The monoidal coherence theorem

In this file, we prove the monoidal coherence theorem, stated in the following form: the free monoidal category over any type C is thin.

We follow a proof described by Ilya Beylin and Peter Dybjer, which has been previously formalized in the proof assistant ALF. The idea is to declare a normal form (with regard to association and adding units) on objects of the free monoidal category and consider the discrete subcategory of objects that are in normal form. A normalization procedure is then just a functor fullNormalize : FreeMonoidalCategory C ⥤ Discrete (NormalMonoidalObject C), where functoriality says that two objects which are related by associators and unitors have the same normal form. Another desirable property of a normalization procedure is that an object is isomorphic (i.e., related via associators and unitors) to its normal form. In the case of the specific normalization procedure we use we not only get these isomorphisms, but also that they assemble into a natural isomorphism 𝟭 (FreeMonoidalCategory C) ≅ fullNormalize ⋙ inclusion. But this means that any two parallel morphisms in the free monoidal category factor through a discrete category in the same way, so they must be equal, and hence the free monoidal category is thin.

References

• [Ilya Beylin and Peter Dybjer, Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids][beylin1996]

inductive CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject (C : Type u) : Type u

• Unit: {C : Type u} → CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C
• tensor: {C : Type u} → CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C → C → CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C

We say an object in the free monoidal category is in normal form if it is of the form (((𝟙_ C) ⊗ X₁) ⊗ X₂) ⊗ ⋯.

instance CategoryTheory.FreeMonoidalCategory.instSubsingletonHomCompTypeDiscreteNormalMonoidalObjectToQuiverToCategoryStructInstForAllCategoryCompTypeDiscreteCategory {C : Type u} (x : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) (y : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) : Subsingleton (x ⟶ y)

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def CategoryTheory.FreeMonoidalCategory.inclusionObj {C : Type u} : CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C → CategoryTheory.FreeMonoidalCategory C

Auxiliary definition for inclusion.

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• CategoryTheory.FreeMonoidalCategory.inclusionObj CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject.Unit = CategoryTheory.FreeMonoidalCategory.Unit

def CategoryTheory.FreeMonoidalCategory.inclusion {C : Type u} : CategoryTheory.Functor ((CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) (CategoryTheory.FreeMonoidalCategory C)

The discrete subcategory of objects in normal form includes into the free monoidal category.

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• CategoryTheory.FreeMonoidalCategory.inclusion = CategoryTheory.Discrete.functor CategoryTheory.FreeMonoidalCategory.inclusionObj

def CategoryTheory.FreeMonoidalCategory.normalizeObj {C : Type u} : CategoryTheory.FreeMonoidalCategory C → CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C → (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C

Auxiliary definition for normalize.

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• CategoryTheory.FreeMonoidalCategory.normalizeObj CategoryTheory.FreeMonoidalCategory.Unit x = { as := x }
• CategoryTheory.FreeMonoidalCategory.normalizeObj (CategoryTheory.FreeMonoidalCategory.of X) x = { as := CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject.tensor x X }

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theorem CategoryTheory.FreeMonoidalCategory.normalizeObj_unitor {C : Type u} (n : CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C) : CategoryTheory.FreeMonoidalCategory.normalizeObj (CategoryTheory.MonoidalCategory.tensorUnit (CategoryTheory.FreeMonoidalCategory C)) n = { as := n }

@[simp]

theorem CategoryTheory.FreeMonoidalCategory.normalizeObj_tensor {C : Type u} (X : CategoryTheory.FreeMonoidalCategory C) (Y : CategoryTheory.FreeMonoidalCategory C) (n : CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C) : CategoryTheory.FreeMonoidalCategory.normalizeObj (CategoryTheory.MonoidalCategory.tensorObj X Y) n = CategoryTheory.FreeMonoidalCategory.normalizeObj Y (CategoryTheory.FreeMonoidalCategory.normalizeObj X n).as

def CategoryTheory.FreeMonoidalCategory.normalizeMapAux {C : Type u} {X : CategoryTheory.FreeMonoidalCategory C} {Y : CategoryTheory.FreeMonoidalCategory C} : CategoryTheory.FreeMonoidalCategory.Hom X Y → (CategoryTheory.Discrete.functor (CategoryTheory.FreeMonoidalCategory.normalizeObj X) ⟶ CategoryTheory.Discrete.functor (CategoryTheory.FreeMonoidalCategory.normalizeObj Y))

Auxiliary definition for normalize. Here we prove that objects that are related by associators and unitors map to the same normal form.

def CategoryTheory.FreeMonoidalCategory.normalize (C : Type u) : CategoryTheory.Functor (CategoryTheory.FreeMonoidalCategory C) (CategoryTheory.Functor ((CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) ((CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C))

Our normalization procedure works by first defining a functor F C ⥤ (N C ⥤ N C) (which turns out to be very easy), and then obtain a functor F C ⥤ N C by plugging in the normal object 𝟙_ C.

