiskerRight f.hom.hom I.X

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@[simp]

theorem CategoryTheory.Grp.whiskerRight_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) {H I : Grp C} (f : H ⟶ I) :

(MonoidalCategoryStruct.whiskerLeft G f).hom.hom = MonoidalCategoryStruct.whiskerLeft G.X f.hom.hom

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@[simp]

theorem CategoryTheory.AddGrp.whiskerRight_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : AddGrp C) {H I : AddGrp C} (f : H ⟶ I) :

(MonoidalCategoryStruct.whiskerLeft G f).hom.hom = MonoidalCategoryStruct.whiskerLeft G.X f.hom.hom

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@[simp]

theorem CategoryTheory.Grp.leftUnitor_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.leftUnitor G).hom.hom.hom = (MonoidalCategoryStruct.leftUnitor G.X).hom

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@[simp]

theorem CategoryTheory.AddGrp.leftUnitor_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : AddGrp C) :

(MonoidalCategoryStruct.leftUnitor G).hom.hom.hom = (MonoidalCategoryStruct.leftUnitor G.X).hom

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@[simp]

theorem CategoryTheory.Grp.leftUnitor_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.leftUnitor G).inv.hom.hom = (MonoidalCategoryStruct.leftUnitor G.X).inv

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@[simp]

theorem CategoryTheory.AddGrp.leftUnitor_neg_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : AddGrp C) :

(MonoidalCategoryStruct.leftUnitor G).inv.hom.hom = (MonoidalCategoryStruct.leftUnitor G.X).inv

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@[simp]

theorem CategoryTheory.Grp.rightUnitor_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.rightUnitor G).hom.hom.hom = (MonoidalCategoryStruct.rightUnitor G.X).hom

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@[simp]

theorem CategoryTheory.AddGrp.rightUnitor_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : AddGrp C) :

(MonoidalCategoryStruct.rightUnitor G).hom.hom.hom = (MonoidalCategoryStruct.rightUnitor G.X).hom

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@[simp]

theorem CategoryTheory.Grp.rightUnitor_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.rightUnitor G).inv.hom.hom = (MonoidalCategoryStruct.rightUnitor G.X).inv

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@[simp]

theorem CategoryTheory.AddGrp.rightUnitor_neg_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : AddGrp C) :

(MonoidalCategoryStruct.rightUnitor G).inv.hom.hom = (MonoidalCategoryStruct.rightUnitor G.X).inv

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@[simp]

theorem CategoryTheory.Grp.associator_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H I : Grp C) :

(MonoidalCategoryStruct.associator G H I).hom.hom.hom = (MonoidalCategoryStruct.associator G.X H.X I.X).hom

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@[simp]

theorem CategoryTheory.AddGrp.associator_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H I : AddGrp C) :

(MonoidalCategoryStruct.associator G H I).hom.hom.hom = (MonoidalCategoryStruct.associator G.X H.X I.X).hom

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@[simp]

theorem CategoryTheory.Grp.associator_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H I : Grp C) :

(MonoidalCategoryStruct.associator G H I).inv.hom.hom = (MonoidalCategoryStruct.associator G.X H.X I.X).inv

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@[simp]

theorem CategoryTheory.AddGrp.associator_neg_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H I : AddGrp C) :

(MonoidalCategoryStruct.associator G H I).inv.hom.hom = (MonoidalCategoryStruct.associator G.X H.X I.X).inv

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@[deprecated CategoryTheory.Grp.whiskerLeft_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.whiskerLeft_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H : Grp C} (f : G ⟶ H) (I : Grp C) :

(MonoidalCategoryStruct.whiskerRight f I).hom.hom = MonoidalCategoryStruct.whiskerRight f.hom.hom I.X

Alias of CategoryTheory.Grp.whiskerLeft_hom_hom.

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@[deprecated CategoryTheory.Grp.whiskerRight_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.whiskerRight_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) {H I : Grp C} (f : H ⟶ I) :

(MonoidalCategoryStruct.whiskerLeft G f).hom.hom = MonoidalCategoryStruct.whiskerLeft G.X f.hom.hom

Alias of CategoryTheory.Grp.whiskerRight_hom_hom.

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@[deprecated CategoryTheory.Grp.leftUnitor_hom_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.leftUnitor_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.leftUnitor G).hom.hom.hom = (MonoidalCategoryStruct.leftUnitor G.X).hom

Alias of CategoryTheory.Grp.leftUnitor_hom_hom_hom.

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@[deprecated CategoryTheory.Grp.leftUnitor_inv_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.leftUnitor_inv_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.leftUnitor G).inv.hom.hom = (MonoidalCategoryStruct.leftUnitor G.X).inv

Alias of CategoryTheory.Grp.leftUnitor_inv_hom_hom.

