nLab coherence theorem for symmetric monoidal categories

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.

References

• Saunders Mac Lane. “Natural associativity and commutativity”. Rice University Studies 49, 28-46 (1963). (Rice Digital Scholarship Archive)

• Saunders Mac Lane. Categories for the Working Mathematician (Chapter 11).

• Andre Joyal and Ross Street. “Braided tensor categories”. Adv. Math 1993

See also section 5 of

• Saunders Mac Lane, Topology and Logic as a Source of Algebra (Retiring Presidential Address), Bulletin of the AMS 82:1, January 1976. (euclid)