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def CategoryTheory.FreeMonoidalCategory.normalize' (C : Type u) : CategoryTheory.Functor (CategoryTheory.FreeMonoidalCategory C) (CategoryTheory.Functor ((CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) (CategoryTheory.FreeMonoidalCategory C))

A variant of the normalization functor where we consider the result as an object in the free monoidal category (rather than an object of the discrete subcategory of objects in normal form).

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def CategoryTheory.FreeMonoidalCategory.fullNormalize (C : Type u) : CategoryTheory.Functor (CategoryTheory.FreeMonoidalCategory C) ((CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C)

The normalization functor for the free monoidal category over C.

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def CategoryTheory.FreeMonoidalCategory.tensorFunc (C : Type u) : CategoryTheory.Functor (CategoryTheory.FreeMonoidalCategory C) (CategoryTheory.Functor ((CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) (CategoryTheory.FreeMonoidalCategory C))

Given an object X of the free monoidal category and an object n in normal form, taking the tensor product n ⊗ X in the free monoidal category is functorial in both X and n.

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theorem CategoryTheory.FreeMonoidalCategory.tensorFunc_map_app (C : Type u) {X : CategoryTheory.FreeMonoidalCategory C} {Y : CategoryTheory.FreeMonoidalCategory C} (f : X ⟶ Y) (n : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) : ((CategoryTheory.FreeMonoidalCategory.tensorFunc C).map f).app n = CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id (CategoryTheory.FreeMonoidalCategory.inclusion.obj { as := n.as })) f

theorem CategoryTheory.FreeMonoidalCategory.tensorFunc_obj_map (C : Type u) (Z : CategoryTheory.FreeMonoidalCategory C) {n : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C} {n' : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C} (f : n ⟶ n') : ((CategoryTheory.FreeMonoidalCategory.tensorFunc C).obj Z).map f = CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.FreeMonoidalCategory.inclusion.map f) (CategoryTheory.CategoryStruct.id Z)

def CategoryTheory.FreeMonoidalCategory.normalizeIsoApp (C : Type u) (X : CategoryTheory.FreeMonoidalCategory C) (n : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) : ((CategoryTheory.FreeMonoidalCategory.tensorFunc C).obj X).obj n ≅ ((CategoryTheory.FreeMonoidalCategory.normalize' C).obj X).obj n

Auxiliary definition for normalizeIso. Here we construct the isomorphism between n ⊗ X and normalize X n.

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theorem CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_tensor (C : Type u) (X : CategoryTheory.FreeMonoidalCategory C) (Y : CategoryTheory.FreeMonoidalCategory C) (n : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) : CategoryTheory.FreeMonoidalCategory.normalizeIsoApp C (CategoryTheory.MonoidalCategory.tensorObj X Y) n = (CategoryTheory.MonoidalCategory.associator (CategoryTheory.FreeMonoidalCategory.inclusion.obj { as := n.as }) X Y).symm ≪≫ (CategoryTheory.FreeMonoidalCategory.normalizeIsoApp C X n ⊗ CategoryTheory.Iso.refl Y) ≪≫ CategoryTheory.FreeMonoidalCategory.normalizeIsoApp C Y (CategoryTheory.FreeMonoidalCategory.normalizeObj X n.as)

@[simp]

theorem CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_unitor (C : Type u) (n : (CategoryTheory.Discrete ∘ CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject) C) : CategoryTheory.FreeMonoidalCategory.normalizeIsoApp C (CategoryTheory.MonoidalCategory.tensorUnit (CategoryTheory.FreeMonoidalCategory C)) n = CategoryTheory.MonoidalCategory.rightUnitor (CategoryTheory.FreeMonoidalCategory.inclusion.obj { as := n.as })

def CategoryTheory.FreeMonoidalCategory.normalizeIsoAux (C : Type u) (X : CategoryTheory.FreeMonoidalCategory C) : (CategoryTheory.FreeMonoidalCategory.tensorFunc C).obj X ≅ (CategoryTheory.FreeMonoidalCategory.normalize' C).obj X

Auxiliary definition for normalizeIso.