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@[deprecated CategoryTheory.Grp.rightUnitor_hom_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.rightUnitor_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.rightUnitor G).hom.hom.hom = (MonoidalCategoryStruct.rightUnitor G.X).hom

Alias of CategoryTheory.Grp.rightUnitor_hom_hom_hom.

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@[deprecated CategoryTheory.Grp.rightUnitor_inv_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.rightUnitor_inv_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G : Grp C) :

(MonoidalCategoryStruct.rightUnitor G).inv.hom.hom = (MonoidalCategoryStruct.rightUnitor G.X).inv

Alias of CategoryTheory.Grp.rightUnitor_inv_hom_hom.

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@[deprecated CategoryTheory.Grp.associator_hom_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.associator_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H I : Grp C) :

(MonoidalCategoryStruct.associator G H I).hom.hom.hom = (MonoidalCategoryStruct.associator G.X H.X I.X).hom

Alias of CategoryTheory.Grp.associator_hom_hom_hom.

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@[deprecated CategoryTheory.Grp.associator_inv_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.associator_inv_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H I : Grp C) :

(MonoidalCategoryStruct.associator G H I).inv.hom.hom = (MonoidalCategoryStruct.associator G.X H.X I.X).inv

Alias of CategoryTheory.Grp.associator_inv_hom_hom.

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@[implicit_reducible]

instance CategoryTheory.Grp.instMonoidalCategory {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

MonoidalCategory (Grp C)

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@[implicit_reducible]

instance CategoryTheory.AddGrp.instMonoidalCategory {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

MonoidalCategory (AddGrp C)

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@[implicit_reducible]

instance CategoryTheory.Grp.instCartesianMonoidalCategory {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

CartesianMonoidalCategory (Grp C)

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@[implicit_reducible]

instance CategoryTheory.AddGrp.instCartesianMonoidalCategory {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

CartesianMonoidalCategory (AddGrp C)

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@[simp]

theorem CategoryTheory.Grp.lift_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H₁ H₂ : Grp C} (f : G ⟶ H₁) (g : G ⟶ H₂) :

(CartesianMonoidalCategory.lift f g).hom = CartesianMonoidalCategory.lift f.hom g.hom

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@[simp]

theorem CategoryTheory.AddGrp.lift_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] {G H₁ H₂ : AddGrp C} (f : G ⟶ H₁) (g : G ⟶ H₂) :

(CartesianMonoidalCategory.lift f g).hom = CartesianMonoidalCategory.lift f.hom g.hom

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@[simp]

theorem CategoryTheory.Grp.fst_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(SemiCartesianMonoidalCategory.fst G H).hom.hom = SemiCartesianMonoidalCategory.fst G.X H.X

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@[simp]

theorem CategoryTheory.AddGrp.fst_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

(SemiCartesianMonoidalCategory.fst G H).hom.hom = SemiCartesianMonoidalCategory.fst G.X H.X

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@[simp]

theorem CategoryTheory.Grp.snd_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(SemiCartesianMonoidalCategory.snd G H).hom.hom = SemiCartesianMonoidalCategory.snd G.X H.X

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@[simp]

theorem CategoryTheory.AddGrp.snd_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

(SemiCartesianMonoidalCategory.snd G H).hom.hom = SemiCartesianMonoidalCategory.snd G.X H.X

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@[deprecated CategoryTheory.Grp.fst_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.fst_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(SemiCartesianMonoidalCategory.fst G H).hom.hom = SemiCartesianMonoidalCategory.fst G.X H.X

Alias of CategoryTheory.Grp.fst_hom_hom.

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@[deprecated CategoryTheory.Grp.snd_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.snd_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(SemiCartesianMonoidalCategory.snd G H).hom.hom = SemiCartesianMonoidalCategory.snd G.X H.X

Alias of CategoryTheory.Grp.snd_hom_hom.

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@[implicit_reducible]

instance CategoryTheory.Grp.instMonoidalMonForget₂Mon {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

(forget₂Mon C).Monoidal

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@[implicit_reducible]

instance CategoryTheory.AddGrp.instMonoidalMonForget₂Mon {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

(forget₂Mon C).Monoidal

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@[simp]

theorem CategoryTheory.AddGrp.δ_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

Functor.OplaxMonoidal.δ (forget₂Mon C) G H = CategoryStruct.id ((forget₂Mon C).obj (MonoidalCategoryStruct.tensorObj G H))

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@[simp]

theorem CategoryTheory.Grp.δ_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

Functor.OplaxMonoidal.δ (forget₂Mon C) G H = CategoryStruct.id ((forget₂Mon C).obj (MonoidalCategoryStruct.tensorObj G H))

