nLab coherence and strictification for symmetric monoidal categories

Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The coherence theorem and strictification theorem for symmetric monoidal categories may each take several forms.

Coherence 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 equivalent to an unbiased symmetric monoidal category?.

Strictification theorems

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

  2. 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.

References

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 September 25, 2024 at 09:29:31. See the history of this page for a list of all contributions to it.