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category_theory.monoidal.Mon_

The category of monoids in a monoidal category. #

We define monoids in a monoidal category C and show that the category of monoids is equivalent to the category of lax monoidal functors from the unit monoidal category to C. We also show that if C is braided, then the category of monoids is naturally monoidal.

structure Mon_ (C : Type u₁)  :
Type (max u₁ v₁)

A monoid object internal to a monoidal category.

When the monoidal category is preadditive, this is also sometimes called an "algebra object".

Instances for Mon_
theorem Mon_.one_mul {C : Type u₁} (self : Mon_ C) :
(self.one 𝟙 self.X) self.mul = (λ_ self.X).hom
theorem Mon_.mul_one {C : Type u₁} (self : Mon_ C) :
(𝟙 self.X self.one) self.mul = (ρ_ self.X).hom
@[simp]
theorem Mon_.mul_assoc {C : Type u₁} (self : Mon_ C) :
(self.mul 𝟙 self.X) self.mul = (α_ self.X self.X self.X).hom (𝟙 self.X self.mul) self.mul
theorem Mon_.one_mul_assoc {C : Type u₁} (self : Mon_ C) {X' : C} (f' : self.X X') :
(self.one 𝟙 self.X) self.mul f' = (λ_ self.X).hom f'
theorem Mon_.mul_one_assoc {C : Type u₁} (self : Mon_ C) {X' : C} (f' : self.X X') :
(𝟙 self.X self.one) self.mul f' = (ρ_ self.X).hom f'
@[simp]
theorem Mon_.mul_assoc_assoc {C : Type u₁} (self : Mon_ C) {X' : C} (f' : self.X X') :
(self.mul 𝟙 self.X) self.mul f' = (α_ self.X self.X self.X).hom (𝟙 self.X self.mul) self.mul f'
@[simp]
theorem Mon_.trivial_X (C : Type u₁)  :
def Mon_.trivial (C : Type u₁)  :

The trivial monoid object. We later show this is initial in Mon_ C.

Equations
@[simp]
theorem Mon_.trivial_mul (C : Type u₁)  :
@[simp]
theorem Mon_.trivial_one (C : Type u₁)  :
@[protected, instance]
def Mon_.inhabited (C : Type u₁)  :
Equations
@[simp]
theorem Mon_.one_mul_hom {C : Type u₁} {M : Mon_ C} {Z : C} (f : Z M.X) :
(M.one f) M.mul = (λ_ Z).hom f
@[simp]
theorem Mon_.mul_one_hom {C : Type u₁} {M : Mon_ C} {Z : C} (f : Z M.X) :
(f M.one) M.mul = (ρ_ Z).hom f
theorem Mon_.assoc_flip {C : Type u₁} {M : Mon_ C} :
(𝟙 M.X M.mul) M.mul = (α_ M.X M.X M.X).inv (M.mul 𝟙 M.X) M.mul
theorem Mon_.hom.ext_iff {C : Type u₁} {_inst_1 : category_theory.category C} {_inst_2 : category_theory.monoidal_category C} {M N : Mon_ C} (x y : M.hom N) :
x = y x.hom = y.hom
@[ext]
structure Mon_.hom {C : Type u₁} (M N : Mon_ C) :
Type v₁

A morphism of monoid objects.

Instances for Mon_.hom
theorem Mon_.hom.ext {C : Type u₁} {_inst_1 : category_theory.category C} {_inst_2 : category_theory.monoidal_category C} {M N : Mon_ C} (x y : M.hom N) (h : x.hom = y.hom) :
x = y
@[simp]
theorem Mon_.hom.one_hom {C : Type u₁} {M N : Mon_ C} (self : M.hom N) :
M.one self.hom = N.one
@[simp]
theorem Mon_.hom.mul_hom {C : Type u₁} {M N : Mon_ C} (self : M.hom N) :
M.mul self.hom = (self.hom self.hom) N.mul
@[simp]
theorem Mon_.hom.mul_hom_assoc {C : Type u₁} {M N : Mon_ C} (self : M.hom N) {X' : C} (f' : N.X X') :
M.mul self.hom f' = (self.hom self.hom) N.mul f'
@[simp]
theorem Mon_.hom.one_hom_assoc {C : Type u₁} {M N : Mon_ C} (self : M.hom N) {X' : C} (f' : N.X X') :
M.one self.hom f' = N.one f'
def Mon_.id {C : Type u₁} (M : Mon_ C) :
M.hom M

The identity morphism on a monoid object.