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@[simp]

theorem CategoryTheory.Grp.η_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

Functor.OplaxMonoidal.η (forget₂Mon C) = CategoryStruct.id ((forget₂Mon C).obj (MonoidalCategoryStruct.tensorUnit (Grp C)))

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@[simp]

theorem CategoryTheory.Grp.μ_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

Functor.LaxMonoidal.μ (forget₂Mon C) G H = CategoryStruct.id (MonoidalCategoryStruct.tensorObj ((forget₂Mon C).obj G) ((forget₂Mon C).obj H))

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@[simp]

theorem CategoryTheory.AddGrp.η_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

Functor.OplaxMonoidal.η (forget₂Mon C) = CategoryStruct.id ((forget₂Mon C).obj (MonoidalCategoryStruct.tensorUnit (AddGrp C)))

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@[simp]

theorem CategoryTheory.AddGrp.ε_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

Functor.LaxMonoidal.ε (forget₂Mon C) = CategoryStruct.id (MonoidalCategoryStruct.tensorUnit (AddMon C))

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@[simp]

theorem CategoryTheory.Grp.ε_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

Functor.LaxMonoidal.ε (forget₂Mon C) = CategoryStruct.id (MonoidalCategoryStruct.tensorUnit (Mon C))

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@[simp]

theorem CategoryTheory.AddGrp.μ_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

Functor.LaxMonoidal.μ (forget₂Mon C) G H = CategoryStruct.id (MonoidalCategoryStruct.tensorObj ((forget₂Mon C).obj G) ((forget₂Mon C).obj H))

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@[implicit_reducible]

instance CategoryTheory.Grp.instBraidedCategory {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

BraidedCategory (Grp C)

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@[implicit_reducible]

instance CategoryTheory.AddGrp.instBraidedCategory {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] :

BraidedCategory (AddGrp C)

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@[simp]

theorem CategoryTheory.Grp.braiding_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(β_ G H).hom.hom.hom = (β_ G.X H.X).hom

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@[simp]

theorem CategoryTheory.AddGrp.braiding_hom_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

(β_ G H).hom.hom.hom = (β_ G.X H.X).hom

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@[simp]

theorem CategoryTheory.Grp.braiding_inv_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(β_ G H).inv.hom.hom = (β_ G.X H.X).inv

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@[simp]

theorem CategoryTheory.AddGrp.braiding_neg_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : AddGrp C) :

(β_ G H).inv.hom.hom = (β_ G.X H.X).inv

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@[deprecated CategoryTheory.Grp.braiding_hom_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.braiding_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(β_ G H).hom.hom.hom = (β_ G.X H.X).hom

Alias of CategoryTheory.Grp.braiding_hom_hom_hom.

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@[deprecated CategoryTheory.Grp.braiding_inv_hom_hom (since := "2025-12-18")]

theorem CategoryTheory.Grp.braiding_inv_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] [BraidedCategory C] (G H : Grp C) :

(β_ G H).inv.hom.hom = (β_ G.X H.X).inv

Alias of CategoryTheory.Grp.braiding_inv_hom_hom.

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@[reducible, inline]

abbrev CategoryTheory.Functor.grpObjObj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] {G : C} [GrpObj G] :

GrpObj (F.obj G)

The image of a group object under a monoidal functor is a group object.

Equations

CategoryTheory.Functor.grpObjObj = { toMonObj := CategoryTheory.Functor.monObjObj G, inv := F.map CategoryTheory.GrpObj.inv, left_inv := ⋯, right_inv := ⋯ }

Instances For

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@[reducible, inline]

abbrev CategoryTheory.Functor.addGrpObjObj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] {G : C} [AddGrpObj G] :

AddGrpObj (F.obj G)

The image of an additive group object under a monoidal functor is an additive group object.

Equations

CategoryTheory.Functor.addGrpObjObj = { toAddMonObj := CategoryTheory.Functor.addMonObjObj G, neg := F.map CategoryTheory.AddGrpObj.neg, left_neg := ⋯, right_neg := ⋯ }

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theorem CategoryTheory.Functor.obj.ι_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] {G : C} [GrpObj G] :

GrpObj.inv = F.map GrpObj.inv

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@[simp]

theorem CategoryTheory.Functor.obj.neg_def {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] {G : C} [AddGrpObj G] :

AddGrpObj.neg = F.map AddGrpObj.neg

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theorem CategoryTheory.Functor.obj.ι_def_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] {G : C} [GrpObj G] {Z : D} (h : F.obj G ⟶ Z) :

CategoryStruct.comp GrpObj.inv h = CategoryStruct.comp (F.map GrpObj.inv) h

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theorem CategoryTheory.Functor.obj.neg_def_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] {G : C} [AddGrpObj G] {Z : D} (h : F.obj G ⟶ Z) :

CategoryStruct.comp AddGrpObj.neg h = CategoryStruct.comp (F.map AddGrpObj.neg) h

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def CategoryTheory.Functor.mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] :

Functor (Grp C) (Grp D)

A finite-product-preserving functor takes group objects to group objects.

