Documentation

Mathlib.CategoryTheory.Monoidal.Free.Coherence

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) ≅ fullNormalizeinclusion. 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 #

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

Instances For

    Auxiliary definition for inclusion.

    Equations
    Instances For

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

      Equations
      Instances For

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

        Instances For

          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.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For

            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).

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For

              The normalization functor for the free monoidal category over C.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For

                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.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For

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

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For

                    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.

                    Equations
                    Instances For

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

                      Equations
                      • One or more equations did not get rendered due to their size.
                      Instances For
                        Equations
                        • One or more equations did not get rendered due to their size.