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category_theory.monoidal.free.basic

The free monoidal category over a type #

Given a type C, the free monoidal category over C has as objects formal expressions built from (formal) tensor products of terms of C and a formal unit. Its morphisms are compositions and tensor products of identities, unitors and associators.

In this file, we construct the free monoidal category and prove that it is a monoidal category. If D is a monoidal category, we construct the functor free_monoidal_category C ⥤ D associated to a function C → D.

The free monoidal category has two important properties: it is a groupoid and it is thin. The former is obvious from the construction, and the latter is what is commonly known as the monoidal coherence theorem. Both of these properties are proved in the file coherence.lean.

inductive category_theory.free_monoidal_category (C : Type u) :
Type u

Given a type C, the free monoidal category over C has as objects formal expressions built from (formal) tensor products of terms of C and a formal unit. Its morphisms are compositions and tensor products of identities, unitors and associators.

@[nolint]

Formal compositions and tensor products of identities, unitors and associators. The morphisms of the free monoidal category are obtained as a quotient of these formal morphisms by the relations defining a monoidal category.

inductive category_theory.free_monoidal_category.hom_equiv {C : Type u} {X Y : category_theory.free_monoidal_category C} :
X.hom YX.hom Y → Prop

The morphisms of the free monoidal category satisfy 21 relations ensuring that the resulting category is in fact a category and that it is monoidal.

@[instance]

We say that two formal morphisms in the free monoidal category are equivalent if they become equal if we apply the relations that are true in a monoidal category. Note that we will prove that there is only one equivalence class -- this is the monoidal coherence theorem.

Equations
@[simp]
theorem category_theory.free_monoidal_category.mk_tensor {C : Type u} {X₁ Y₁ X₂ Y₂ : category_theory.free_monoidal_category C} (f : X₁.hom Y₁) (g : X₂.hom Y₂) :
@[simp]

Auxiliary definition for free_monoidal_category.project.

Equations