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def CategoryTheory.Functor.mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] :

Functor (AddGrp C) (AddGrp D)

A finite-product-preserving functor takes additive group objects to additive group objects.

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theorem CategoryTheory.Functor.mapAddGrp_obj_addGrp_neg {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : AddGrp C) :

AddGrpObj.neg = F.map AddGrpObj.neg

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@[simp]

theorem CategoryTheory.Functor.mapGrp_obj_X {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : Grp C) :

(F.mapGrp.obj A).X = F.obj A.X

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@[simp]

theorem CategoryTheory.Functor.mapGrp_obj_grp_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : Grp C) :

MonObj.one = CategoryStruct.comp (LaxMonoidal.ε F) (F.map MonObj.one)

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@[simp]

theorem CategoryTheory.Functor.mapAddGrp_obj_addGrp_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : AddGrp C) :

AddMonObj.zero = CategoryStruct.comp (LaxMonoidal.ε F) (F.map AddMonObj.zero)

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@[simp]

theorem CategoryTheory.Functor.mapAddGrp_obj_addGrp_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : AddGrp C) :

AddMonObj.add = CategoryStruct.comp (LaxMonoidal.μ F A.X A.X) (F.map AddMonObj.add)

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@[simp]

theorem CategoryTheory.Functor.mapGrp_obj_grp_inv {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : Grp C) :

GrpObj.inv = F.map GrpObj.inv

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@[simp]

theorem CategoryTheory.Functor.mapGrp_map_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] {X✝ Y✝ : Grp C} (f : X✝ ⟶ Y✝) :

(F.mapGrp.map f).hom.hom = F.map f.hom.hom

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@[simp]

theorem CategoryTheory.Functor.mapGrp_obj_grp_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : Grp C) :

MonObj.mul = CategoryStruct.comp (LaxMonoidal.μ F A.X A.X) (F.map MonObj.mul)

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@[simp]

theorem CategoryTheory.Functor.mapAddGrp_map_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] {X✝ Y✝ : AddGrp C} (f : X✝ ⟶ Y✝) :

(F.mapAddGrp.map f).hom.hom = F.map f.hom.hom

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@[simp]

theorem CategoryTheory.Functor.mapAddGrp_obj_X {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : Functor C D) [F.Monoidal] (A : AddGrp C) :

(F.mapAddGrp.obj A).X = F.obj A.X

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instance CategoryTheory.Functor.Faithful.mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] [F.Faithful] :

F.mapGrp.Faithful

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instance CategoryTheory.Functor.Faithful.mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] [F.Faithful] :

F.mapAddGrp.Faithful

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def CategoryTheory.Functor.FullyFaithful.mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) :

F.mapGrp.FullyFaithful

If F : C ⥤ D is a fully faithful monoidal functor, then F.mapGrp : Grp C ⥤ Grp D is fully faithful too.

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def CategoryTheory.Functor.FullyFaithful.mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) :

F.mapAddGrp.FullyFaithful

If F : C ⥤ D is a fully faithful monoidal functor, then F.mapAddGrp : AddGrp C ⥤ AddGrp D is fully faithful too.

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instance CategoryTheory.Functor.Full.mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] [F.Full] [F.Faithful] :

F.mapGrp.Full

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instance CategoryTheory.Functor.Full.mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] [F.Full] [F.Faithful] :

F.mapAddGrp.Full

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@[simp]

theorem CategoryTheory.Functor.mapGrp_id_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

MonObj.one = CategoryStruct.comp (CategoryStruct.id (MonoidalCategoryStruct.tensorUnit C)) MonObj.one

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theorem CategoryTheory.Functor.mapAddGrp_id_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

AddMonObj.zero = CategoryStruct.comp (CategoryStruct.id (MonoidalCategoryStruct.tensorUnit C)) AddMonObj.zero

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theorem CategoryTheory.Functor.mapGrp_id_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : Grp C) :

MonObj.mul = CategoryStruct.comp (CategoryStruct.id (MonoidalCategoryStruct.tensorObj A.X A.X)) MonObj.mul

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theorem CategoryTheory.Functor.mapAddGrp_id_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (A : AddGrp C) :

AddMonObj.add = CategoryStruct.comp (CategoryStruct.id (MonoidalCategoryStruct.tensorObj A.X A.X)) AddMonObj.add

