nLab coherence theorem for symmetric monoidal categories

Context

Monoidal categories

monoidal categories

In higher category theory

Higher category theory

higher category theory

Contents

Idea

The coherence theorem for symmetric monoidal categories, like many coherence theorems, has several forms (or, alternatively, refers to several different theorems):

1. Every diagram in a free symmetric monoidal category made up of associators and unitors and symmetries (braidings), and in which both sides have the same underlying permutation, commutes.

2. The free symmetric monoidal category on some given data is equivalent to the free symmetric strict monoidal category on the same data.

3. Every symmetric monoidal category is symmetric-monoidally equivalent to a symmetric strict monoidal category.

4. Every symmetric monoidal category is equivalent to an unbiased symmetric monoidal category?.

5. The forgetful 2-functor $SymStrMonCat \to SymMonCat$ has a strict left adjoint and the components of the unit are equivalences in $SymMonCat$.

Note that in a symmetric strict monoidal category, the associators and unitors are identities, but the symmetry is not in general.