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• CategoryTheory.FreeMonoidalCategory.normalizeIsoAux C X = CategoryTheory.NatIso.ofComponents (CategoryTheory.FreeMonoidalCategory.normalizeIsoApp C X)

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theorem CategoryTheory.FreeMonoidalCategory.discrete_functor_obj_eq_as {D : Type u} [CategoryTheory.Category.{u, u} D] {I : Type u} (f : I → D) (X : CategoryTheory.Discrete I) : (CategoryTheory.Discrete.functor f).obj X = f X.as

@[simp]

theorem CategoryTheory.FreeMonoidalCategory.discrete_functor_map_eq_id {D : Type u} [CategoryTheory.Category.{u, u} D] {I : Type u} (f : I → D) (X : CategoryTheory.Discrete I) (g : X ⟶ X) : (CategoryTheory.Discrete.functor f).map g = CategoryTheory.CategoryStruct.id ((CategoryTheory.Discrete.functor f).obj X)

def CategoryTheory.FreeMonoidalCategory.normalizeIso (C : Type u) : CategoryTheory.FreeMonoidalCategory.tensorFunc C ≅ CategoryTheory.FreeMonoidalCategory.normalize' C

The isomorphism between n ⊗ X and normalize X n is natural (in both X and n, but naturality in n is trivial and was "proved" in normalizeIsoAux). This is the real heart of our proof of the coherence theorem.

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• CategoryTheory.FreeMonoidalCategory.normalizeIso C = CategoryTheory.NatIso.ofComponents (CategoryTheory.FreeMonoidalCategory.normalizeIsoAux C)

def CategoryTheory.FreeMonoidalCategory.fullNormalizeIso (C : Type u) : CategoryTheory.Functor.id (CategoryTheory.FreeMonoidalCategory C) ≅ CategoryTheory.Functor.comp (CategoryTheory.FreeMonoidalCategory.fullNormalize C) CategoryTheory.FreeMonoidalCategory.inclusion

The isomorphism between an object and its normal form is natural.

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instance CategoryTheory.FreeMonoidalCategory.subsingleton_hom {C : Type u} : Quiver.IsThin (CategoryTheory.FreeMonoidalCategory C)

The monoidal coherence theorem.

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• @CategoryTheory.FreeMonoidalCategory.subsingleton_hom = @CategoryTheory.FreeMonoidalCategory.subsingleton_hom.proof_1

def CategoryTheory.FreeMonoidalCategory.inverseAux {C : Type u} {X : CategoryTheory.FreeMonoidalCategory C} {Y : CategoryTheory.FreeMonoidalCategory C} : CategoryTheory.FreeMonoidalCategory.Hom X Y → CategoryTheory.FreeMonoidalCategory.Hom Y X

Auxiliary construction for showing that the free monoidal category is a groupoid. Do not use this, use IsIso.inv instead.

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• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.id x) = CategoryTheory.FreeMonoidalCategory.Hom.id x
• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.α_hom X Y Z) = CategoryTheory.FreeMonoidalCategory.Hom.α_inv X Y Z
• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.α_inv X Y Z) = CategoryTheory.FreeMonoidalCategory.Hom.α_hom X Y Z
• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.ρ_hom x) = CategoryTheory.FreeMonoidalCategory.Hom.ρ_inv x
• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.ρ_inv x) = CategoryTheory.FreeMonoidalCategory.Hom.ρ_hom x
• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.l_hom x) = CategoryTheory.FreeMonoidalCategory.Hom.l_inv x
• CategoryTheory.FreeMonoidalCategory.inverseAux (CategoryTheory.FreeMonoidalCategory.Hom.l_inv x) = CategoryTheory.FreeMonoidalCategory.Hom.l_hom x

instance CategoryTheory.FreeMonoidalCategory.instGroupoidFreeMonoidalCategory {C : Type u} : CategoryTheory.Groupoid (CategoryTheory.FreeMonoidalCategory C)

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