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@[simp]

theorem CategoryTheory.Functor.comp_mapGrp_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : Grp C) :

MonObj.one = CategoryStruct.comp (LaxMonoidal.ε (F.comp G)) ((F.comp G).map MonObj.one)

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theorem CategoryTheory.Functor.comp_mapAddGrp_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : AddGrp C) :

AddMonObj.zero = CategoryStruct.comp (LaxMonoidal.ε (F.comp G)) ((F.comp G).map AddMonObj.zero)

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theorem CategoryTheory.Functor.comp_mapAddGrp_zero_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : AddGrp C) {Z : E} (h : ((F.comp G).mapAddGrp.obj A).X ⟶ Z) :

CategoryStruct.comp AddMonObj.zero h = CategoryStruct.comp (CategoryStruct.comp (LaxMonoidal.ε (F.comp G)) (G.map (F.map AddMonObj.zero))) h

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theorem CategoryTheory.Functor.comp_mapGrp_one_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : Grp C) {Z : E} (h : ((F.comp G).mapGrp.obj A).X ⟶ Z) :

CategoryStruct.comp MonObj.one h = CategoryStruct.comp (CategoryStruct.comp (LaxMonoidal.ε (F.comp G)) (G.map (F.map MonObj.one))) h

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theorem CategoryTheory.Functor.comp_mapGrp_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : Grp C) :

MonObj.mul = CategoryStruct.comp (LaxMonoidal.μ (F.comp G) A.X A.X) ((F.comp G).map MonObj.mul)

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@[simp]

theorem CategoryTheory.Functor.comp_mapAddGrp_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : AddGrp C) :

AddMonObj.add = CategoryStruct.comp (LaxMonoidal.μ (F.comp G) A.X A.X) ((F.comp G).map AddMonObj.add)

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theorem CategoryTheory.Functor.comp_mapGrp_mul_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : Grp C) {Z : E} (h : ((F.comp G).mapGrp.obj A).X ⟶ Z) :

CategoryStruct.comp MonObj.mul h = CategoryStruct.comp (CategoryStruct.comp (LaxMonoidal.μ (F.comp G) A.X A.X) (G.map (F.map MonObj.mul))) h

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theorem CategoryTheory.Functor.comp_mapAddGrp_add_assoc {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (A : AddGrp C) {Z : E} (h : ((F.comp G).mapAddGrp.obj A).X ⟶ Z) :

CategoryStruct.comp AddMonObj.add h = CategoryStruct.comp (CategoryStruct.comp (LaxMonoidal.μ (F.comp G) A.X A.X) (G.map (F.map AddMonObj.add))) h

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def CategoryTheory.Functor.mapGrpIdIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(Functor.id C).mapGrp ≅ Functor.id (Grp C)

The identity functor is also the identity on group objects.

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def CategoryTheory.Functor.mapAddGrpIdIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] :

(Functor.id C).mapAddGrp ≅ Functor.id (AddGrp C)

The identity functor is also the identity on additive group objects.

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theorem CategoryTheory.Functor.mapAddGrpIdIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (X : AddGrp C) :

(mapAddGrpIdIso.hom.app X).hom.hom = CategoryStruct.id X.X

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theorem CategoryTheory.Functor.mapAddGrpIdIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (X : AddGrp C) :

(mapAddGrpIdIso.inv.app X).hom.hom = CategoryStruct.id X.X

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theorem CategoryTheory.Functor.mapGrpIdIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (X : Grp C) :

(mapGrpIdIso.inv.app X).hom.hom = CategoryStruct.id X.X

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@[simp]

theorem CategoryTheory.Functor.mapGrpIdIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] (X : Grp C) :

(mapGrpIdIso.hom.app X).hom.hom = CategoryStruct.id X.X

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def CategoryTheory.Functor.mapGrpCompIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] :

(F.comp G).mapGrp ≅ F.mapGrp.comp G.mapGrp

The composition functor is also the composition on group objects.

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def CategoryTheory.Functor.mapAddGrpCompIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] :

(F.comp G).mapAddGrp ≅ F.mapAddGrp.comp G.mapAddGrp

The composition functor is also the composition on additive group objects.

