coherence theorem for symmetric monoidal categories



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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 SymStrMonCatSymMonCatSymStrMonCat \to SymMonCat has a strict left adjoint and the components of the unit are equivalences in SymMonCatSymMonCat.

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


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)

Last revised on December 16, 2016 at 00:36:56. See the history of this page for a list of all contributions to it.