Equations
@[simp]
theorem Mon_.id_hom {C : Type u₁} (M : Mon_ C) :
M.id.hom = 𝟙 M.X
@[protected, instance]
def Mon_.hom_inhabited {C : Type u₁} (M : Mon_ C) :
Equations
def Mon_.comp {C : Type u₁} {M N O : Mon_ C} (f : M.hom N) (g : N.hom O) :
M.hom O

Composition of morphisms of monoid objects.

Equations
@[simp]
theorem Mon_.comp_hom {C : Type u₁} {M N O : Mon_ C} (f : M.hom N) (g : N.hom O) :
g).hom = f.hom g.hom
@[protected, instance]
def Mon_.category_theory.category {C : Type u₁}  :
Equations
@[simp]
theorem Mon_.id_hom' {C : Type u₁} (M : Mon_ C) :
(𝟙 M).hom = 𝟙 M.X
@[simp]
theorem Mon_.comp_hom' {C : Type u₁} {M N K : Mon_ C} (f : M N) (g : N K) :
(f g).hom = f.hom g.hom
@[simp]
theorem Mon_.forget_obj (C : Type u₁) (A : Mon_ C) :
(Mon_.forget C).obj A = A.X
@[simp]
theorem Mon_.forget_map (C : Type u₁) (A B : Mon_ C) (f : A B) :
def Mon_.forget (C : Type u₁)  :
Mon_ C C

The forgetful functor from monoid objects to the ambient category.

Equations
Instances for Mon_.forget
@[protected, instance]
def Mon_.forget_faithful {C : Type u₁}  :
@[protected, instance]
def Mon_.quiver.hom.hom.category_theory.is_iso {C : Type u₁} {A B : Mon_ C} (f : A B) [e : category_theory.is_iso ((Mon_.forget C).map f)] :
@[protected, instance]

The forgetful functor from monoid objects to the ambient category reflects isomorphisms.

def Mon_.iso_of_iso {C : Type u₁} {M N : Mon_ C} (f : M.X N.X) (one_f : M.one f.hom = N.one) (mul_f : M.mul f.hom = (f.hom f.hom) N.mul) :
M N

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

Equations
@[protected, instance]
def Mon_.unique_hom_from_trivial {C : Type u₁} (A : Mon_ C) :
Equations
@[protected, instance]
def category_theory.lax_monoidal_functor.map_Mon {C : Type u₁} {D : Type u₂}  :

A lax monoidal functor takes monoid objects to monoid objects.

That is, a lax monoidal functor F : C ⥤ D induces a functor Mon_ C ⥤ Mon_ D.

Equations
@[simp]
theorem category_theory.lax_monoidal_functor.map_Mon_obj_one {C : Type u₁} {D : Type u₂} (A : Mon_ C) :
@[simp]
theorem category_theory.lax_monoidal_functor.map_Mon_map_hom {C : Type u₁} {D : Type u₂} (A B : Mon_ C) (f : A B) :
@[simp]
theorem category_theory.lax_monoidal_functor.map_Mon_obj_mul {C : Type u₁} {D : Type u₂} (A : Mon_ C) :
(F.map_Mon.obj A).mul = F.μ A.X A.X F.to_functor.map A.mul
@[simp]
theorem category_theory.lax_monoidal_functor.map_Mon_obj_X {C : Type u₁} {D : Type u₂} (A : Mon_ C) :
def category_theory.lax_monoidal_functor.map_Mon_functor (C : Type u₁) (D : Type u₂)  :

map_Mon is functorial in the lax monoidal functor.

Equations

Implementation of Mon_.equiv_lax_monoidal_functor_punit.

Equations
@[simp]
@[simp]

Implementation of Mon_.equiv_lax_monoidal_functor_punit.

Equations
@[simp]
@[simp]
@[simp]

Implementation of Mon_.equiv_lax_monoidal_functor_punit.

Equations
@[simp]

Implementation of Mon_.equiv_lax_monoidal_functor_punit.

Equations
@[simp]
@[simp]
@[simp]
@[simp]
@[simp]
@[simp]

Monoid objects in C are "just" lax monoidal functors from the trivial monoidal category to C.

Equations

In this section, we prove that the category of monoids in a braided monoidal category is monoidal.

Given two monoids M and N in a braided monoidal category C, the multiplication on the tensor product M.X ⊗ N.X is defined in the obvious way: it is the tensor product of the multiplications on M and N, except that the tensor factors in the source come in the wrong order, which we fix by pre-composing with a permutation isomorphism constructed from the braiding.