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theorem CategoryTheory.Functor.mapGrpCompIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (X : Grp C) :

(mapGrpCompIso.hom.app X).hom.hom = CategoryStruct.id (G.obj (F.obj X.X))

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@[simp]

theorem CategoryTheory.Functor.mapAddGrpCompIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (X : AddGrp C) :

(mapAddGrpCompIso.hom.app X).hom.hom = CategoryStruct.id (G.obj (F.obj X.X))

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@[simp]

theorem CategoryTheory.Functor.mapAddGrpCompIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (X : AddGrp C) :

(mapAddGrpCompIso.inv.app X).hom.hom = CategoryStruct.id (G.obj (F.obj X.X))

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@[simp]

theorem CategoryTheory.Functor.mapGrpCompIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {E : Type u₃} [Category.{v₃, u₃} E] [CartesianMonoidalCategory E] {F : Functor C D} {G : Functor D E} [F.Monoidal] [G.Monoidal] (X : Grp C) :

(mapGrpCompIso.inv.app X).hom.hom = CategoryStruct.id (G.obj (F.obj X.X))

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def CategoryTheory.Functor.mapGrpNatTrans {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (f : F ⟶ F') :

F.mapGrp ⟶ F'.mapGrp

Natural transformations between functors lift to group objects.

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def CategoryTheory.Functor.mapAddGrpNatTrans {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (f : F ⟶ F') :

F.mapAddGrp ⟶ F'.mapAddGrp

Natural transformations between functors lift to additive group objects.

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theorem CategoryTheory.Functor.mapGrpNatTrans_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (f : F ⟶ F') (X : Grp C) :

((mapGrpNatTrans f).app X).hom.hom = f.app X.X

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theorem CategoryTheory.Functor.mapAddGrpNatTrans_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (f : F ⟶ F') (X : AddGrp C) :

((mapAddGrpNatTrans f).app X).hom.hom = f.app X.X

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def CategoryTheory.Functor.mapGrpNatIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (e : F ≅ F') :

F.mapGrp ≅ F'.mapGrp

Natural isomorphisms between functors lift to group objects.

Equations

CategoryTheory.Functor.mapGrpNatIso e = CategoryTheory.NatIso.ofComponents (fun (X : CategoryTheory.Grp C) => CategoryTheory.Grp.mkIso (e.app X.X) ⋯ ⋯) ⋯

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def CategoryTheory.Functor.mapAddGrpNatIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (e : F ≅ F') :

F.mapAddGrp ≅ F'.mapAddGrp

Natural isomorphisms between functors lift to additive group objects.

Equations

CategoryTheory.Functor.mapAddGrpNatIso e = CategoryTheory.NatIso.ofComponents (fun (X : CategoryTheory.AddGrp C) => CategoryTheory.AddGrp.mkIso (e.app X.X) ⋯ ⋯) ⋯

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@[simp]

theorem CategoryTheory.Functor.mapAddGrpNatIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (e : F ≅ F') (X : AddGrp C) :

((mapAddGrpNatIso e).hom.app X).hom.hom = e.hom.app X.X

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@[simp]

theorem CategoryTheory.Functor.mapGrpNatIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (e : F ≅ F') (X : Grp C) :

((mapGrpNatIso e).inv.app X).hom.hom = e.inv.app X.X

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@[simp]

theorem CategoryTheory.Functor.mapGrpNatIso_hom_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (e : F ≅ F') (X : Grp C) :

((mapGrpNatIso e).hom.app X).hom.hom = e.hom.app X.X

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@[simp]

theorem CategoryTheory.Functor.mapAddGrpNatIso_inv_app_hom_hom {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F F' : Functor C D} [F.Monoidal] [F'.Monoidal] (e : F ≅ F') (X : AddGrp C) :

((mapAddGrpNatIso e).inv.app X).hom.hom = e.inv.app X.X

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noncomputable def CategoryTheory.Functor.mapGrpFunctor {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] :

Functor (C ⥤ₗ D) (Functor (Grp C) (Grp D))

mapGrp is functorial in the left-exact functor.

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One or more equations did not get rendered due to their size.

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noncomputable def CategoryTheory.Functor.mapAddGrpFunctor {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] :

Functor (C ⥤ₗ D) (Functor (AddGrp C) (AddGrp D))

mapAddGrp is functorial in the left-exact functor.

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One or more equations did not get rendered due to their size.

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@[simp]

theorem CategoryTheory.Functor.mapAddGrpFunctor_map_app {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F G : C ⥤ₗ D} (α : F ⟶ G) (A : AddGrp C) :

(mapAddGrpFunctor.map α).app A = AddGrp.homMk'' (α.hom.app A.X) ⋯ ⋯

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@[simp]

theorem CategoryTheory.Functor.mapGrpFunctor_obj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : C ⥤ₗ D) :

mapGrpFunctor.obj F = F.obj.mapGrp

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@[simp]

theorem CategoryTheory.Functor.mapAddGrpFunctor_obj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (F : C ⥤ₗ D) :

mapAddGrpFunctor.obj F = F.obj.mapAddGrp

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@[simp]

theorem CategoryTheory.Functor.mapGrpFunctor_map_app {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F G : C ⥤ₗ D} (α : F ⟶ G) (A : Grp C) :

(mapGrpFunctor.map α).app A = Grp.homMk'' (α.hom.app A.X) ⋯ ⋯

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@[reducible, inline]

abbrev CategoryTheory.Functor.FullyFaithful.grpObj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [GrpObj (F.obj X)] :

GrpObj X

Pullback a group object along a fully faithful monoidal functor.