A more conceptual way of understanding this definition is the following: The braiding on C gives rise to a monoidal structure on the tensor product functor from C × C to C. A pair of monoids in C gives rise to a monoid in C × C, which the tensor product functor by being monoidal takes to a monoid in C. The permutation isomorphism appearing in the definition of the multiplication on the tensor product of two monoids is an instance of a more general family of isomorphisms which together form a strength that equips the tensor product functor with a monoidal structure, and the monoid axioms for the tensor product follow from the monoid axioms for the tensor factors plus the properties of the strength (i.e., monoidal functor axioms). The strength tensor_μ of the tensor product functor has been defined in category_theory.monoidal.braided. Its properties, stated as independent lemmas in that module, are used extensively in the proofs below. Notice that we could have followed the above plan not only conceptually but also as a possible implementation and could have constructed the tensor product of monoids via map_Mon, but we chose to give a more explicit definition directly in terms of tensor_μ.

To complete the definition of the monoidal category structure on the category of monoids, we need to provide definitions of associator and unitors. The obvious candidates are the associator and unitors from C, but we need to prove that they are monoid morphisms, i.e., compatible with unit and multiplication. These properties translate to the monoidality of the associator and unitors (with respect to the monoidal structures on the functors they relate), which have also been proved in category_theory.monoidal.braided.

theorem Mon_.one_associator {C : Type u₁} {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ((λ_ (𝟙_ C)).inv (M.one N.one) P.one)) (α_ M.X N.X P.X).hom = (λ_ (𝟙_ C)).inv (M.one (λ_ (𝟙_ C)).inv (N.one P.one))
theorem Mon_.one_left_unitor {C : Type u₁} {M : Mon_ C} :
((λ_ (𝟙_ C)).inv (𝟙 (𝟙_ C) M.one)) (λ_ M.X).hom = M.one
theorem Mon_.one_right_unitor {C : Type u₁} {M : Mon_ C} :
((λ_ (𝟙_ C)).inv (M.one 𝟙 (𝟙_ C))) (ρ_ M.X).hom = M.one
theorem Mon_.Mon_tensor_one_mul {C : Type u₁} (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv (M.one N.one) 𝟙 (M.X N.X)) (M.X, N.X) (M.X, N.X) (M.mul N.mul) = (λ_ (M.X N.X)).hom
theorem Mon_.Mon_tensor_mul_one {C : Type u₁} (M N : Mon_ C) :
(𝟙 (M.X N.X) (λ_ (𝟙_ C)).inv (M.one N.one)) (M.X, N.X) (M.X, N.X) (M.mul N.mul) = (ρ_ (M.X N.X)).hom
theorem Mon_.Mon_tensor_mul_assoc {C : Type u₁} (M N : Mon_ C) :
(M.X, N.X) (M.X, N.X) (M.mul N.mul) 𝟙 (M.X N.X)) (M.X, N.X) (M.X, N.X) (M.mul N.mul) = (α_ (M.X N.X) (M.X N.X) (M.X N.X)).hom (𝟙 (M.X N.X) (M.X, N.X) (M.X, N.X) (M.mul N.mul)) (M.X, N.X) (M.X, N.X) (M.mul N.mul)
theorem Mon_.mul_associator {C : Type u₁} {M N P : Mon_ C} :
(M.X N.X, P.X) (M.X N.X, P.X) (M.X, N.X) (M.X, N.X) (M.mul N.mul) P.mul)) (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom (α_ M.X N.X P.X).hom) (M.X, N.X P.X) (M.X, N.X P.X) (M.mul (N.X, P.X) (N.X, P.X) (N.mul P.mul))
theorem Mon_.mul_left_unitor {C : Type u₁} {M : Mon_ C} :
(𝟙_ C _inst_2, M.X) (𝟙_ C _inst_2, M.X) ((λ_ (𝟙_ C)).hom M.mul)) (λ_ M.X).hom = ((λ_ M.X).hom (λ_ M.X).hom) M.mul
theorem Mon_.mul_right_unitor {C : Type u₁} {M : Mon_ C} :
(M.X, 𝟙_ C _inst_2) (M.X, 𝟙_ C _inst_2) (M.mul (λ_ (𝟙_ C)).hom)) (ρ_ M.X).hom = ((ρ_ M.X).hom (ρ_ M.X).hom) M.mul
@[protected, instance]
Equations

Projects:

• Check that Mon_ Mon ≌ CommMon, via the Eckmann-Hilton argument. (You'll have to hook up the cartesian monoidal structure on Mon first, available in #3463)
• Check that Mon_ Top ≌ [bundled topological monoids].
• Check that Mon_ AddCommGroup ≌ Ring. (We've already got Mon_ (Module R) ≌ Algebra R, in category_theory.monoidal.internal.Module.)
• Can you transport this monoidal structure to Ring or Algebra R? How does it compare to the "native" one?
• Show that when C is braided, the forgetful functor Mon_ C ⥤ C is monoidal.
• Show that when F is a lax braided functor C ⥤ D, the functor map_Mon F : Mon_ C ⥤ Mon_ D is lax monoidal.