Equations

hF.grpObj X = { toMonObj := hF.monObj X, inv := hF.preimage CategoryTheory.GrpObj.inv, left_inv := ⋯, right_inv := ⋯ }

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@[reducible, inline]

abbrev CategoryTheory.Functor.FullyFaithful.addGrpObj {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [AddGrpObj (F.obj X)] :

AddGrpObj X

Pullback an additive group object along a fully faithful monoidal functor.

Equations

hF.addGrpObj X = { toAddMonObj := hF.addMonObj X, neg := hF.preimage CategoryTheory.AddGrpObj.neg, left_neg := ⋯, right_neg := ⋯ }

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@[simp]

theorem CategoryTheory.Functor.FullyFaithful.addGrpObj_add {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [AddGrpObj (F.obj X)] :

AddMonObj.add = hF.preimage (CategoryStruct.comp (OplaxMonoidal.δ F X X) AddMonObj.add)

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@[simp]

theorem CategoryTheory.Functor.FullyFaithful.addGrpObj_zero {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [AddGrpObj (F.obj X)] :

AddMonObj.zero = hF.preimage (CategoryStruct.comp (OplaxMonoidal.η F) AddMonObj.zero)

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@[simp]

theorem CategoryTheory.Functor.FullyFaithful.addGrpObj_neg {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [AddGrpObj (F.obj X)] :

AddGrpObj.neg = hF.preimage AddGrpObj.neg

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@[simp]

theorem CategoryTheory.Functor.FullyFaithful.grpObj_one {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [GrpObj (F.obj X)] :

MonObj.one = hF.preimage (CategoryStruct.comp (OplaxMonoidal.η F) MonObj.one)

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@[simp]

theorem CategoryTheory.Functor.FullyFaithful.grpObj_mul {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [GrpObj (F.obj X)] :

MonObj.mul = hF.preimage (CategoryStruct.comp (OplaxMonoidal.δ F X X) MonObj.mul)

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@[simp]

theorem CategoryTheory.Functor.FullyFaithful.grpObj_inv {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] (hF : F.FullyFaithful) (X : C) [GrpObj (F.obj X)] :

GrpObj.inv = hF.preimage GrpObj.inv

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@[simp]

theorem CategoryTheory.Functor.essImage_mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] [F.Full] [F.Faithful] {G : Grp D} :

F.mapGrp.essImage G ↔ F.essImage G.X

The essential image of a full and faithful functor between cartesian-monoidal categories is the same on group objects as on objects.

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@[simp]

theorem CategoryTheory.Functor.essImage_mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} [F.Monoidal] [F.Full] [F.Faithful] {G : AddGrp D} :

F.mapAddGrp.essImage G ↔ F.essImage G.X

The essential image of a full and faithful functor between cartesian-monoidal categories is the same on additive group objects as on objects.

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@[implicit_reducible]

noncomputable instance CategoryTheory.Functor.mapGrp.instMonoidal {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] [BraidedCategory C] [BraidedCategory D] (F : Functor C D) [F.Braided] :

F.mapGrp.Monoidal

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One or more equations did not get rendered due to their size.

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@[implicit_reducible]

noncomputable instance CategoryTheory.Functor.mapAddGrp.instMonoidal {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] [BraidedCategory C] [BraidedCategory D] (F : Functor C D) [F.Braided] :

F.mapAddGrp.Monoidal

Equations

One or more equations did not get rendered due to their size.

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@[implicit_reducible]

noncomputable instance CategoryTheory.Functor.mapGrp.instBraided {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] [BraidedCategory C] [BraidedCategory D] (F : Functor C D) [F.Braided] :

F.mapGrp.Braided

Equations

CategoryTheory.Functor.mapGrp.instBraided F = { toMonoidal := CategoryTheory.Functor.mapGrp.instMonoidal F, braided := ⋯ }

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@[implicit_reducible]

noncomputable instance CategoryTheory.Functor.mapAddGrp.instBraided {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] [BraidedCategory C] [BraidedCategory D] (F : Functor C D) [F.Braided] :

F.mapAddGrp.Braided

Equations

CategoryTheory.Functor.mapAddGrp.instBraided F = { toMonoidal := CategoryTheory.Functor.mapAddGrp.instMonoidal F, braided := ⋯ }

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def CategoryTheory.Adjunction.mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Monoidal] [G.Monoidal] :

F.mapGrp ⊣ G.mapGrp

An adjunction of monoidal functors lifts to an adjunction of their lifts to group objects.

Equations

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def CategoryTheory.Adjunction.mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Monoidal] [G.Monoidal] :

F.mapAddGrp ⊣ G.mapAddGrp

An adjunction of monoidal functors lifts to an adjunction of their lifts to additive group objects.

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@[simp]

theorem CategoryTheory.Adjunction.mapGrp_counit {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Monoidal] [G.Monoidal] :

a.mapGrp.counit = CategoryStruct.comp Functor.mapGrpCompIso.inv (CategoryStruct.comp (Functor.mapGrpNatTrans a.counit) Functor.mapGrpIdIso.hom)

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@[simp]

theorem CategoryTheory.Adjunction.mapGrp_unit {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Monoidal] [G.Monoidal] :

a.mapGrp.unit = CategoryStruct.comp Functor.mapGrpIdIso.inv (CategoryStruct.comp (Functor.mapGrpNatTrans a.unit) Functor.mapGrpCompIso.hom)

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@[simp]

theorem CategoryTheory.Adjunction.mapAddGrp_counit {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Monoidal] [G.Monoidal] :

a.mapAddGrp.counit = CategoryStruct.comp Functor.mapAddGrpCompIso.inv (CategoryStruct.comp (Functor.mapAddGrpNatTrans a.counit) Functor.mapAddGrpIdIso.hom)

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@[simp]

theorem CategoryTheory.Adjunction.mapAddGrp_unit {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] {F : Functor C D} {G : Functor D C} (a : F ⊣ G) [F.Monoidal] [G.Monoidal] :

a.mapAddGrp.unit = CategoryStruct.comp Functor.mapAddGrpIdIso.inv (CategoryStruct.comp (Functor.mapAddGrpNatTrans a.unit) Functor.mapAddGrpCompIso.hom)

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def CategoryTheory.Equivalence.mapGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

Grp C ≌ Grp D

An equivalence of categories lifts to an equivalence of their group objects.

Equations

One or more equations did not get rendered due to their size.

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def CategoryTheory.Equivalence.mapAddGrp {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

AddGrp C ≌ AddGrp D

An equivalence of categories lifts to an equivalence of their additive group objects.

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One or more equations did not get rendered due to their size.

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@[simp]

theorem CategoryTheory.Equivalence.mapGrp_counitIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapGrp.counitIso = Functor.mapGrpCompIso.symm ≪≫ Functor.mapGrpNatIso e.counitIso ≪≫ Functor.mapGrpIdIso

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@[simp]

theorem CategoryTheory.Equivalence.mapAddGrp_unitIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapAddGrp.unitIso = Functor.mapAddGrpIdIso.symm ≪≫ Functor.mapAddGrpNatIso e.unitIso ≪≫ Functor.mapAddGrpCompIso

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@[simp]

theorem CategoryTheory.Equivalence.mapAddGrp_inverse {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapAddGrp.inverse = e.inverse.mapAddGrp

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@[simp]

theorem CategoryTheory.Equivalence.mapGrp_functor {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapGrp.functor = e.functor.mapGrp

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@[simp]

theorem CategoryTheory.Equivalence.mapGrp_unitIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapGrp.unitIso = Functor.mapGrpIdIso.symm ≪≫ Functor.mapGrpNatIso e.unitIso ≪≫ Functor.mapGrpCompIso

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@[simp]

theorem CategoryTheory.Equivalence.mapAddGrp_counitIso {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapAddGrp.counitIso = Functor.mapAddGrpCompIso.symm ≪≫ Functor.mapAddGrpNatIso e.counitIso ≪≫ Functor.mapAddGrpIdIso

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@[simp]

theorem CategoryTheory.Equivalence.mapGrp_inverse {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapGrp.inverse = e.inverse.mapGrp

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@[simp]

theorem CategoryTheory.Equivalence.mapAddGrp_functor {C : Type u₁} [Category.{v₁, u₁} C] [CartesianMonoidalCategory C] {D : Type u₂} [Category.{v₂, u₂} D] [CartesianMonoidalCategory D] (e : C ≌ D) [e.functor.Monoidal] [e.inverse.Monoidal] :

e.mapAddGrp.functor = e.functor.mapAddGrp

Documentation

Mathlib.CategoryTheory.Monoidal.Grp

The category of groups in a Cartesian monoidal category #

`Grp C` is cartesian-monoidal #

The category of groups in a Cartesian monoidal category #

Grp C is cartesian-monoidal #

`Grp C` is cartesian-monoidal #